Lectures on the Geometry of Manifolds - Liviu I. Nicolaescu

March 23, 2018 | Author: Henry Rojas | Category: Differentiable Manifold, Geometry, Vector Space, Manifold, Differential Geometry


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Lectures on the Geometry of ManifoldsLiviu I. Nicolaescu July 27, 2007 Introduction Shape is a fascinating and intriguing subject which has stimulated the imagination of many people. It suffices to look around to become curious. Euclid did just that and came up with the first pure creation. Relying on the common experience, he created an abstract world that had a life of its own. As the human knowledge progressed so did the ability of formulating and answering penetrating questions. In particular, mathematicians started wondering whether Euclid’s “obvious” absolute postulates were indeed obvious and/or absolute. Scientists realized that Shape and Space are two closely related concepts and asked whether they really look the way our senses tell us. As Felix Klein pointed out in his Erlangen Program, there are many ways of looking at Shape and Space so that various points of view may produce different images. In particular, the most basic issue of “measuring the Shape” cannot have a clear cut answer. This is a book about Shape, Space and some particular ways of studying them. Since its inception, the differential and integral calculus proved to be a very versatile tool in dealing with previously untouchable problems. It did not take long until it found uses in geometry in the hands of the Great Masters. This is the path we want to follow in the present book. In the early days of geometry nobody worried about the natural context in which the methods of calculus “feel at home”. There was no need to address this aspect since for the particular problems studied this was a non-issue. As mathematics progressed as a whole the “natural context” mentioned above crystallized in the minds of mathematicians and it was a notion so important that it had to be given a name. The geometric objects which can be studied using the methods of calculus were called smooth manifolds. Special cases of manifolds are the curves and the surfaces and these were quite well understood. B. Riemann was the first to note that the low dimensional ideas of his time were particular aspects of a higher dimensional world. The first chapter of this book introduces the reader to the concept of smooth manifold through abstract definitions and, more importantly, through many we believe relevant examples. In particular, we introduce at this early stage the notion of Lie group. The main geometric and algebraic properties of these objects will be gradually described as we progress with our study of the geometry of manifolds. Besides their obvious usefulness in geometry, the Lie groups are academically very friendly. They provide a marvelous testing ground for abstract results. We have consistently taken advantage of this feature throughout this book. As a bonus, by the end of these lectures the reader will feel comfortable manipulating basic Lie theoretic concepts. To apply the techniques of calculus we need “things to derivate and integrate”. These i ii “things” are introduced in Chapter 2. The reason why smooth manifolds have many differentiable objects attached to them is that they can be locally very well approximated by linear spaces called tangent spaces . Locally, everything looks like traditional calculus. Each point has a tangent space attached to it so that we obtain a “bunch of tangent spaces” called the tangent bundle. We found it appropriate to introduce at this early point the notion of vector bundle. It helps in structuring both the language and the thinking. Once we have “things to derivate and integrate” we need to know how to explicitly perform these operations. We devote the Chapter 3 to this purpose. This is perhaps one of the most unattractive aspects of differential geometry but is crucial for all further developments. To spice up the presentation, we have included many examples which will found applications in later chapters. In particular, we have included a whole section devoted to the representation theory of compact Lie groups essentially describing the equivalence between representations and their characters. The study of Shape begins in earnest in Chapter 4 which deals with Riemann manifolds. We approach these objects gradually. The first section introduces the reader to the notion of geodesics which are defined using the Levi-Civita connection. Locally, the geodesics play the same role as the straight lines in an Euclidian space but globally new phenomena arise. We illustrate these aspects with many concrete examples. In the final part of this section we show how the Euclidian vector calculus generalizes to Riemann manifolds. The second section of this chapter initiates the local study of Riemann manifolds. Up to first order these manifolds look like Euclidian spaces. The novelty arises when we study “second order approximations ” of these spaces. The Riemann tensor provides the complete measure of how far is a Riemann manifold from being flat. This is a very involved object and, to enhance its understanding, we compute it in several instances: on surfaces (which can be easily visualized) and on Lie groups (which can be easily formalized). We have also included Cartan’s moving frame technique which is extremely useful in concrete computations. As an application of this technique we prove the celebrated Theorema Egregium of Gauss. This section concludes with the first global result of the book, namely the Gauss-Bonnet theorem. We present a proof inspired from [25] relying on the fact that all Riemann surfaces are Einstein manifolds. The Gauss-Bonnet theorem will be a recurring theme in this book and we will provide several other proofs and generalizations. One of the most fascinating aspects of Riemann geometry is the intimate correlation “local-global”. The Riemann tensor is a local object with global effects. There are currently many techniques of capturing this correlation. We have already described one in the proof of Gauss-Bonnet theorem. In Chapter 5 we describe another such technique which relies on the study of the global behavior of geodesics. We felt we had the moral obligation to present the natural setting of this technique and we briefly introduce the reader to the wonderful world of the calculus of variations. The ideas of the calculus of variations produce remarkable results when applied to Riemann manifolds. For example, we explain in rigorous terms why “very curved manifolds” cannot be “too long” . In Chapter 6 we leave for a while the “differentiable realm” and we briefly discuss the fundamental group and covering spaces. These notions shed a new light on the results of Chapter 5. As a simple application we prove Weyl’s theorem that the semisimple Lie groups with definite Killing form are compact and have finite fundamental group. Chapter 7 is the topological core of the book. We discuss in detail the cohomology iii of smooth manifolds relying entirely on the methods of calculus. In writing this chapter we could not, and would not escape the influence of the beautiful monograph [17], and this explains the frequent overlaps. In the first section we introduce the DeRham cohomology and the Mayer-Vietoris technique. Section 2 is devoted to the Poincar´ e duality, a feature which sets the manifolds apart from many other types of topological spaces. The third section offers a glimpse at homology theory. We introduce the notion of (smooth) cycle and then present some applications: intersection theory, degree theory, Thom isomorphism and we prove a higher dimensional version of the Gauss-Bonnet theorem at the cohomological level. The fourth section analyzes the role of symmetry in restricting the ´ Cartan’s old result that the cohomology topological type of a manifold. We prove Elie of a symmetric space is given by the linear space of its bi-invariant forms. We use this technique to compute the lower degree cohomology of compact semisimple Lie groups. We conclude this section by computing the cohomology of complex grassmannians relying on Weyl’s integration formula and Schur polynomials. The chapter ends with a fifth section ˇ containing a concentrated description of Cech cohomology. Chapter 8 is a natural extension of the previous one. We describe the Chern-Weil construction for arbitrary principal bundles and then we concretely describe the most important examples: Chern classes, Pontryagin classes and the Euler class. In the process, we compute the ring of invariant polynomials of many classical groups. Usually, the connections in principal bundles are defined in a global manner, as horizontal distributions. This approach is geometrically very intuitive but, at a first contact, it may look a bit unfriendly in concrete computations. We chose a local approach build on the reader’s experience with connections on vector bundles which we hope will attenuate the formalism shock. In proving the various identities involving characteristic classes we adopt an invariant theoretic point of view. The chapter concludes with the general Gauss-Bonnet-Chern theorem. Our proof is a variation of Chern’s proof. Chapter 9 is the analytical core of the book. Many objects in differential geometry are defined by differential equations and, among these, the elliptic ones play an important role. This chapter represents a minimal introduction to this subject. After presenting some basic notions concerning arbitrary partial differential operators we introduce the Sobolev spaces and describe their main functional analytic features. We then go straight to the core of elliptic theory. We provide an almost complete proof of the elliptic a priori estimates (we left out only the proof of the Calderon-Zygmund inequality). The regularity results are then deduced from the a priori estimates via a simple approximation technique. As a first application of these results we consider a Kazhdan-Warner type equation which recently found applications in solving the Seiberg-Witten equations on a K¨ ahler manifold. We adopt a variational approach. The uniformization theorem for compact Riemann surfaces is then a nice bonus. This may not be the most direct proof but it has an academic advantage. It builds a circle of ideas with a wide range of applications. The last section of this chapter is devoted to Fredholm theory. We prove that the elliptic operators on compact manifolds are Fredholm and establish the homotopy invariance of the index. These are very general Hodge type theorems. The classical one follows immediately from these results. We conclude with a few facts about the spectral properties of elliptic operators. The last chapter is entirely devoted to a very important class of elliptic operators iv namely the Dirac operators. The important role played by these operators was singled out in the works of Atiyah and Singer and, since then, they continue to be involved in the most dramatic advances of modern geometry. We begin by first describing a general notion of Dirac operators and their natural geometric environment, much like in [11]. We then isolate a special subclass we called geometric Dirac operators. Associated to each such operator is a very concrete Weitzenb¨ ock formula which can be viewed as a bridge between geometry and analysis, and which is often the source of many interesting applications. The abstract considerations are backed by a full section describing many important concrete examples. In writing this book we had in mind the beginning graduate student who wants to specialize in global geometric analysis in general and gauge theory in particular. The second half of the book is an extended version of a graduate course in differential geometry we taught at the University of Michigan during the winter semester of 1996. The minimal background needed to successfully go through this book is a good knowledge of vector calculus and real analysis, some basic elements of point set topology and linear algebra. A familiarity with some basic facts about the differential geometry of curves of surfaces would ease the understanding of the general theory, but this is not a must. Some parts of Chapter 9 may require a more advanced background in functional analysis. The theory is complemented by a large list of exercises. Quite a few of them contain technical results we did not prove so we would not obscure the main arguments. There are however many non-technical results which contain additional information about the subjects discussed in a particular section. We left hints whenever we believed the solution is not straightforward. Personal note It has been a great personal experience writing this book, and I sincerely hope I could convey some of the magic of the subject. Having access to the remarkable science library of the University of Michigan and its computer facilities certainly made my job a lot easier and improved the quality of the final product. I learned differential equations from Professor Viorel Barbu, a very generous and enthusiastic person who guided my first steps in this field of research. He stimulated my curiosity by his remarkable ability of unveiling the hidden beauty of this highly technical subject. My thesis advisor, Professor Tom Parker, introduced me to more than the fundamentals of modern geometry. He played a key role in shaping the manner in which I regard mathematics. In particular, he convinced me that behind each formalism there must be a picture, and uncovering it, is a very important part of the creation process. Although I did not directly acknowledge it, their influence is present throughout this book. I only hope the filter of my mind captured the full richness of the ideas they so generously shared with me. My friends Louis Funar and Gheorghe Ionesei 1 read parts of the manuscript. I am grateful to them for their effort, their suggestions and for their friendship. I want to thank Arthur Greenspoon for his advice, enthusiasm and relentless curiosity which boosted my spirits when I most needed it. Also, I appreciate very much the input I received from the 1 He passed away in 2006. He was the ultimate poet of mathematics. i graduate students of my “Special topics in differential geometry” course at the University of Michigan which had a beneficial impact on the style and content of this book. At last, but not the least, I want to thank my family who supported me from the beginning to the completion of this project. Ann Arbor, 1996. Preface to the second edition Rarely in life is a man given the chance to revisit his “youthful indiscretions”. With this second edition I have been given this opportunity, and I have tried to make the best of it. The first edition was generously sprinkled with many typos, which I can only attribute to the impatience of youth. In spite of this problem, I have received very good feedback from a very indulgent and helpful audience from all over the world. In preparing the new edition, I have been engaged on a massive typo hunting, supported by the wisdom of time, and the useful comments that I have received over the years from many readers. I can only say that the number of typos is substantially reduced. However, experience tells me that Murphy’s Law is still at work, and there are still typos out there which will become obvious only in the printed version. The passage of time has only strengthened my conviction that, in the words of Isaac Newton, “in learning the sciences examples are of more use than precepts”. The new edition continues to be guided by this principle. I have not changed the old examples, but I have polished many of my old arguments, and I have added quite a large number of new examples and exercises. The only major addition to the contents is a new chapter on classical integral geometry. This is a subject that captured my imagination over the last few years, and since the first edition of this book developed all the tools needed to understand some of the juiciest results in this area of geometry, I could not pass the chance to share with a curious reader my excitement about this line of thought. One novel feature in our presentation of integral geometry is the use of tame geometry. This is a recent extension of the better know area of real algebraic geometry which allowed us to avoid many heavy analytical arguments, and present the geometric ideas in as clear a light as possible. Notre Dame, 2007. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Manifolds 1.1 Preliminaries . . . . . . . . . . . . . . §1.1.1 Space and Coordinatization . . §1.1.2 The implicit function theorem . 1.2 Smooth manifolds . . . . . . . . . . . §1.2.1 Basic definitions . . . . . . . . §1.2.2 Partitions of unity . . . . . . . §1.2.3 Examples . . . . . . . . . . . . §1.2.4 How many manifolds are there? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 1 . 1 . 1 . 3 . 5 . 5 . 8 . 9 . 18 . . . . . . . . . . . . . . . . . . . . . 21 21 21 24 26 30 34 38 38 43 50 53 61 65 65 66 70 76 76 76 78 83 86 2 Natural Constructions on Manifolds 2.1 The tangent bundle . . . . . . . . . . . . . . . . §2.1.1 Tangent spaces . . . . . . . . . . . . . . §2.1.2 The tangent bundle . . . . . . . . . . . §2.1.3 Sard’s Theorem . . . . . . . . . . . . . . §2.1.4 Vector bundles . . . . . . . . . . . . . . §2.1.5 Some examples of vector bundles . . . . 2.2 A linear algebra interlude . . . . . . . . . . . . §2.2.1 Tensor products . . . . . . . . . . . . . §2.2.2 Symmetric and skew-symmetric tensors §2.2.3 The “super” slang . . . . . . . . . . . . §2.2.4 Duality . . . . . . . . . . . . . . . . . . §2.2.5 Some complex linear algebra . . . . . . 2.3 Tensor fields . . . . . . . . . . . . . . . . . . . . §2.3.1 Operations with vector bundles . . . . . §2.3.2 Tensor fields . . . . . . . . . . . . . . . §2.3.3 Fiber bundles . . . . . . . . . . . . . . . 3 Calculus on Manifolds 3.1 The Lie derivative . . . . §3.1.1 Flows on manifolds §3.1.2 The Lie derivative §3.1.3 Examples . . . . . 3.2 Derivations of Ω∗ (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii CONTENTS §3.2.1 The exterior derivative . . . . . . . . . . . . . . . . . . §3.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . Connections on vector bundles . . . . . . . . . . . . . . . . . §3.3.1 Covariant derivatives . . . . . . . . . . . . . . . . . . . §3.3.2 Parallel transport . . . . . . . . . . . . . . . . . . . . . §3.3.3 The curvature of a connection . . . . . . . . . . . . . . §3.3.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . §3.3.5 The Bianchi identities . . . . . . . . . . . . . . . . . . §3.3.6 Connections on tangent bundles . . . . . . . . . . . . Integration on manifolds . . . . . . . . . . . . . . . . . . . . . §3.4.1 Integration of 1-densities . . . . . . . . . . . . . . . . . §3.4.2 Orientability and integration of differential forms . . . §3.4.3 Stokes’ formula . . . . . . . . . . . . . . . . . . . . . . §3.4.4 Representations and characters of compact Lie groups §3.4.5 Fibered calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 86 91 92 92 97 98 101 104 105 107 107 111 118 123 129 134 134 134 137 142 145 150 160 160 164 166 169 175 184 184 184 190 193 194 197 204 205 205 209 210 210 212 3.3 3.4 4 Riemannian Geometry 4.1 Metric properties . . . . . . . . . . . . . . . . . . . . . . §4.1.1 Definitions and examples . . . . . . . . . . . . . §4.1.2 The Levi-Civita connection . . . . . . . . . . . . §4.1.3 The exponential map and normal coordinates . . §4.1.4 The length minimizing property of geodesics . . §4.1.5 Calculus on Riemann manifolds . . . . . . . . . . 4.2 The Riemann curvature . . . . . . . . . . . . . . . . . . §4.2.1 Definitions and properties . . . . . . . . . . . . . §4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . §4.2.3 Cartan’s moving frame method . . . . . . . . . . §4.2.4 The geometry of submanifolds . . . . . . . . . . §4.2.5 The Gauss-Bonnet theorem for oriented surfaces 5 Elements of the Calculus of Variations 5.1 The least action principle . . . . . . . . . . . . . . . §5.1.1 The 1-dimensional Euler-Lagrange equations §5.1.2 Noether’s conservation principle . . . . . . . 5.2 The variational theory of geodesics . . . . . . . . . . §5.2.1 Variational formulæ . . . . . . . . . . . . . . §5.2.2 Jacobi fields . . . . . . . . . . . . . . . . . . . 6 The Fundamental group and Covering 6.1 The fundamental group . . . . . . . . §6.1.1 Basic notions . . . . . . . . . . §6.1.2 Of categories and functors . . . 6.2 Covering Spaces . . . . . . . . . . . . §6.2.1 Definitions and examples . . . §6.2.2 Unique lifting property . . . . Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv CONTENTS §6.2.3 Homotopy lifting property . . . . . . . . . . . . . . . . . . . . . . . . 213 §6.2.4 On the existence of lifts . . . . . . . . . . . . . . . . . . . . . . . . . 214 §6.2.5 The universal cover and the fundamental group . . . . . . . . . . . . 216 7 Cohomology 7.1 DeRham cohomology . . . . . . . . . . . . . . . . . . . . . . . §7.1.1 Speculations around the Poincar´ e lemma . . . . . . . ˇ §7.1.2 Cech vs. DeRham . . . . . . . . . . . . . . . . . . . . §7.1.3 Very little homological algebra . . . . . . . . . . . . . §7.1.4 Functorial properties of the DeRham cohomology . . . §7.1.5 Some simple examples . . . . . . . . . . . . . . . . . . §7.1.6 The Mayer-Vietoris principle . . . . . . . . . . . . . . §7.1.7 The K¨ unneth formula . . . . . . . . . . . . . . . . . . 7.2 The Poincar´ e duality . . . . . . . . . . . . . . . . . . . . . . . §7.2.1 Cohomology with compact supports . . . . . . . . . . §7.2.2 The Poincar´ e duality . . . . . . . . . . . . . . . . . . . 7.3 Intersection theory . . . . . . . . . . . . . . . . . . . . . . . . §7.3.1 Cycles and their duals . . . . . . . . . . . . . . . . . . §7.3.2 Intersection theory . . . . . . . . . . . . . . . . . . . . §7.3.3 The topological degree . . . . . . . . . . . . . . . . . . §7.3.4 Thom isomorphism theorem . . . . . . . . . . . . . . . §7.3.5 Gauss-Bonnet revisited . . . . . . . . . . . . . . . . . 7.4 Symmetry and topology . . . . . . . . . . . . . . . . . . . . . §7.4.1 Symmetric spaces . . . . . . . . . . . . . . . . . . . . . §7.4.2 Symmetry and cohomology . . . . . . . . . . . . . . . §7.4.3 The cohomology of compact Lie groups . . . . . . . . §7.4.4 Invariant forms on Grassmannians and Weyl’s integral §7.4.5 The Poincar´ e polynomial of a complex Grassmannian ˇ 7.5 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . §7.5.1 Sheaves and presheaves . . . . . . . . . . . . . . . . . ˇ §7.5.2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . 8 Characteristic classes 8.1 Chern-Weil theory . . . . . . . . . . . . . §8.1.1 Connections in principal G-bundles §8.1.2 G-vector bundles . . . . . . . . . . §8.1.3 Invariant polynomials . . . . . . . §8.1.4 The Chern-Weil Theory . . . . . . 8.2 Important examples . . . . . . . . . . . . §8.2.1 The invariants of the torus T n . . §8.2.2 Chern classes . . . . . . . . . . . . §8.2.3 Pontryagin classes . . . . . . . . . §8.2.4 The Euler class . . . . . . . . . . . §8.2.5 Universal classes . . . . . . . . . . 8.3 Computing characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 . 218 . 218 . 222 . 224 . 231 . 234 . 236 . 239 . 242 . 242 . 246 . 250 . 250 . 255 . 260 . 262 . 265 . 269 . 269 . 272 . 275 . 276 . 284 . 290 . 290 . 294 305 . 305 . 305 . 311 . 312 . 315 . 319 . 320 . 320 . 323 . 325 . 328 . 334 CONTENTS v §8.3.1 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 §8.3.2 The Gauss-Bonnet-Chern theorem . . . . . . . . . . . . . . . . . . . 340 9 Classical Integral Geometry 349 9.1 The integral geometry of real Grassmannians . . . . . . . . . . . . . . . . . 349 §9.1.1 Co-area formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 §9.1.2 Invariant measures on linear Grassmannians . . . . . . . . . . . . . . 360 §9.1.3 Affine Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . 370 9.2 Gauss-Bonnet again?!? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 §9.2.1 The shape operator and the second fundamental form of a submanifold in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 §9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space. 375 §9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space . . . . . . 380 9.3 Curvature measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 §9.3.1 Tame geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 §9.3.2 Invariants of the orthogonal group . . . . . . . . . . . . . . . . . . . 389 §9.3.3 The tube formula and curvature measures . . . . . . . . . . . . . . . 393 §9.3.4 Tube formula =⇒ Gauss-Bonnet formula for arbitrary submanifolds 403 §9.3.5 Curvature measures of domains in an Euclidean space . . . . . . . . 405 §9.3.6 Crofton Formulæ for domains of an Euclidean space . . . . . . . . . 408 §9.3.7 Crofton formulæ for submanifolds of an Euclidean space . . . . . . . 418 10 Elliptic Equations on Manifolds 10.1 Partial differential operators: algebraic aspects . . . . . . . §10.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . §10.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . §10.1.3 Formal adjoints . . . . . . . . . . . . . . . . . . . . . 10.2 Functional framework . . . . . . . . . . . . . . . . . . . . . §10.2.1 Sobolev spaces in RN . . . . . . . . . . . . . . . . . §10.2.2 Embedding theorems: integrability properties . . . . §10.2.3 Embedding theorems: differentiability properties . . §10.2.4 Functional spaces on manifolds . . . . . . . . . . . . 10.3 Elliptic partial differential operators: analytic aspects . . . §10.3.1 Elliptic estimates in RN . . . . . . . . . . . . . . . . §10.3.2 Elliptic regularity . . . . . . . . . . . . . . . . . . . . §10.3.3 An application: prescribing the curvature of surfaces 10.4 Elliptic operators on compact manifolds . . . . . . . . . . . §10.4.1 The Fredholm theory . . . . . . . . . . . . . . . . . . §10.4.2 Spectral theory . . . . . . . . . . . . . . . . . . . . . §10.4.3 Hodge theory . . . . . . . . . . . . . . . . . . . . . . 426 426 426 432 434 440 440 447 451 455 459 460 465 470 480 480 489 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Dirac Operators 498 11.1 The structure of Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . 498 §11.1.1 Basic definitions and examples . . . . . . . . . . . . . . . . . . . . . 498 §11.1.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 CONTENTS §11.1.3 Clifford modules: the even case §11.1.4 Clifford modules: the odd case §11.1.5 A look ahead . . . . . . . . . . §11.1.6 Spin . . . . . . . . . . . . . . . §11.1.7 Spinc . . . . . . . . . . . . . . . §11.1.8 Low dimensional examples . . . §11.1.9 Dirac bundles . . . . . . . . . . 11.2 Fundamental examples . . . . . . . . . §11.2.1 The Hodge-DeRham operator . §11.2.2 The Hodge-Dolbeault operator §11.2.3 The spin Dirac operator . . . . §11.2.4 The spinc Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 505 509 510 512 520 523 528 532 532 536 542 548 1. Euclid’s work is a masterpiece of mathematics. more precisely. It is safe to say that geometry as a rigorous science is a creation of ancient Greeks. These coordinates are obtained using a very special method (in this case using three concurrent. We refer of course to the axiomatic method. The first and most important mathematical abstraction is the notion of number. z (P )).Chapter 1 Manifolds 1. the most complicated shapes one could reasonably study using this method are the conics and/or quadrics. the aim of this book is to illustrate how these two concepts. A line or a plane becomes via coordinatization an algebraic object. He then postulated certain (natural) relations between them.1 Space and Coordinatization Mathematics is a natural science with a special modus operandi. and the Greeks certainly did this. All the other properties were derived from these simple axioms. and he distinguished some basic objects in the space such as lines.1 Preliminaries §1. So far. but it has its own limitations. and it has produced many interesting results. We call such a process coordinatization. P −→ (x(P ). y (P ). What is important here is that they produced a one-to-one mapping Euclidian Space → R3 . planes etc. Loosely speaking. The corresponding map is called (in this case) Cartesian system of coordinates. Descartes’ idea of producing what is now commonly referred to as Cartesian coordinates is familiar to any undergraduate. Numbers were put to work in the study of Space. 1 . an equation. It replaces concrete natural objects with mental abstractions which serve as intermediaries. A major breakthrough in geometry was the discovery of coordinates by Ren´ e Descartes in the 17th century. Space and Number. this approach proved to be very productive. For example. The most visible natural object is the Space. One studies the properties of these abstractions in the hope they reflect facts of life. each one endowed with an orientation and a unit length standard. He viewed the Space as a collection of points. pairwise perpendicular lines. the place where all things happen. Euclid proposed a method of research that was later adopted by the entire mathematics. fit together. The space associated with this problem consists of all pairs (position. As science progressed. Clearly. ∞) × (−π.1) A remarkable feature of (1. Fortunately. There are many ways to coordinatize the configuration space of a motion of a particle.1. MANIFOLDS r θ Figure 1. We say the configuration space is 6-dimensional. For example.1: Polar coordinates In general. (1. we know from the principles of Newtonian mechanics that the motion of a particle in the ambient space can be completely described if we know the position and the velocity of the particle at a given moment. The shift from geometry to numbers is beneficial to geometry as long as one has efficient tools do deal with numbers and equations.1). This choice is related to the Cartesian choice by the well known formulae x = r cos θ y = r sin θ. the polar coordinates represent another coordinatization of (a piece of the plane) (see Figure 1. any coordinatization replaces geometry by algebra and we get a two-way correspondence Study of Space ←→ Study of Equations. about the same time with the introduction of coordinates.1. Isaac Newton created the differential and integral calculus and opened new horizons in the study of equations. We can coordinatize this space using six functions: three of them will describe the position. i. but it helps to think of it as an Euclidian space. P → (r(P ). θ(P )) ∈ (0.1) is that x(P ) and y (P ) depend smoothly upon r(P ) and θ(P ). . One can think of Space as a configuration set. velocity) a particle can possibly have.e. The Cartesian system of coordinates is by no means the unique. Concrete problems dictate other choices. For example.. and the other three of them will describe the velocity. so did the notion of Space.2 CHAPTER 1. We cannot visualize this space. only “roomier”. or the most useful coordinatization. the collection of all possible states of a certain phenomenon. and for each choice of coordinates we get a different description of the motion. π ). For example. and the intention of this book is to introduce the reader to the problems and the methods which arise in the study of manifolds. Let X and Y be two Banach spaces and denote by L(X. . and each of them is then naturally coordinatized.1. . the operator T in the above definition is uniquely determined by Th = 1 d |t=0 F (u0 + th) = lim (F (u0 + th) − F (u0 )) . . The map F is differentiable at a point p = (p1 . a continuous function is differentiable at a point if. t→0 t dt We will use the notation T = Du0 F and we will call T the Fr´ echet derivative of F at u0 . sailors used it to travel from one point to another on Earth.1. The coordinatization process had been used by people centuries before mathematicians accepted it as a method.1.1. . it admits a “ best approximation ” by a linear map. One can define inductively C k and C ∞ (or smooth ) maps. if the map u → Du F ∈ L(X. For example. . Let F : U ⊂ X → Y be a continuous function (U is an open subset of X ). near that point. . For more details one can consult [26]. since they all reflect the same phenomenon. xn ). . . PRELIMINARIES 3 all these descriptions must “agree” in some sense. . Y ) such that F (u0 + h) − F (u0 ) − T h Y = o( h X ) as h → 0. Y ) is continuous. Y = Rm . xn ) in Rn and u = (u1 . This coordinatization is not a global one. then L(X. Y ) the space of bounded linear operators X → Y . . Consider F : U ⊂ Rn → Rm . .1. . Loosely speaking. Then F is said to be of class C 1 . with smooth correspondence on overlaps. §1. In other words.2.1. Using Cartesian coordinates x = (x1 . . Note that the points on the Equator or the Greenwich meridian admit two different coordinatizations which are smoothly related. . The map F is said to be (Fr´ echet) differentiable at u ∈ U if there exists T ∈ L(X. . . Assume that the map F : U → Y is differentiable at each point u ∈ U . . Each point has a latitude and a longitude that completely determines its position on Earth. a collection of classical analytical facts. Definition 1. . um = um (x1 . . . if X = Rn . . . Differential geometry studies the objects which are independent of coordinates. We will review the implicit function theorem which will be one of the basic tools for detecting manifolds. F is said to be of class C 2 if u → Du F is of class C 1 . xn ). There exist four domains delimited by the Equator and the Greenwich meridian. The Fr´ echet derivative of F at p is . . pn ) ∈ U if and only if the functions ui are differentiable at p in the usual sense of calculus. with only sketchy proofs. . The next section is a technical interlude. The manifolds are precisely those spaces which can be piecewise coordinatized. Y ) can be identified with the space of m × n matrices with real entries. um ) in Rm we can think of F as a collection of m functions on U u1 = u1 (x1 . Example 1. When F is differentiable at u0 ∈ U . these descriptions should be independent of coordinates.2 The implicit function theorem We gather here. . (b) Let F : U → U be defined as A → A−1 . i.   . Z be Banach spaces. then there exits a neighborhood U1 of u0 in U such that F (U1 ) is an open neighborhood of v0 = F (u0 ) in Y and F : U1 → F (U1 ) is bijective. ··· ∂u1 ∂xn (p) ∂u2 ∂xn (p) ∂um ∂xn (p)     . ∀H ∈ L(Rn .4 (Inverse function theorem). dt Theorem 1. If at a point u0 ∈ U the derivative Du0 F ∈ L(X. um )   Dp F = =  ∂ (x1 . The theorem states that. and F : X × Y → Z a smooth map. with smooth inverse. MANIFOLDS the linear operator Dp F : Rn → Rm given by the Jacobian matrix   ∂ (u1 . Show that U is an open set. . y0 ) ∈ X × Y . Exercise 1. equivalently. .5 (Implicit function theorem). . the equation F (u) = v has a unique solution u = u0 + h. . y ). . for every v sufficiently close to v0 . .1. . (a) Let U ⊂ L(Rn . and a smooth map G : U → V such that . Rn ) → R. V of y0 ∈ Y . More formally. and F : U ⊂ X → Y a smooth function.. where r(h) = o( h ) as h → 0. (c) Show that the Fr´ echet derivative of the map det : L(Rn .1. A → det A.3. for v − v0 Y sufficiently small. Show that DA F (H ) = −A−1 HA−1 for any n × n matrix H . Y be two Banach spaces. To prove the theorem one has to show that. with h very small. .e. the equation below v0 + T h + r(h) = v has a unique solution. Theorem 1. This last equation is a fixed point problem that can be approached successfully via the Banach fixed point theorem. The spirit of the theorem is very clear: the invertibility of the derivative Du0 F “propagates” locally to F because Du0 F is a very good local approximation for F . Let (x0 . Then there exist neighborhoods U of x0 ∈ X . Y . . Rn ) denote the set of invertible n × n matrices. F2 (y ) = F (x0 . . . R ) is given by tr H . The map F is smooth if and only if the functions ui (x) are smooth. .1. . . . Set F2 : Y → Z . then F (u0 + h) = F (u0 ) + T h + r(h). xn )  ∂u1 (p) ∂x1 ∂u2 (p) ∂x1 ∂um (p) ∂x1 ∂u1 (p) ∂x2 ∂u2 (p) ∂x2 ∂um (p) ∂x2 ··· ··· . . if we set T = Du0 F . Z ) is invertible. y0 ). Rn ). Assume that Dy0 F2 ∈ L(Y. at n n A = Rn ∈ L(R . as h = T −1 (v − v0 − r(h)). Y ) is invertible.4 CHAPTER 1. Let X . d |t=0 det(Rn + tH ) = tr H. and set z0 = F (x0 . Let X . . We can rewrite the above equation as T h = v − v0 − r(h) or. y ) = z0 . z0 ) in a neighborhood of (x0 . (b) A collection of continuous. y ) . (a) An open cover {Ui }i∈I of M .1 Basic definitions We now introduce the object which will be the main focus of this book. paracompact Hausdorff space M together with the following collection of data (henceforth called atlas or smooth structure ) consisting of the following. z0 ) has a unique solution (˜ x.. Definition 1. y0 ). namely. and at ξ0 = (x0 . 1. y ) → (x. y )) = (x. Consider the map H : X × Y → X × Z.1. y ) = z0 } is locally the graph of x → H −1 (x. (Dξ0 H )−1 =  −1 −1 − (Dξ0 F2 ) ◦ (Dξ0 F1 ) (Dξ0 F2 ) Thus. .e. (x. It formalizes the general principles outlined in Subsection 1. the set {(x. F (x. ξ = (x. SMOOTH MANIFOLDS 5 the set S of solution (x. y ˜) = H −1 (x. the equation (x. y ˜) = z0 . y ) = z0 which lie inside U × V can be identified with the graph of G. Proof. In pre-Bourbaki times.1. The linear operator Dξ0 H is invertible. DF1 (respectively DF2 ) denotes the derivative of x → F (x. y ) ∈ U × V . y ) of the equation F (x. i. i ∈ I (called charts or local coordinates ) such that. the concept of (smooth) manifold. then the transition map 1 m m Ψj ◦ Ψ− i : Ψi (Ui ∩ Uj ) ⊂ R → Ψj (Ui ∩ Uj ) ⊂ R is smooth.2). injective maps Ψi : Ui → Rm .2. y ) = z0 = (x.2. F (x. F (x. It obviously satisfies x ˜ = x and F (˜ x. Hence. F (x. and if Ui ∩ Uj = ∅. G(x)) ∈ U × V .2. and its inverse has the block decomposition   X 0 . z0 ). y )). (We say the various charts are smoothly compatible . x ∈ U . the classics regarded the coordinate y as a function of x defined implicitly by the equality F (x. Above. Ψi (Ui ) is open in Rm . The map H is a smooth map. A smooth manifold of dimension m is a locally compact. y0 ) (respectively the derivative of y → F (x0 .2 Smooth manifolds §1. by the inverse function theorem.1. see Figure 1. y0 ) its derivative Dξ0 H : X × Y → X × Z has the block decomposition Dξ0 H = X Dξ0 F1 0 Dξ0 F2 . y )).1. . . .2.2. The map t → t3 is a homeomorphism R → R but it is not a diffeomorphism! If M is a smooth manifold we will denote by C ∞ (M ) the linear space of all smooth functions M → R. . ψ ) can be added to the original atlas and the new smooth structure is diffeomorphic with the . xm ) → y 1 (x1 . . Remark 1. ∞).2. Similarly. . . Let U be an open subset of the smooth manifold M (dim M = m) and ψ : U → Rm a smooth.3. A continuous map f : M → N is said to be smooth if. . . . To construct more examples we will use the implicit function theorem . . .4. . . Then ψ defines local coordinates over U compatible with the existing atlas of M . . . y m (x1 . xm ). xm ) on Ui . Thus (U. xm ) . The natural smooth structure consists of an atlas with a single chart. . (b) A smooth map f : M → N is called a diffeomorphism if it is invertible and its inverse is also a smooth map. The first and the most important example of manifold is Rn itself. .2: Transition maps R m Each chart Ψi can be viewed as a collection of m functions (x1 . . N be two smooth manifolds of dimensions m and respectively n. The 1 transition map Ψj ◦ Ψ− can then be interpreted as a collection of maps i (x1 . Definition 1. (a) Let M .6 CHAPTER 1. Example 1. . . we can view another chart Ψj as another collection of functions (y 1 . . Rn : Rn → Rn . The map t → et is a diffeomorphism (−∞. y m ). for any local charts φ on M and ψ on N . MANIFOLDS Ui Uj ψ i ψ ψ ψ −1 j i j R m Figure 1. one-to one map with open image and smooth inverse.2. . ∞) → (0. . the composition ψ ◦ f ◦ φ−1 (whenever this makes sense) is a smooth map Rm → Rn . fk are functionally independent along Z. Thus. Let M be a smooth manifold of dimension m and f1 . Historically. . . Y = Rk . SMOOTH MANIFOLDS 7 initial one. Step 1: Constructing the charts. ∂x ∂fk ∂fk ∂fk · · · ∂xm x1 =···=xm =0 ∂x1 ∂x2 has rank k . Using Zermelo’s Axiom we can produce a maximal atlas (no more compatible local chart can be added to it).. Define Z = Z(f1 . xm ) defined in a neighborhood of p in M such that xi (p) = 0. f1 (p) = · · · = fk (p) = 0 . . We are now in the setting of the implicit function theorem with X = Rm−k . . . . fk as defined on an open subset U of Rm . i. . . . . . this is how manifolds entered mathematics. . . . . Z = Rk . x := (xm−k+1 . . . . . . xm ). DF2 = ∂x our nonzero minor.2. . . xm ) local coordinates near p0 such that xi (p0 ) = 0. . . m. .  . for each p ∈ Z. . . We can think of the functions f1 . . . . and set x := (x1 . . .1. and denote by (x1 . . |p :=  . . . . Proof. there exist local coordinates (x1 . .5. . . Proposition 1. . . . . . k . x→ ∈R .e.  ∂x ∂fk ∂fk ∂fk · · · ∂x m ∂x1 ∂x2 x1 =···=xm =0 is nonzero. Assume this minor is determined by the last k columns (and all the k lines). One of the k × k minors of the matrix  ∂f1 ∂f1 ∂f1  · · · ∂x m ∂x1 ∂x2 ∂f  . the set Z is the graph of some function  g : Up0 ⊂ Rm−k −→ Up0 ⊂ Rk . Split m R as Rm−k × Rk . . in a product neighborhood Up0 = Up0 × Up0 of p0 . . . . fk (x)) ∂F : Rk → Rk is invertible since its determinant corresponds to In this case. . and F : X × Y → Z given by  f1 (x)   .2. Let p0 ∈ Z. . |p :=  .  . and the matrix  ∂f1 ∂f1 ∂f1  · · · 1 2 ∂xm ∂x ∂x ∂f  .  . . . Then Z has a natural structure of smooth manifold of dimension m − k . Assume that the functions f1 . Our next result is a general recipe for producing manifolds. fk ∈ C ∞ (M ). . . fk ) = p ∈ M . xm−k ). . i = 1. Proposition1.2. Step 2. i. We now define ψp0 : Z ∩ Up0 → Rm−k by ψp The map ψp0 is a local chart of Z near p0 . MANIFOLDS (x . The transition maps for the charts constructed above are smooth. . A codimension k submanifold of M is a subset N ⊂ M locally defined as the common zero locus of k functionally independent functions f1 .5 shows that any submanifold N ⊂ M has a natural smooth structure so it becomes a manifold per se. then we can find a partition of unity in which B = A and φ = A . Let M be a smooth manifold and S ⊂ M a closed submanifold.e.2. Let M be a m-dimensional manifold. (a) For any open cover U = (Uα )α∈A of a smooth manifold M there exists at least one smooth partition of unity (fβ )β ∈B subordinated to U such that supp fβ is compact for any β . .2. x ∈ Up0 . the inclusion map i : N → M is smooth.7.2 Partitions of unity This is a very brief technical subsection describing a trick we will extensively use in this book. A (smooth) partition of unity subordinated to this cover is a family (fβ )β ∈B ⊂ C ∞ (M ) satisfying the following conditions. fk ∈ C ∞ (M ). (iv) β fβ (x) = 1 for all x ∈ M . §1. (i) 0 ≤ fβ ≤ 1.9. Exercise 1. Moreover. Proposition 1.2.2. |x | small . any point x ∈ M admits an open neighborhood intersecting only finitely many of the supports supp fβ . .2. Definition 1. Prove that the restriction map r : C ∞ (M ) → C ∞ (S ) f → f |S is surjective. Definition 1..2.8 i. Exercise 1.e. (b) If we do not require compact supports. Complete Step 2 in the proof of Proposition1. The details are left to the reader.6. Z ∩ Up0 = CHAPTER 1. 0 ( x . For a proof we refer to [95]..8. g (x ) ) −→ x ∈ Rm−k .10. (iii) The family (supp fβ ) is locally finite. . We include here for the reader’s convenience the basic existence result concerning partitions of unity. . (ii) ∃φ : B → A such that supp fβ ⊂ Uφ(β ) .5. g (x ) ) ∈ Rm−k × Rk . Let M be a smooth manifold and (Uα )α∈A an open cover of M .2. Note that T 1 is diffeomorphic with the 1-dimensional sphere S 1 (unit circle).2. y n (P ) → z 1 (P ). Consider the open sets UN = S n \ {N } and US = S n \ {S }. . . yn ) defined as the zero locus 2 2 2 x2 1 + y1 = · · · = x n + yn = 1 . . . for any P ∈ UN . .2. . Exercise 1. S n is indeed a smooth manifold. . . Show that the functions y i . −1) ∈ Rn+1 ). σN ). many problems in mathematics find their most natural presentation using the language of manifolds.. . The above example suggests the following general construction. y1 . . . and in fact. to many physical phenomena which can be modelled mathematically one can naturally associate a manifold. As a set T n is a direct product of n circles T n = S 1 × · · · × S 1 (see Figure 1. z j constructed in the above example satisfy zi = yi n j 2 j =1 (y ) . Example 1. ∀i = 1. so by Proposition 1.2. Hence {(UN . we present here a short list of examples of manifolds. z n (P ) . n. xn . .. . along the sphere.2. S = (0. South pole of S n (N = (0. The construction relies on stereographic projections.2.3 Examples Manifolds are everywhere. To give the reader an idea of the scope and extent of modern geometry. x = (x0 .1. Example 1. . . . In this case one can explicitly construct an atlas (consisting of two charts) which is useful in many applications. .2.13. 1) ∈ Rn+1 . They form an open cover of S n . . . One checks that. (US . The stereographic projection from the South pole is defined similarly. . 0. . . . The stereographic projection from the North pole is the map σN : UN → Rn such that. A simple argument shows the map y 1 (P ). P ∈ UN ∩ US . (The n-dimensional sphere). 0. . .3). the point σN (P ) is the intersection of the line N P with the hyperplane {xn = 0} ∼ = Rn . . This list will be enlarged as we enter deeper into the study of manifolds. and for Q ∈ US . On the other hand. . z n (Q)) the coordinates of σS (Q). SMOOTH MANIFOLDS 9 §1. . (The n-dimensional torus). . . xn ) ∈ Rn+1 . . Let N and S denote the North and resp. .5. .12. is smooth (see the exercise below). y n (P )) the coordinates of σN (P ). the differential of |x|2 is nowhere zero. This is the codimension 1 submanifold of Rn+1 given by the equation n |x| = i=0 2 (xi )2 = r2 . For P ∈ UN we denote by (y 1 (P ). This is the codimension n submanifold of R2n (x1 . σS )} defines a smooth structure on S n . . .11. we denote by (z 1 (Q). MANIFOLDS Figure 1. −1 Now “glue” V1 to V2 using the “prescription” given by ψ2 ◦ φ ◦ ψ1 : V1 → V2 . Then their topological direct product has a natural structure of smooth manifold of dimension m + n. In this way we obtain a new topological space with a natural smooth structure induced by the smooth structures on Mi .3: The 2-dimensional torus V 1 V 2 Figure 1. Consider φ: 1/2 < |x| < 2 → 1/2 < |x| < 2 .4: Connected sum of tori Example 1. Example 1.10 CHAPTER 1.14. |x|2 The action of φ is clear: it switches the two boundary components of {1/2 < |x| < 2}. Up to a diffeomeorphism. Let M1 and M2 be two manifolds of the same dimension m. the new manifold thus obtained is . Let Vi ⊂ Ui correspond (via ψi ) to the annulus {1/2 < |x| < 2} ⊂ Rm .2. Let M and N be smooth manifolds of dimension m and respectively n. Pick pi ∈ Mi (i = 1. choose small open neighborhoods Ui of pi in Mi and then local charts ψi identifying each of these neighborhoods with B2 (0). φ(x) = x . 2).15. (The connected sum of two manifolds).2. and reverses the orientation of the radial directions. the ball of radius 2 in Rm . Set ψk ([x0 . xk+1 /xk . xn ) ∈ Rn+1 \ {0} is usually denoted by [x0 . . xk−1 /xk .17. . . . One way to see this is to observe that the projective space can be alternatively described as the quotient space of S n modulo the equivalence relation which identifies antipodal points .4). xn ] ∈ RPn . . ξm−2 = ηm−1 /ηk . xn ] → (x0 /xk . . . . Note This shows the map ψk ◦ ψm 1 that when n = 1.1) so that CPn is a smooth manifold of dimension 2n. . . . . one attaches a point to each direction in Rn+1 . xn ]) = (ξ1 . The equality [x0 . (The complex projective space CPn ). The space RPn+1 has a natural structure of smooth manifold. . . . . . As a topological space RPn is the quotient of Rn+1 modulo the equivalence relation x ∼ y ⇐⇒ ∃λ ∈ R∗ : x = λy. . . ξn ). ηn ] immediately implies (assume k < m)  ξk+1 = ηk  ξ1 = η1 /ηk . . . so that RPn can be thought as the collection of all points at infinity along all directions. ξn ] = [η1 . . (The real projective space RPn ). ψm ([x0 . . . . ξm−1 = 1/ηk  ξm = ηm ηk . . . . The definition is formally identical to that of RPn . xk = 0 . xn ] = [ξ1 . . . ξk = ηk+1 /ηk . They satisfy transition rules similar to (1. . . xn ]) = (η1 . . and it is called the connected sum of M1 and M2 and is denoted by M1 #M2 (see Figure 1. . .2. the so called “point at infinity” along that direction. . . xn ). The equivalence class of x = (x0 . ηm−1 . . . def The maps ψk define local coordinates on the projective space. Example 1. . xk xm = 0} can be easily described. . . . . . To describe it consider the sets Uk = [x0 . 1. Example 1. . The transition map on the overlap region Uk ∩ Um = {[x0 . Alternatively. . ξn = ηn /ηk (1. . xn ]. . . RPn is the set of all lines (directions) in Rn+1 . . ξk . . . The open sets Uk are defined similarly and so are the local charts ψk : Uk → Cn . . xn ] .2. . .1. . . . SMOOTH MANIFOLDS 11 independent of the choices of local coordinates ([19]). . . ξk−1 = ηk−1 /ηk . n. Traditionally.2. . Now define ψ k : Uk → Rn [x0 . . ηm . . ηn ). RP is diffeomorphic with S 1 . .2.16. . .2. . . . CPn is the quotient space of Cn+1 \ {0} modulo the equivalence relation def x ∼ y ⇐⇒ ∃λ ∈ C∗ : x = λy. . . . .1) −1 is smooth and proves that RPn is a smooth manifold. 1. . . . k = 0. . ξk−1 . . . We want to show that Grk (V ) has a natural structure of smooth manifold compatible with this topology. We will say that Grk (V ) is the linear Grassmannian of k -planes in E . MANIFOLDS In the above example we encountered a special (and very pleasant) situation: the −1 : U → V gluing maps not only are smooth. . Prove that CP1 is diffeomorphic to S 2 . Any k -dimensional subspace L ⊂ V is uniquely determined by the orthogonal projection onto L which we will denote by PL .2.18. Our next example naturally generalizes the projective spaces described above. L → PL with inverse P → Range (P ). To do this it is very convenient to fix an Euclidean metric on V .19. Example 1. Grk (Cn )). We would like to give several equivalent descriptions of the natural structure of smooth manifold on Grk (V ). B ∈ End+ (V ). L) := U ∈ Grk (V ).20.2. ψi ) : Ui → Cn } such that all transition maps are holomorphic. rank P = k . A complex manifold is a smooth. The set Projk (V ) is a subset of the vector space of symmetric endomorphisms End+ (V ) := A ∈ End(V ). Thus we can identify Grk (V ) with the set of rank k projectors Projk (V ) := P : V → V . 2ndimensional manifold M which admits an atlas {(Ui . A∗ = A . We have a natural map P : Grk (V ) → Projk (V ). For every 0 ≤ k ≤ n we denote by Grk (V ) the set of k -dimensional vector subspaces of V . P ∗ = P = P 2 . 2 (1. The bijection P : Grk (V ) → Projk (V ) induces a topology on Grk (V ). Definition 1. B ) := 1 tr(AB ). This type of gluing induces a “rigidity” in the underlying manifold and it is worth distinguishing this situation. (The real and complex Grassmannians Grk (Rn ). The complex projective space is a complex manifold.2.2. (Complex manifolds).n (R) instead of Grk (Rn ). To see this. ∀A. they are also holomorphic as maps ψk ◦ ψm n where U and V are open sets in C .12 Exercise 1. CHAPTER 1. The sapce End+ (V ) is equipped with a natural inner product (A. we define for every L ⊂ Grk (V ) the set Grk (V. U ∩ L⊥ = 0 . Suppose V is a real vector space of dimension n.2) The set Projk (V ) is a closed and bounded subset of End+ (V ). When V = Rn we will write Grk. L⊥ ) → Grk (V. Since K ∈ Grk (V. L) is an open subset of Grk (V ). Since L ∈ Grk (V. It suffices to show that ker( − PL + PU ) = 0. L) we deduce from (a) that the map  − PL − PK : V → V is an isomorphism. for every K ∈ Grk (V. Proof. The space of isomorphisms of V is an open subset of End(V ). the endomorphism ( − PL − PU ) is an isomorphism. Conversely.1. (a) Note first that a dimension count implies that U ∩ L⊥ = 0 ⇐⇒ U + L⊥ = V ⇐⇒ U ⊥ ∩ L = 0.2. (b) The set Grk (V. there exists ε > 0 such that any U satisfying PU − PK < ε intersects L⊥ trivially. we will show that if  − PL + PU = PL⊥ + PU is onto. (b) We have to show that. L). ∀L ∈ Grk (V ). From the equality ( − PL − PU )v = 0 we also deduce ( − PL )v = 0 so that v ∈ L.3) Let us show that U ∩ L⊥ = 0 implies that  − PL + PL is an isomorphism. Suppose v ∈ ker( − PL + PU ). Then U ∩ L⊥ = 0 ⇐⇒  − PL + PU : V → V is an isomorphism. for any subspace U satisfying PU − PK < ε. let v ∈ V . (a) Let L ∈ Grk (V ). then U + L⊥ = V . L). Then 0 = PL ( − PL + PU )v = PL PU v = 0 =⇒ PU v ∈ U ∩ ker PL = U ∩ L⊥ = 0. .2. Note that for every L ∈ Grk (V ) we have a natural map Γ : Hom(L. Note that ( − PL − PK ) − ( − PL − PU ) = PK − PU .2.21. Hence PU v = 0. Indeed. so that v ∈ U ⊥ . We now conclude using part (a). SMOOTH MANIFOLDS Lemma 1. Hence v ∈ U ⊥ ∩ L = 0. 13 (1. Then there exists x ∈ V such that v = PL⊥ x + PU x ∈ L⊥ + U. L). L). we have an open cover of Grk (V ) Grk (V ) = L∈Grk (V ) Grk (V. Hence there exists ε > 0 such that. 5) ΓS = {x + Sx ∈ L + L⊥ = V .5). y ∈ L Let v = PL v + PL⊥ v = vL + vL+ ∈ V (see Figure 1. Then the above linear system can be rewritten as S· Now observe that S2 = Hence S is invertible. ∗ ⊥ ⊥ Γ⊥ . More precisely. x ∈ L ⇐⇒ v − (x + Sx) ∈ Γ⊥ S ⇐⇒ ∃x ∈ L.5: Subspaces as graphs of linear operators. Then PΓS v = x + Sx. which associates to each linear map S : L → L⊥ its graph (see Figure 1. L + S ∗ S 0 0 L⊥ + SS ∗ . x ∈ L}. MANIFOLDS v v L Sx Γ S v L x L Figure 1. and S−1 = (L + S ∗ S )−1 0 0 (L⊥ + SS ∗ )−1 ·S x y = vL vL⊥ . S = Γ−S ∗ = y − S y ∈ L + L = V . Sx − y = vL⊥ Consider the operator S : L ⊕ L⊥ → L ⊕ L⊥ which has the block decomposition S= L S∗ S −⊥ L . .14 L CHAPTER 1. We will show that this is a homeomorphism by providing an explicit description of the orthogonal projection PΓS Observe first that the orthogonal complement of ΓS is the graph of −S ∗ : L⊥ → L. y ∈ L⊥ such that x + S ∗ y = vL . 4) Note that if U ∈ Grk (V. 1. PΓS v = Hence PΓS has the block decomposition PΓS = L S 15 . i = 0. where for every subspace K → V we denoted by IK : K → V the canonical inclusion. L⊥ ) S → ΓS ∈ Grk (V ) is a homeomorphism. L⊥ ) = k (n − k ). the above formulæ show that if U ∈ Grk (V. SMOOTH MANIFOLDS = We deduce x = (L + S ∗ S )−1 vL + (L + S ∗ S )−1 S ∗ vL⊥ and.1.2.2. then with respect to the decomposition V = L + L⊥ the projector PU has the block form PU = A B C D = PL PU IL PL PU IL⊥ PL⊥ PU IL PL L⊥ PU IL⊥ . This shows that the graph map Hom(L. and Gr1 (Cn ) ∼ = CPn−1 . (L + S ∗ S )−1 (L + S ∗ S )−1 S ∗ ∗ − 1 (L⊥ + SS ) S −(L⊥ + SS ∗ )−1 x Sx . L1 ). Grk (Cn ) is defined as the space of complex k -dimensional subspaces of Cn . where S = CA−1 . . L⊥ i ). · [(L + S ∗ S )−1 (L + S ∗ S )−1 S ∗ ] = (L + S ∗ S )−1 (L + S ∗ S )−1 S ∗ ∗ − 1 S (L + S S ) S (L + S ∗ S )−1 S ∗ . then U = ΓS . Show that Grk (Cn ) is a complex manifold of complex dimension k (n − k )). It can be structured as above as a smooth manifold of dimension 2k (n − k ). U = ΓS0 = ΓS1 . This shows that Grk (V ) has a natural structure of smooth manifold of dimension dim Grk (V ) = dim Hom(L.2. L0 ) ∩ Grk (V. (1. Si ∈ Hom(Li . Note that Gr1 (Rn ) ∼ = RPn−1 . The Grassmannians have important applications in many classification problems.22. and the correspondence S0 → S1 is smooth. then we can represent U in two ways. Moreover. L). Exercise 1. To describe its smooth structure we will use the Cayley transform trick as in [84] (see also the classical [100]). T · O(n)# . (a) (Rn . One can show that any connected commutative Lie group has the from T n × Rm . (A# )# = A for any A ∈ Mn (R)# . (d) The orthogonal group O(n) is the group of real n × n matrices satisfying T · T t = . Set Mn (R)# := T ∈ Mn (R) . MANIFOLDS Example 1. This is a commutative group. More generally. K) defined as the group of invertible n × n matrices with entries in the field K = R. A Lie group is a smooth manifold G together with a group structure on it such that the map G×G→G (g. K).3) in the linear space of n × n matrices with entries in K. Any T ∈ O(n) defines a self-homeomorphism of O(n) by left translation in the group LT : O(n) → O(n) We obtain an open cover of O(n): O(n) = T ∈O(n) 1 S → LT (S ) = T · S. The matrices in Mn (R)# are called non exceptional.2.1. the torus T n is a Lie group as a direct product of n circles1 . GL(n. matrices. det( + T ) = 0 . The Cayley transform is the map # : Mn (R)# → Mn (R) defined by A → A# = ( − A)( + A)−1 . (ii) # is involutive. These structures provide an excellent way to formalize the notion of symmetry. (i) A# ∈ Mn (R)# for every A ∈ Mn (R)# . o(n). The Cayley transform has some very nice properties. K) is an open subset (see Exercise 1. if A ∈ Mn (R)# is skew-symmetric then A# ∈ O(n). i. Indeed. (b) The unit circle S 1 can be alternatively described as the set of complex numbers of norm one and the complex multiplication defines a Lie group structure on it. h) → g · h−1 is smooth.. (iii) For every T ∈ O(n)# the matrix T # is skew-symmetric.23. and conversely. (Lie groups). The latter space is an open subset in the linear space of real n × n skew-symmetric matrices.e. It has dimension dK n2 . where dK is the dimension of K as a linear space over R. skew-symmetric. C is a Lie group. . We will often use the alternate notation GL(Kn ) when referring to GL(n. +) is a commutative Lie group.16 CHAPTER 1. Thus the Cayley transform is a homeomorphism from O(n)# to the space of nonexceptional. Clearly  ∈ O(n)# = O(n) ∩ Mn (R)# so that O(n)# is a nonempty open subset of O(n). (c) The general linear group GL(n. the kernel of the group homomorphism det : U (n) → S 1 . tr A = 0}. Inside O(n) lies a normal subgroup (the special orthogonal group ) SO(n) = {T ∈ O(n) . K) = {A ∈ Mn×n (K) .25. K) denote the group of n × n matrices of determinant 1 with entries in the field K = R. C) . K) is a smooth manifold modeled on the linear space sl(n. (b) Prove that U (n) and SU (n) are manifolds. SO(n). (e) The unitary group U (n) is defined as U (n) = {T ∈ GL(n. In particular. where dK = dimR K.1. tr A = 0}. ΨT T ∈O(n) defines a smooth structure on O(n). ΨT T ∈O(n) 17 defines a smooth structure on O(n). we coordinatize the group using the space u(n) of skew-adjoint (skew-Hermitian) n × n complex matrices (A = −A∗ ). where T ∗ denotes the conjugate transpose (adjoint) of T . . Using the Cayley trick show that SL(n. (d) Prove that SU (2) is diffeomorphic with S 3 (Hint: think of S 3 as the group of unit quaternions. T · T ∗ = }. SU (n) is also called the special unitary group . In particular. det T = 1}. This time. Let SL(n.2. SU (n) are compact spaces. One can show that the collection T · O(n)# . C.2. Thus U (n) is a smooth manifold of dimension dim U (n) = dim u(n) = n2 . (a) Prove the properties (i)-(iii) of the Cayley transform.) Exercise 1. it has dimension dK (n2 − 1). SMOOTH MANIFOLDS Define ΨT : T · O(n)# → o(n) by S → (T −1 · S )# .2. Inside U (n) sits the normal subgroup SU (n). (c) Show that O(n). we deduce dim O(n) = n(n − 1)/2. and then show that T · O(n)# . To prove that U (n) is a manifold one uses again the Cayley transform trick. U (n). In fact the Cayley transform trick allows one to coordinatize SU (n) using the space su(n) = {A ∈ u(n) . This a smooth manifold of dimension n2 − 1.24. Exercise 1. The group SO(n) is a Lie group as well and dim SO(n) = dim O(n). 2. we can ask Question 1: Which are the compact.26. 95].28. and by ΓT the graph of T . Exercise 1. §1. T ∗ : V1 → V0 . If G is a Lie group. RT = 2PΓT − . RT = RT .27. ΓT = (v0 . Show that 1 ∈ G lies in the closure of {g n . and T : V0 → V1 is a linear map. and ker( − RT ) = ΓT .18 CHAPTER 1.2. In view of Exercise 1. Prove that the closure of H is also a subgroup of G. We have to be more specific. where PΓT denotes the orthogonal projection onto ΓT . Show that RT = CT R0 is an orthogonal involution. If H is the subgroup algebraically generated by U show that H is dense in G. 2 ∗ RT = . MANIFOLDS Exercise 1. We denote by CT the Cayley transform of X . (b) Let G be a compact Lie group and g ∈ G. this fact allows us to produce many examples of Lie groups. connected manifolds of a given dimension d? . and H is an abstract subgroup of G..2. This is too general a question to expect a clear cut answer. For example. Exercise 1. We form the skew-symmetric operator X : V0 ⊕ V1 → V0 ⊕ V1 . RT is the orthogonal reflection in the subspace ΓT . V1 are two real. finite dimensional Euclidean space. We denote by T ∗ is adjoint. In other words. and H is a closed subgroup of G. (Quillen). CT = ( − X )( + X )−1 .27. n ∈ Z \ {0}}. (a) Let G be a connected Lie group and denote by U a neighborhood of 1 ∈ G.2. Remark 1. then H is in fact a smooth submanifold of G. X · v0 v1 = 0 −T T∗ 0 · v0 v1 . and by R0 : V0 ⊕ V1 → V0 ⊕ V1 the reflection R0 = V0 0 0 −V1 .4 How many manifolds are there? The list of examples in the previous subsection can go on for ever. v1 ) ∈ V0 ⊕ V1 .2. so one may ask whether there is any coherent way to organize the collection of all possible manifolds.e. and with respect to this smooth structure H is a Lie group. i.29. Suppose V0 .2. v1 = T v0 l . For a proof we refer to [44. Suppose G is a Lie group. Property X may refer to more than the (differential) topology of a manifold. A classical result in topology says that all compact surfaces can be obtained in this way (see [68]). (Can you prove this?) We can raise the stakes and try the same problem for d = 2. In many instances one can give fairly accurate answers.6 Figure 1. the projective plane RP2 . but there has been considerable progress due to G. Real life situations suggest the study of manifolds with additional structure. Unfortunately. and we naturally ask whether it looks like one of the surfaces constructed above. for example. One reason is that. We can connect sum two tori. a torus has two faces: an inside face and an outside face (think of a car rubber tube). we cannot visualize the real projective plane (one can prove rigorously it does not have enough room to exist inside our 3-dimensional Universe). Perelman’s proof of the Poincar´ e conjecture. 2 Things are still not settled in 2007. which we cannot visualize and we have yet no idea if they are pairwise distinct. Question 2 Can we wrap a planar piece of canvas around a metal sphere in a oneto-one fashion? (The canvas is flexible but not elastic). This is not the end of the story. To kill the suspense. We have to decide this question using a little more than the raw geometric intuition provided by a picture. The following problem may give the reader a taste of the types of problems we will be concerned with in this book. and look for all the manifolds with a given property X . Already the situation is more elaborate. We can reconsider our goals. RP2 has a weird behavior: it has “no inside” and “no outside”. but in the above list some manifolds are diffeomorphic. In dimension 3 things are not yet settled2 and.1. It has only one side! One says the torus is orientable while the projective plane is not. three tori or any number g of tori. and we have to describe which. They clearly look different but we have not yet proved rigorously that they are indeed not diffeomorphic.6: Connected sum of 3 tori Again we face the same question: do we get non-diffeomorphic surfaces for different choices of g ? Figure 1. We obtain doughnut-shaped surface as in Figure 1. . SMOOTH MANIFOLDS 19 For d = 1 the answer is very simple: the only compact connected 1-dimensional manifold is the circle S 1 . We know at least two surfaces: the sphere S 2 and the torus T 2 . We know another example of compact surface. We can now connect sum any numbers of RP2 ’s to any donut an thus obtain more and more surfaces. to make things look hopeless. in dimension ≥ 4 Question 1 is algorithmically undecidable .2.6 suggests that this may be the case but this is no rigorous argument. we mention that RP2 does not belong to the family of donuts. 20 CHAPTER 1.]. and it may take a while to reveal itself. it is there. one asks for a rigorous explanation of what goes wrong. Matter. It might comfort the reader during this less than glamorous journey to carry in the back of his mind Hermann Weyl’s elegantly phrased advise. Time. There are many other structures Nature forced us into studying them. . nevertheless. our object being to get an insight into the nature of space [. so here too the only way we can lessen the burden of formulæ is to master the technique of tensor analysis to such a degree that we can turn to real problems that concern us without feeling any encumbrance. since a (canvas) sphere is curved in a different way than a plane canvas.. A word to the reader. Naturally. Most of the constructions the reader will have to “endure” in the next two chapters constitute not just some difficult to “swallow” formalism. and for such objects there is a rigorous way to measure “how curved” are they. The best explanation of this phenomenon is contained in the celebrated Theorema Egregium (Golden Theorem) of Gauss. it cannot be avoided.. “It is certainly regrettable that we have to enter into the purely formal aspect in such detail and to give it so much space but. Whoever sets out in quest of these goals must possess a perfect mathematical equipment from the outset. but they may not be so easily described in elementary terms. Weyl: Space. The next two chapters are probably the most arid in geometry but.” H. but the basic language of geometry. even if we do not always have the time to show it to the reader. One then realizes that the problem in Question 2 is impossible. MANIFOLDS A simple do-it-yourself experiment is enough to convince anyone that this is not possible. keep in mind that. Canvas surfaces have additional structure (they are made of a special material). behind each construction lies a natural motivation and. Just as anyone who wishes to give expressions to his thoughts with ease must spend laborious hours learning language and writing. . and |γ (t)|2 = 1. it is the horizontal affine plane through the North pole. The linear approximation of x2 + y 2 + z 2 near N seems like the best candidate.1.Chapter 2 Natural Constructions on Manifolds The goal of this chapter is to introduce the basic terminology used in differential geometry. There is one natural way to fix this problem. We want to find the plane passing through the North pole N (0. Example 2.1 The tangent bundle §2. where O(2) denotes a quadratic error.e. it relies on objects “outside” the manifold S 2 . Hence. the osculator plane is z = 1. Look at a smooth path γ (t) on S 2 passing through N at t = 0. t → γ (t) ∈ R3 . The classics would refer to such a plane as an osculator plane. The natural candidate for this osculator plane would be a plane given by a linear equation that best approximates the defining equation x2 + y 2 + z 2 = 1 in a neighborhood of the North pole. The above construction has one deficiency: it is not intrinsic. 0. 21 (2.1 Tangent spaces We begin with a simple example which will serve as a motivation for the abstract definitions.1.1. 1) that is “closest” to the sphere.1. Consider the sphere (S 2 ) : x2 + y 2 + z 2 = 1 in R3 . The key concept is that of tangent space at a point which is a first order approximation of the manifold near that point. i. We have x2 + y 2 + z 2 − 1 = 2(z − 1) + O(2).1) . The linear subspace {z = 0} ⊂ R3 is called the tangent space to S 2 at N . Geometrically. 2. Hence. We will be able to transport many notions in linear analysis to manifolds via the tangent space. . this is a non-issue.1. . We write this α ∼1 β . . . . . . However. If φ(t) is another curve through N given in local coordinates by t → (u(t). Tp M has a natural structure of vector space.1. xm ) and (y ) = (y 1 . The binary relation ∼1 is obviously reflexive and symmetric. The transitivity follows from the equality i i y ˙γ (0) = y ˙β (0) = j ∂y i j x ˙ (0) = ∂xj β j ∂y i j j x ˙ (0) = y ˙α (0). if we use the stereographic projection from the South pole we get local coordinates (u. and the equivalence classes of “∼”.4. Proof. β : (−ε. We deduce that the tangent space consists of the tangents to the curves on S 2 passing through N . . Fortunately. . . .22 CHAPTER 2. v ˙ (0) . γ ˙ (0) ⊥ γ (0). v ) near N . . 0).1. Thus. v ) plane. . In the above local coordinates the curves α. ∼1 is an equivalence relation.1. v ˙ (0) = u ˙ (0).e. The set of these equivalence classes is denoted by Tp M . . Lemma 2. Lemma 2. . A tangent vector to M at p is a first-order-contact equivalence class of curves through p. Let M m be a smooth manifold and p0 a point in M . Let α ∼1 β and β ∼1 γ . This equivalence relation is now intrinsic modulo one problem: “∼” may depend on the choice of the local coordinates. .1. there is a bijective correspondence between the tangents to the curves through N . as we are going to see. Thus there exist local coordinates (x) = (x1 . . Two smooth paths α. . ∂xj α Definition 2. and is called the tangent space to M at p. xm ) near p such that xi (p) = 0. m 1 m where α(t) = (xα (t)) = (x1 α (t). i. .5. Sketch of proof. y m ) near p0 such that (x ˙ α (0)) = (x ˙ β (0)) and (y ˙ β (0)) = (y ˙ γ (0)). so we only have to check the transitivity. γ (0)) = 0.3. . Choose local coordinates (x1 . This is apparently no major conceptual gain since we still regard the tangent space as a subspace of R3 . xm ) near p0 such that x ˙ α (0) = x ˙ β (0). . then ˙ (0) ⇐⇒ γ ˙ (0) = φ u ˙ (0). and let α and β be two smooth curves through p.2. xβ (t)). xβ (t) . v (t)) in the (u.1) at t = 0 we get (γ ˙ (0). . and this is still an extrinsic description. and β (t) = (xβ (t)) = (xβ (t). ∀i. v (t)). . and any curve γ (t) as above can be viewed as a curve t → (u(t). The right hand side of the above equality defines an equivalence relation ∼ on the set of smooth curves passing trough (0. xα (t)). so that γ ˙ (0) lies in the linear subspace z = 0. Construct a new curve γ through p given by i i xi γ (t) = xα (t) + xβ (t) . . .. β become i xi α (t) . ε) → M such that α(0) = β (0) = p0 are said to have a first order contact at p0 if there exist local coordinates (x) = (x1 . The equivalence class of a curve α(t) such that α(0) = p will be temporarily denoted by [α ˙ (0)]. NATURAL CONSTRUCTIONS ON MANIFOLDS If we differentiate (2. Definition 2. Let F : RN → Rk be a smooth map. . . THE TANGENT BUNDLE 23 ˙ (0)] := [γ Set [α ˙ (0)] + [β ˙ (0)].1.5.1. . . (b) rank Dx F = k . . xm ) be coordinates near p ∈ M such that xi (p) = 0.1. . so the above equality states that this tangent The matrix T space lies inside the space of skew-symmetric matrices. . T ˙ (0) defines a vector in T O(n). Exercise 2. xm ). all the above notions admit a nice description using local coordinates. m. .9. i. Let (x1 . (a) G = O(n). Example 2.1.1. On the other hand.2. Let (−ε. ..2) Note that these vectors depend on the local coordinates (x1 . k = 1. It follows from the obvious fact that any path through the origin in Rm has first order contact with a linear one t → (a1 t.2. i denotes Kronecker’s delta symbol. . . . ∂xk (2. when the point p is clear from the context. We want to describe T G. ˙1 (0)] = [β ˙2 (0)] then (b) If [α ˙1 (0)] = [α ˙2 (0)] and [β ˙1 (0)] = [α ˙2 (0)]. T O(n) ⊂ o(n). Lemma 2. am t). Then T t (s) · T (s) = . ε) s → T (s) be a smooth path of orthogonal matrices such that T (0) = . . Often. ∀x ∈ M. tδk ).6. Then M is a smooth manifold of dimension N − k and Tx M = ker Dx F. for all x ∈ M . Thus.7.2. we proved in Section 1. Exercise 2.1. Assume that (a) M = F −1 (0) = ∅. We set where δj ∂ (p) := e ˙ k (0). .8. Differentiating this equality at s = 0 we get ˙ t (0) + T ˙ (0) = 0. .e. ∂ (p) 1≤k≤m ∂xk is a basis of Tp M . Consider the curves 1 m ek (t) = (tδk .2 that dim G = dim o(n) so that T O(n) = o(n). For this operation to be well defined one has to check two things. . ∀i. Finish the proof of the Lemma 2. [α ˙ (0)] will be written simply as α ˙ (0). where G is one of the Lie groups discussed in Section 1. Proof. As one expects. . . . (a) The equivalence class [γ ˙ (0)] is independent of coordinates. we will omit it in the above notation. . From this point on we will omit the brackets [ – ] in the notation of a tangent vector.2. .1. [α ˙1 (0)] + [β ˙2 (0)] + [β We let the reader supply the routine details. 10. R) we deduce T SL(n. Let (−ε. tr T Thus. R) = sl(n. There is a natural surjection π : TM = p∈M Tp M → M. Then det T (s) = 1 and differentiating this equality at s = 0 we get (see Exercise 1. i. Exercise 2. i. ε) s → T (s) be a smooth path in SL(n. R) = dim sl(n.2 The tangent bundle In the previous subsection we have naturally associated to an arbitrary point p on a manifold M a vector space Tp M . R) such that T (0) = . ∂ (p) . for any p ∈ U . ∂xi X i (v ) We thus have a bijection Ψx : T U → U x × Rm ⊂ Rm × Rm . T SL(n. Since (according to Exercise 1.1. π (v ) = p ⇐⇒ v ∈ Tp M. NATURAL CONSTRUCTIONS ON MANIFOLDS (b) G = SL(n. pick a different coordinate system y = (y i ) on U . R). §2.e. In particular. R).1. an element v ∈ T U = p∈U Tp M is completely determined if we know • which tangent space does it belong to. we have to make sure this topology is independent of local coordinates. • the coordinates of v in the basis v= i ∂ (p) ∂xi . R). Show that T U (n) = u(n) and T SU (n) = su(n). we want to give a rigorous meaning to the intuitive fact that Tp M depends smoothly upon p.25) dim SL(n. we know p = π (v ))..1.24 CHAPTER 2. where U x is the image of U in Rm via the coordinates (xi ). Again. We can now use the map Ψx to transfer the topology on Rm × Rm to T U . the tangent space at  lies inside the space of traceless matrices.2. We will organize the disjoint union of all tangent spaces as a smooth manifold T M . Thus. To see this. Any local coordinate system x = (xi ) defined over an open set U ⊂ M produces a natural ∂ basis ∂x i (p) of Tp M .3) ˙ (0) = 0. It is the goal of the present subsection to coherently organize the family of tangent spaces (Tp M )p∈M . The coordinate independence referred to above is equivalent to the statement that the transition map 1 x m y m Ψy ◦ Ψ− x : U × R −→ T U −→ U × R is a homeomorphism. . R) ⊂ sl(n.e. so that Df is well defined. . and (y 1 . The natural topology of T M is obtained by patching together the topologies of T Uγ . ∂xi (2.1. since any two choices differ (infinitesimally) by a linear substitution. Recall that Tp M is the space of tangent vectors to curves through p. . A smooth map f : M → N induces a smooth map Df : T M → T N such that (a) Df (Tp M ) ⊂ Tf (p) N . . and yα (t) is the description of the path α(t) in j j the coordinates y . while a basis of Tq N is given by ∂y . If α(t) is such a curve (α(0) = p). . β : (−ε.3) 1 This proves that Ψy ◦ Ψ− x is actually smooth. ∂ ∂ A basis in Tp M is given by ∂x . . y n ) near q . . the natural projection π : T M → M is a smooth map. . Applying the chain rule we deduce j Yj =y ˙α (0) = i ∂y j i x ˙ (0) = ∂xi i ∂y j i X. where (Uγ . Df (α ˙ (0)) := β One checks easily that if α1 ∼1 α2 . Definition 2. . . Y .2. xβ (t)). . then in local coordinates they have the description m 1 m α(t) = (x1 α (t).1. . THE TANGENT BUNDLE 25 1 Let A := (x. . x (A) = y (x). . . Moreover. . . φγ )γ is an atlas of M . and one often uses the alternate notation f∗ = Df . then β (t) = f (α(t)) is a smooth curve through q = f (p).1. . ∀p ∈ M (b) The restriction to each tangent space Dp F : Tp M → Tf (p) N is linear. y n (x1 . Then 1 1 m Ψy ◦ Ψ− . To prove that the map Df : Tp M → Tq N is linear it suffices to verify this in any particular local coordinates (x1 . xm ) → (y 1 (x1 . . . . . . The map Df is called the differential of f . xα (t)). . . . . α is a path through p given in the x coordinates as α(t) = x + tX. y j (q ) = 0. xm ) near p. Proof. xm )).12. and α(t) ⊂ U x (A) = (p. . xm ). Then Ψ− ˙ (0)). Y j ∂ where α ˙ (0) = (y ˙α (0)) = Y j ∂y (p). Ψγ ) a defining atlas. . β (t) = (xβ (t). X ) ∈ U x × Rm . . . i j If α. Proposition 2. . such that xi (p) = 0. The smooth manifold T M described above is called the tangent bundle of M . where x(p) = x. we can regard f as a collection of maps (x1 . A set D ⊂ T M is open if its intersection with any T Uγ is open in T Uγ . The above argument shows that T M is a smooth manifold with (T Uγ . .11. ε) → M are two smooth paths such that α(0) = β (0) = p. . .1. and we define ˙ (0). . . then f (α1 ) ∼1 f (α2 ). . Hence. Denote by F : U x → U y the transition map x → y . j . ∀i. then for most real numbers h the level set f −1 (h) does not contain a point where the differential of f is zero. where X is a smooth manifold. In particular. Define Z := (x. β (0) = (x β (0). Suppose M is a submanifold of dimension 2 in R3 . Y . c). . The points in ∆F are called the critical values of F .. a simple thought experiment suggests that most horizontal planes will not be tangent to M . . ax2 + bx + c = 0. for any coordinate chart of Y . .1. Ψ : U → Rm . i.14. . a. y ( xα (t) + xβ (t) ) . Iif we denote by f the restriction of the function z to M . We describe it first in a special case. so that most level sets f −1 (h) are smooth submanifolds of M of codimension 1. a smooth map f : X → Y . b. Exercise 2. x β (0) ) Then n i 1 i n i F (α(t)) = y 1 ( xi α (t) ). We will refer to ∆F as the discriminant set of F . smooth curves on M .1. . Given two smooth manifolds X. DF β i ∂x ∂yj n j =1 n j =1 m i=1 m i=1 ∂y j i ∂ x ˙β . . the set Ψ(S ∩ U ) ⊂ Rm has Lebesgue measure zero in Rm . . Compute the discriminant set of π . known as Sard’s theorem. . F (β (t)) = y ( xβ (t) ). NATURAL CONSTRUCTIONS ON MANIFOLDS ˙ α ˙ (0) = (x ˙1 ˙m ˙1 ˙m α (0). this implies that Df is also smooth. (a) We say that a subset S ⊂ Y is negligible if. . connected manifold of dimension m. and by ∆F ⊂ Y its image via F . n m i=1 DF α ˙ (0) = j =1 ∂ ∂y j i ˙ (0) = x ˙α .13. . if the differential Dx F : Tx X → TF (x) Y is not surjective. a. . Definition 2. . . Suppose that Y is a smooth. We can ask a more general question.1. . c) −→ (a. Equivalently. 1≤i≤m . §2. A point x ∈ X is called a critical point of F . a = 0 .e. y ( xβ (t) ) . i ∂x ∂yj ˙ (0) = d |t=0 F α(t) + β (t) = DF α ˙ (0) + β dt ∂ ∂y j i (x ˙ +x ˙i β) ∂xi α ∂yj ˙ (0) . . .26 CHAPTER 2. = DF α ˙ (0) + DF β This shows that Df : Tp M → Tq N is the linear operator given in these bases by the matrix ∂y j ∂xi 1≤j ≤n. y ( xα (t) ) . . is it true that for “most” y ∈ Y the level set f −1 (y ) is a smooth submanifold of X of codimension dim Y ? This question has a positive answer. . . We denote by Cr F the set of critical points of F . i n i i F (α(t) + β (t) ) = y 1 ( xi α (t) + xβ (t) ). π (b) Define π : Z → R3 by (x. x α (0)). b. b. c) ∈ R4 . (a) Prove that Z is a smooth submanifold of R4 .3 Sard’s Theorem In this subsection we want to explain rigorously a phenomenon with which the reader may already be intuitively acquainted. . Then. . (b) Suppose F : X → Y is a smooth map. . . m. Using Fubini’s theorem we deduce that N ∩ Cr F has trivial ndimensional Lebesgue measure. . . . Proof. and any m. The inductive step is divided into three intermediary steps. and the statement is true for any n < n. . Since Cr F is separable. . ∀j = 1. . . . it suffices to prove that every point u ∈ Cr F admits an open neighborhood N such that F N ∩ Cr F is negligible. Assume first that there exist local coordinates (x1 . Observe that N ∩ Cr F = t Cr Gt . . Suppose u0 ∈ Cr F . k+1 k Step 2. Let f : O → V be a smooth map as above. Theorem 2. . or a submanifold of M of codimension dim N . xn ). . . x1 = t . The set F (Cr F \ Cr 1 F ) is negligible.15. . V are finite dimensional real Euclidean vector spaces. and F : O → V is a smooth map. . x2 . . . . Set Cr F := Cr F \ Cr 1 F . y j (v0 ) = 0. . The set F (Cr F \ Cr F ) is negligible for all k ≥ 1.1. . and the restriction of F to N is described by functions y j = y j (x1 . The inductive assumption implies that the sets Cr Gt have trivial (n − 1)-dimensional Lebesgue measure. . . and we define Gt : Nt → Rm−1 . We obtain a decreasing filtration of closed sets 2 Cr F ⊃ Cr 1 F ⊃ Cr F ⊃ · · · . . xi (u0 ) = 0. . Show that for every q ∈ N \ ∆F the fiber f −1 (q ) is either empty. . For every t ∈ R we set Nt := (x1 . We will argue inductively on the dimension n. xn ) . y m (t. y m ) near v0 = F (u0 ) such that. . . x2 . Step 3. n. xn ) ∈ N. Set n = dim U . and m = dim V . . The set F (Cr k F ) is negligible for some sufficiently large k . and local coordinates (y 1 . THE TANGENT BUNDLE 27 Suppose U. Step 1. (t. . xn ) → y 2 (t.2. xm ) such that y 1 = x1 . . O ⊂ U is an open subset. Pontryagin [82]. We will show that there exists a countable open cover Oj j ≥1 of Cr F such that F (Oj ∩ Cr F is negligible for all j ≥ 1. . .1. xn ) defined in a neighborhood N of u0 . . ∀i = 1. We follow the elegant approach of J. x2 . . The case n = 0 is trivially true so we may assume n > 0. Then the discriminant set ∆F is negligible. .16 (Sard). . . Step 1. Then a point u ∈ O is a critical point of F if and only if rank (Du F : U → V ) < dim V. Milnor [74] and L. . Exercise 2. .1. For every positive integer k we denote by Cr k F ⊂ Cr F the set of points u ∈ O such that all the partial derivatives of F up to order k vanish at u. . F (Cr F ∩N) = G(Cr G ). . The map F is then locally described by a collection of functions y j (s1 . such that. Now define ∂s1 x1 = y 1 (s1 . The implicit function theorem shows that the collection of functions (x1 . . 1) and λ0 > 0. . ∀i = 2. . Then the collection (xi ) defines smooth local coordinates on an open neighborhood N 1 of u0 . By covering Cr F with a countably many open neighborhood {N } >1 such that F (Cr k F ∩N ) is negligible we (k) conclude that F (Cr F ) is negligible. . Then k k k Cr k G ∩N = Cr G . . We regard y j as functions of xi . . . . . and the induction assumption implies that G(Cr k G ) is negligible. . x2 . . depending only on S . ∂ j y1 (u0 ) = 0. . m. . . . ∀i = 2. and Cr k F ∩N is contained in the hyperplane {x = 0}. . . ∀i = 1. . To see this. . n. . . Since u0 ∈ Cr F . Since u ∈ Cr F . . ∂ (s1 )k Define and set xi := si . . . ∂ (s1 )k+1 x1 (s) = ∂ k y1 . . . . . . . . sn ) near u0 and coordinates (y 1 . . then diam F (C ) < λ0 rk .28 CHAPTER 2. . the set F (S ∩ Cr k F ) is negligible. k+1 Step 2. . ∀i = 1. we can find local coordinates (s1 . . ∂ (s1 )j and ∂ k+1 y 1 (u0 ) = 0. if C is a cube with edge r < r0 which intersects Cr k F ∩S . xm ) = F (0. . . m. . xn ) defines a coordinate system in a neighborhood of u0 . Define G : N ∩ {x1 = 0} → V . y m ) near v0 such that si (u0 ) = 0. . . n. choose local coordinates (s1 . n. n Step 3. From the definition we deduce y 1 = x1 . . (k ) . . . . ∀j = 1. n. ∀j = 1. . . . k. More precisely. y m ) near v0 such that (k) (k) si (u0 ) = 0. Suppose k > m . y j (v0 ) = 0. j = 1. From the Taylor expansion around points in Cr k ∩ S we deduce that there exist numF bers r0 ∈ (0. sn ). sn ) near u0 and coordinates (y 1 . . . n. . . . y j (v0 ) = 0. We will prove that F (Cr k F ) is negligible. for every compact subset S ⊂ O. . . after an eventual re-labelling of coordinates. sn ). xn ). . . NATURAL CONSTRUCTIONS ON MANIFOLDS To conclude Step 1 is suffices to prove that the above simplifying assumption concerning the existence of nice coordinates is always fulfilled. we can assume. . . xi = si . 1 that ∂y (u0 ) = 0. . ∀j = 1. . . Set Cr F := Cr k F \ Cr F . we will show that. . G(x2 . x) → Fλ (x) ∈ Y. X. and dim M ≥ dim N .1. of edges < r0 . such that their interiors are disjoint. a1 . For every positive integer P . Suppose F : M → N is a smooth map. Exercise 2. a2 ∈ A . THE TANGENT BUNDLE where for every set A ⊂ V we define diam(A) := sup |a1 − a2 |. A smooth map f : M → N is called an embedding if it is an injective immersion. Cover Cr k F ∩S by finitely many cubes {C }1≤ ≤L .1. . connected manifolds. L. 29 In particular. Suppose F : X → Y is a smooth map between two smooth manifolds. Suppose Λ. Prove that if F is transversal to S . Then its discriminant set is negligible. we deduce that there exists a constant λ1 > 0 such that µm (F (C ) ≤ λ1 rmk = λ1 µn (C )mk/n . For every sub-cube C σ which intersects Cr k F we have µm (C σ ) ≤ λ1 µn (C σ )mk/n = We deduce that µm (C ∩ Cr k F) ≤ σ n−mk µm (C σ ∩ Cr k µn (C ). and F : Λ × X → Y is a smooth map Λ × X (λ. We say that F is transversal to S if for every x ∈ F −1 (S ) we have TF (x) N = TF (x) S + Dx F (Tx M ). Definition 2.17 (Sard).1. . we deduce that when k > µm (C ∩ Cr k F ) = 0. . .2. F) ≤ P n m λ1 µn (C ).1. A smooth map f : M → N is called immersion (resp.1. Suppose F : M → N is a smooth map.1.16 admits the following immediate generalization. if µm denotes the m-dimensional Lebesgue measure on V . surjective). ∀ = 1.20. subdivide each of the cubes C into P n sub-cubes C i of equal sizes. and S ⊂ N is a smooth submanifold of N . Theorem 2. P mk If we let P → ∞ in the above inequality. Exercise 2. Then F is a submersion if and only if the discriminant set ∆F is empty. then F −1 (S ) is a submanifold of M whose codimension is equal to the codimension of S in N .18. and µn denotes the n-dimensional Lebesgue measure on U . Y are smooth.19. submersion ) if for every p ∈ M the differential Dp f : Tp M → Tf (p) N is injective (resp. we have Corollary 2. . i.j ≤m In the above formula. Prove that Λ0 is contained in the discriminant set of the natural projection Z → Λ. and there exists a surjective submersion π : E → M .1. and the functions (y j ) are local coordinates on V . We denote by Aut(F ) the Lie group GL(n.4 Vector bundles The tangent bundle T M of a manifold M has some special features which makes it a very particular type of manifold. In our special case.1. and for every U ∈ U a diffeomorphism ΨU : E |U → U × F. • π : E → M is a surjective submersion. Then (a) E is a smooth manifold.3) we deduce that there exists a trivializing cover. F ) such that • E . for any p ∈ U ∩ V .21. . A vector bundle is a quadruple (E. M. Definition 2. π.1. The properties (a) and (b) make no mention of the special relationship between E = T M and M . There are many other quadruples (E. (b2) If U.3) A(p) = ∂y j (p) ∂xi . 1≤i. π.. M are smooth manifolds. M. V ∈ U are two trivializing neighborhoods with non empty overlap U ∩ V U −1 : F → F is a linear then. and F := Rm (m = dim M ). In particular. Λ0 = λ ∈ Λ. v → (p = π (v ). the map ΦV U (p) is explicitly defined by the matrix (2.30 CHAPTER 2. NATURAL CONSTRUCTIONS ON MANIFOLDS Suppose S is a submanifold of Y such that F is transversal to S . F ) with these properties and they deserve a special name. Set for brevity E := T M . §2.e. the map ΦV U (p) = ΦV p ◦ (Φp ) isomorphism. We list now the special ingredients which enter into this special structure of T M since they will occur in many instances. Fλ : X → Y is not transversal to S . and moreover. (b) From (2. an open cover U of M . R) of linear automorphisms of F . ΦU p (v )) (b1) Φp is a diffeomorphism Ep → F for any p ∈ U . the map p → ΦV U (p) ∈ Aut(F ) is smooth. For every U ⊂ M we set E |U := π −1 (U ).1. Define Z = F −1 (S ) ⊂ Λ × X. Λ0 must be negligible. the functions (xi ) are local coordinates on U . one can add elements in the same fiber. They satisfy the cocycle condition (a) gαα = F (b) gαβ gβγ gγα = F over Uα ∩ Uβ ∩ Uγ .21. (or simply GL(F ) if there is no ambiguity concerning the field of scalars K) the Lie group of linear automorphisms F → F . Moreover. C. We will say that the map gβα is the transition from the α-trivialization to the β trivialization. Note that the properties (b1) and (b2) imply that the fibers π −1 (p) of a vector bundle have a natural structure of linear space. the addition and scalar multiplication operations on π −1 (p) depend smoothly on p.1. The details are carried out in the exercise below. we can reconstruct a vector bundle by gluing the product bundles Eα = F × Uα on the overlaps Uα ∩ Uβ according to the gluing rules the point (v. Moreover. we can find an open cover (Uα ) of M such that each of the restrictions Eα = E |Uα is isomorphic to a product Ψα : Eα ∼ = F × Uα . and • the conditions (a) and (b) above are satisfied. We will refer to the cover (Uα ) as above as a trivializing cover. π (u) = π (v ) → E. on 1 the overlaps Uα ∩ Uβ . THE TANGENT BUNDLE • F is a vector space over the field K = R. C we denote by GLK (F ). and M is called the base space.1. a vector bundle is a smooth family of vector spaces. Roughly speaking. The smoothness of the scalar multiplication means that it is smooth map R × E → E. x ∈ Eβ ∀x ∈ Uα ∩ Uβ . The smoothness of the addition operation this means that the addition is a smooth map + : E ×M E = (u. To describe it. 31 The manifold E is called the total space. . In particular.2. and a collection of smooth maps gαβ : Uα ∩ Uβ → GL(F ) satisfying the cocycle condition. Conversely. For any vector space F over the field K = R. the transition maps gαβ = Ψα Ψ− can be viewed as smooth maps β gαβ : Uα ∩ Uβ → GL(F ). and its dimension (over the field of scalars K) is called the rank of the bundle. let us introduce a notation. There is an equivalent way of defining vector bundles. A line bundle is a vector bundle of rank one. v ) ∈ E × E . According to Definition 2. given an open cover (Uα ) of M . The vector space F is called the standard fiber. and we will refer to the collection of maps (gβα ) satisfying the cocycle condition as a gluing cocycle. x) ∈ Eα is identified with the point gβα (x)v. 1. and the set of families of smooth maps sα : Uα → F . Set X := α V × Uα × {α}. x. and a gluing cocycle is called a gluing cocycle description of that vector bundle. and gluing cocycle gβα : Uαβ → GL(F ). (b) A section of the tangent bundle of a smooth manifold is called a vector field on that manifold. (u. (c) Prove that (E.32 CHAPTER 2. a vector space V .23. α ∈ A . Find a gluing cocycle description of the tangent bundle of the 2-sphere.1. (b) Show that the projection π : X → M . Prove the above proposition. In the sequel. Consider a smooth manifold M . Suppose E → M is a smooth vector bundle with standard fiber F .27. and we define a relation ∼⊂ X × X by V × Uα × { α } (u. defined by an open cover (Uα )α ∈ A. x.22. and E = X/ ∼ equipped with the quotient topology has a natural structure of smooth manifold. and smooth maps gαβ : Uα ∩ Uβ → GL(V ) satisfying the cocycle condition. NATURAL CONSTRUCTIONS ON MANIFOLDS Exercise 2. π . satisfying the following gluing condition on the overlaps sα (x) = gαβ (x)sβ (x).1. Exercise 2. V ) is naturally a smooth vector bundle. α) ∼ (v.26. v = gβα (x)u. Note that Γ(U. E ) or C ∞ (U. Then there exists a natural bijection between the vector space of smooth sections of E . We topologize X as the disjoint union of the topological spaces Uα × V . The space of smooth sections of E over U will be denoted by Γ(U. Definition 2.24. E ) is naturally a vector space. The space of vector fields over on open subset U of a smooth manifold is denoted by Vect (U ).1. ∀p ∈ U ⇐⇒ π ◦ s = U .1. π. def (a) Show that ∼ is an equivalence relation. M. Definition 2. an open cover (Uα ).25. we will prefer to think of vector bundles in terms of gluing cocycles. Proposition 2. ∀x ∈ Uα ∩ Uβ . α) → x descends to a submersion E → M .1. A description of a vector bundle in terms of a trivializing cover. E ). β ) ∈ V × Uβ × {β } ⇐⇒ x = y. Exercise 2. (a) A section in a vector bundle E → M defined over the open subset u ∈ M is a smooth map s : U → E such that s(p) ∈ Ep = π −1 (p). x. .30. this means that • The map F covers f . Equivalently. A bundle endomorphism of E is a bundle morphism F : E → E . j (x) ∈ C (M ). . i. and respectively F2 . . . F ) the space of bundle maps E → F which cover the identity M .   . then we denote by Hom(E. (b) If E and F are two vector bundles over the same manifold. A vector bundle map consists of a pair of smooth maps f : M1 → M2 and F : E 1 → E 2 satisfying the following properties. F (Ep 1 f (p) the diagram below is commutative E1 π1 F M E2 K M M2 π2 M1 K f 1 → E2 • The induced map F : Ep f (p) is linear.29. . Example 2. Consider the trivial vector bundle Rn M → M over the smooth manifold M . n an 1 (x) a2 (x) · · · π an n (x) The map A is a gauge transformation if and only if det Ax = 0. Such bundle maps are called bundle morphisms. A section of this vector bundle is a smooth map u : M → Rn . . ∀x ∈ M .1.1. For example. The composition of bundle maps is defined in the obvious manner and so is the identity morphism so that one can define the notion of bundle isomorphism in the standard way. (a) Let E i → Mi be two smooth vector bundles.e. ∀p ∈ M . Definition 2.1.28.2. . Suppose E1 . An endomorphism of this vector bundle is a smooth map A : M → EndR (Rn ). Assume that both bundle are defined by a common trivializing cover (Uα )α∈A and gluing cocycles gβα : Uαβ → GL(F1 ).31. Exercise 2. 1 ) ⊂ E 2 . We can think of u as a smooth family of vectors (u(x) ∈ Rn )x∈M . THE TANGENT BUNDLE π 33 i Definition 2. Let E → M be a smooth vector bundle. E2 → M are two smooth vector bundles over the smooth manifold with standard fibers F1 . .1. ai .1. . the differential Df of a smooth map f : M → N is a bundle map Df : T M → T N covering f .. . We can think of A as a smooth family of n × n matrices   1 1 a1 (x) a1 2 (x) · · · an (x)  a2 (x) a2 (x) · · · a2 (x)  n 2   1 ∞ Ax =   . hβα : Uαβ → GL(F2 ). . An automorphism (or gauge transformation) is an invertible endomorphism. Example 2.32. (The tautological vector bundle over a Grassmannian). The real case is completely similar. M a smooth manifold. then for each L ∈ Grk (Cn ) the C fiber over L coincides with the subspace in Cn defined by L.. We consider only the complex case. i. (The tautological line bundle over RPn and CPn ).33. .35. We consider here for brevity only complex Grassmannian Grk (Cn ). Let V be a vector space. C) ∼ = C∗ .e. z belongs to the subspace L ⊂ Cn . The total space of the tautological or universal line bundle over CPn is the space Un = UC n = (z.n .n = (z.36. hαβ : Uα ∩ Uβ → GL(V ) two collections of smooth maps satisfying the cocycle conditions. Note that for every L ∈ CPn . and the set of families of smooth maps Tα : Uα → Hom(F1 .n → Grk (Cn ). z belongs to the line L ⊂ Cn+1 . ∀x ∈ Uαβ . satisfying the gluing conditions −1 Tβ (x) = hβα (x)Tα (x)(x)gβα . there exist smooth maps φα : Uα → GL(V ) such that 1 hαβ = φα gαβ φ− β . L) ∈ Cn+1 × CPn . NATURAL CONSTRUCTIONS ON MANIFOLDS Prove that there exists a bijection between the vector space of bundle morphisms Hom(E.1. L) ∈ Cn × Grk (Cn ) .1.1. Example 2.34. and gαβ . let us recall that a rank one vector bundle is usually called a line bundle. The total space of this bundle is Uk.1. α ∈ A . Describe a gluing cocycle for UC n. Prove the two collections define isomorphic vector bundles if and only they are cohomologous.n = UC k.1. §2. F2 ). Prove that UC n and Uk. If π denotes the natural projection π : Uk.34 CHAPTER 2. the fiber through π −1 (L) = UC n. First. and smooth maps gβα : Uα ∩ Uβ → GL(1. n Let π : UC n → CP denote the projection onto the second component. Example 2.L coincides with the one-dimensional subspace in Cn+1 defined by L. C Exercise 2. F ). A complex line bundle over a smooth manifold M is described by an open cover (Uα )α∈A . Exercise 2.5 Some examples of vector bundles In this section we would like to present some important examples of vector bundles and then formulate some questions concerning the global structure of a bundle. {Uα } an open cover of M .1.n are indeed smooth vector bundles. Note that UC n−1 = U1. Denote as usual by N and S the North and respectively South pole. U1 } and gluing cocycle g10 (z ) = z −n .37. L) ∈ ( Cn+1 \ {0} ) × CPn . The family of vector bundles is very large. Consider the incidence set I := (x.2. In this case. 35 Consider for example the manifold M = S 2 ⊂ R3 . For example. U1 = S 2 \ {N }. U0 = S 2 \ {S }. A smooth section of this line bundle is described by a pair of smooth functions u0 : U0 → C. the pair of functions u0 (z ) = z n . z ∈ L . Exercise 2.1.1. THE TANGENT BUNDLE satisfying the cocycle condition gγα (x) = gγβ (x) · gβα (x). Identify U0 with the complex line C. which along the overlap U0 ∩ U1 satisfy the equality u1 (z ) = z −n u0 (z ).39. and the gluing cocycle ( gαβ ◦ f ). We know that CP1 is diffeomorphic to S 2 . For every n ∈ Z we obtain a complex line bundle Ln → S 2 . so that the North pole becomes the origin z = 0.1. We have an open cover S 2 = U0 ∪ U∞ . UN ∩ US . Definition 2. defined by the open cover {U0 . u1 : U1 → C. Prove that the universal line bundle Un → CP1 is isomorphic with the line bundle L−1 constructed in the above example. ∀z ∈ C∗ = U0 \ {0}. . The pullback of E by f is the vector bundle f ∗ E over X defined by the open cover f −1 (Uα ). Prove that the closure of I in Cn+1 × CPn is a smooth manifold diffeomorphic to the total space of the universal line bundle Un → CPn . The following construction provides a very powerful method of producing vector bundles. ∀p ∈ U1 defines a smooth section of Ln . Exercise 2. This manifold is called the complex blowup of Cn+1 at the origin.1. if n ≥ 0.38. Let f : X → M be a smooth map. and E a vector bundle over M defined by an open cover (Uα ) and gluing cocycle (gαβ ). ∀x ∈ Uα ∩ Uβ ∩ Uγ . u1 (p) = 1. we have only a single nontrivial overlap. . fr (x) ) is a frame of the fiber Ex .41. Thus any vector bundle over a compact manifold is a pullback of some tautological bundle! The notion of vector bundle is trickier than it may look. Definition 2. Its definition may suggest that a vector bundle looks like a direct product manif old × vector space since this happens r at least locally. We have a map F : X → Grk (CN ) defined by x → Sx (a) Prove that F is smooth. and the collection of smooth maps ( sα ). pk ([z0 . . . .44. ur which are pointwise linearly independent. . We can also regard the vectors ei as constant maps M → Kr . Again.e. Assume E is ample. then the pullback f ∗ s is defined by the open cover f −1 (Uα ). sN (x)} spans Ex . i. .43.e. Exercise 2. the collection {s1 (x). .. . A bundle isomorr phism E → KM is called a trivialization of E . . . Let us explain why we refer to a bundle isomorphism ϕ : Kr M → E as a framing. . z1 ]) = [z0 .36.1. while an isomorphism Kr → E is called a framing of E . Let E → X be a rank k (complex) smooth vector bundle over the manifold X . NATURAL CONSTRUCTIONS ON MANIFOLDS One can check easily that the isomorphism class of the pullback of a vector bundle E is independent of the choice of gluing cocycle describing E . Show that any vector bundle over a smooth compact manifold is ample.1. 1 ∼ Show that p∗ k Ln = Lkn . . A rank r vector bundle E → M (over the field K = R. For each x ∈ X we set Sx := v ∈ CN . The isomorphism ϕ determines sections fi = ϕ(ei ) of E with the property that for every x ∈ M the collection (f1 (x). (b) Prove that E is isomorphic with the pullback F ∗ Uk. sN of smooth sections of E such that.40.1. π . or framed bundle. This observation shows that we can regard any framing of a bundle E → M of rank r as a collection of r sections u1 . .42. Note that dim Sx = N − k . there exists a finite family s1 .36 CHAPTER 2. . A pair (trivial vector bundle.N . z1 ]. i ⊥. . . if s ∈ Γ(E ) is defined by the open cover (Uα ). . er ) the canonical basis of Kr . The pullback operation defines a linear map between the space of sections of E and the space of sections of f ∗ E . i. Denote by (e1 . Exercise 2. C) is called trivial or trivializable if there exists a bundle isomorphism E ∼ = Kr M .1. for any x ∈ X . .1. . For every positive integer k consider the map k k pk : CP1 → CP1 . trivialization) is called a trivialized. there is no difficulty to check the above definition is independent of the various choices. as (special) sections of Kr M . v i si (x) = 0 . and the smooth maps ( sα ◦ f ). More precisely.1. where Ln is the complex line bundle Ln → CP defined in Example 2. Exercise 2. . We will denote by Kn M the bundle K × M → M . Remark 2. THE TANGENT BUNDLE 37 One can naively ask the following question.. (g . To see this let n = dim G. Set Ei (g ) = DRg (ei ) ∈ Tg G. (Using the Cayley numbers one can show that T S 7 is also trivial. θ) ∈ Tθ S 1 ∂θ which is easily seen to be a bundle isomorphism. Then ∂ is a globally defined. Moreover. for every g ∈ G. We thus get a map ∂θ RS 1 → T S 1 .1. coincides with the subspace ∂ Re z = 0. We denote by Rg the right translation (by g ) in the group defined by Rg : x → x · g. (s. In particular T S 3 is trivial since the sphere S 3 is a Lie group (unit quaternions). the collection {E1 (g ). h) → g · h is a smooth map we deduce that the vectors Ei (g ) define smooth vector fields over G. θ = 0. X ) → (g . En (g )} is a basis of Tg G so we can define without ambiguity a map 1 n Φ : Rn G → T G. One checks immediately that Φ is a vector bundle isomorphism and this proves the claim. Clearly Rθ is a diffeomorphism. see [83] for details). Example 2. Moreover. Think of S 1 as a Lie group (the group of complex numbers of norm 1).2. . . . X i Ei (g )).46. Is every vector bundle trivial? We can even limit our search to tangent bundles. One has a similar freedom inside a Lie group as we are going to see in the next example. T S 1 ∼ = RS 1 Let θ denote the angular coordinate on the circle. and for each θ we have a linear isomorphism Dθ |θ=0 Rθ : T1 S 1 → Tθ S 1 . Example 2. . Since the multiplication G × G → G. . . The tangent space at z = 1.1. and ∂θ |1 is the unit vertical vector j . X . en a basis of the tangent space at the origin. θ) → (s ∂ .45. . · · · . ∀x ∈ G. and consider e1 . For any Lie group G the tangent bundle T G is trivial. Thus we ask the following question.e. . i. nowhere vanishing vector field on S 1 .1. ∂θ ∂ The existence of the trivializing vector field ∂θ is due to to our ability to “move freely and 1 coherently” inside S . . ∂ = Dθ |θ=0 Rθ j . i = 1. Is it true that for every smooth manifold M the tangent bundle T M is trivial (as a vector bundle)? Let us look at some positive examples. Denote by Rθ the counterclockwise rotation by an angle θ. T G. . n. . . Let us carefully analyze this example. (g. Rg is a diffeomorphism with inverse Rg−1 so that the differential DRg defines a linear isomorphism DRg : T G → Tg G. but we will be able to solve “half” of this problem. NATURAL CONSTRUCTIONS ON MANIFOLDS We see that the tangent bundle T M of a manifold M is trivial if and only if there exist vector fields X1 . first proved by J.1.F. . . Unfortunately.3 and 7. . This suggests the following more refined question. This is perhaps the least glamorous part of geometry. Obviously 0 ≤ v (M ) ≤ dim M and T M is trivial if and only if v (M ) = dim M . In the second part of this book we will describe more efficient ways of measuring the extent of nontriviality of a vector bundle. For a nice presentation we refer to [69]. 2. Xm (m = dim M ) such that for each p ∈ M . . A special instance of this problem is the celebrated vector field problem: compute v (S n ) for any n ≥ 1.47. but we will present them in a form suitable for applications in differential geometry. v (S 2k−1 ) ≥ 1 for any k ≥ 1. This is a highly nontrivial result. Problem Given a manifold M .Adams in [2] using very sophisticated algebraic tools. Xm (p) span Tp M . the maximum number of pointwise linearly independent vector fields over M . these are the only cases when the above equality holds. its computation is highly nontrivial. compute v (M ). . We have seen that v (S n ) = n for n =1. In odd dimensions the situation is far more elaborate (a complete answer can be found in [2]). The quantity v (M ) can be viewed as a measure of nontriviality of a tangent bundle.2 A linear algebra interlude We collect in this section some classical notions of linear algebra. this shows that T S 2n is not trivial.38 CHAPTER 2. Most of them might be familiar to the reader. More precisely we will show that v (S n ) = 0 if and only if n is an even number. . Amazingly. X1 (p). Exercise 2. and unfortunately cannot be avoided. The methods we will develop in this book will not suffice to compute v (S n ) for any n. This fact is related to many other natural questions in algebra. . In particular. . . y ) → 0 if (x. §2. The space T (E.f )∈E ×F K. unless otherwise stated. F ) can be identified with the space of functions c : E × F → K with finite support.2. y ) = (e. f ) . Consider the (infinite) direct sum T (E. Equivalently. f ) δe. F ) has a natural basis consisting of “Dirac functions” 1 if (x.1 Tensor products Let E . All the vector spaces in this section will tacitly be assumed finite dimensional. Convention. y ) = (e. C).f : E × F → K. F be two vector spaces over the field K (K = R. (x. the vector space T (E. F ) = (e. f. The above construction can be iterated. (e. Note that if (ei ) is a basis of E . then (ei ⊗ fj ) is a basis of E ⊗ F . Now define E ⊗K F := T (E.f ). where e.f . Using the universality property of the tensor product prove that there exists a natural isomorphism E ⊗ F ∼ = F ⊗ E uniquely defined by e ⊗ f → f ⊗ e.2. We get a natural map ι : E × F → E ⊗ F. f ∈ F . The tensor product has the following universality property. δe. when the field of scalars is clear from the context. F ). Exercise 2. F ). F ) spanned by λδe. Exercise 2. Proposition 2. e × f → e ⊗ f. F ) → E ⊗ F . Prove there exists a natural isomorphism of K-vector spaces (E1 ⊗ E2 ) ⊗ E3 ∼ = E1 ⊗ (E2 ⊗ E3 ). we will use the simpler notation E ⊗ F . F )/R(E.f − δλe.2. E3 over the same field of scalars K we can construct two triple tensor products: (E1 ⊗ E2 ) ⊗ E3 and E1 ⊗ (E2 ⊗ E3 ).3.f − δe. Set e ⊗ f := π (δe.2.λf . and λ ∈ K. The vector space E ⊗K F is called the tensor product of E and F over K. and (fj ) is a basis of F .1. λδe.f +f − δe. E2 . Given three vector spaces E1 .f − δe.f − δe .2. and therefore dimK E ⊗K F = (dimK E ) · (dimK F ). E×F ι  M E ⊗ F  Φ  K  φ . Inside T (E.2. e ∈ E . and denote by π the canonical projection π : T (E. δe+e .f . we have an injection1 δ : E × F → T (E.2.f . F ) sits the linear subspace R(E.f − δe. A LINEAR ALGEBRA INTERLUDE In particular. Often. 1 A word of caution : δ is not linear! . For any bilinear map φ : E × F → G there exists a unique linear map Φ : E ⊗ F → G such that the diagram below is commutative. G The proof of this result is left to the reader as an exercise.f . f ) → δe. 39 Obviously ι is bilinear. B ∈ Hom(E. Fi . f ∗ ∈ F ∗ . Then they naturally induce a linear operator T = T1 ⊗ T2 : E1 ⊗ E2 → F1 ⊗ F2 . ∀e ∈ E. K) of K-linear maps E → K. A vector v ∈ V admits a decomposition n v= i=1 v i ei . Ae . e := e∗ (e). Suppose Ei . ej = δ j = 1 i=j 0 i = j. More precisely. (a) For any two vector spaces U. we can now define multiple tensor products: E1 ⊗ · · · ⊗ En . Remark 2. S1 ⊗ S2 : F1 ⊗ F2 uniquely defined by T1 ⊗ T2 (e1 ⊗ e2 ) = (T1 e1 ) ⊗ (T2 e2 ). and satisfying (S1 ⊗ S2 ) ◦ (T1 ⊗ T2 ) = (S1 ◦ T1 ) ⊗ (S2 ◦ T2 ). i ) is called the We say that the basis (ei ) is dual to the basis (ej ). Proposition 2.2. The quantity (δj Kronecker symbol. e = f ∗ . The operator A† is called the transpose or adjoint of A. Gi ). For any e∗ ∈ E ∗ and e ∈ E we set e∗ . i = 1. 2 are K-vector spaces. Definition 2. V ). Clearly.6. (A ◦ B )† = B † ◦ A† .2. Fi ). Moreover. i = 1.7. Any basis (ei )1≤i≤n of the n-dimensional K-vector space determines a basis (ei )1≤i≤n of the dual vector space V ∗ uniquely defined by the conditions i ei . G). . NATURAL CONSTRUCTIONS ON MANIFOLDS The above exercise implies that there exists a unique (up to isomorphism) triple tensor product which we denote by E1 ⊗ E2 ⊗ E3 . V over the field K we denote by Hom(U.4.2. F ). ∀A ∈ Hom(F. (b) The dual of a K-linear space E is the linear space E ∗ defined as the space HomK (E.40 CHAPTER 2. or HomK (U. ∀ei ∈ Ei . (b) Any linear operator A : E → F induces a linear operator A† : F ∗ → E ∗ uniquely defined by A† f ∗ . Prove the above proposition. Gi . we have the following result. Let Ti ∈ Hom (Ei . V ) the space of K-linear maps U → V . be two linear operators. Si ∈ Hom(Fi . Exercise 2.5. 2.2. The above constructions are functorial. . is an isomorphism. . y . . ∀x ∈ E. This adjunction formula is known in Bourbaki circles as “formule d’adjonction chˆ er ` a Cartan ”. (b) The adjunction morphism E ∗ ⊗ F → Hom (E. x f ∗ . . Moreover. F ). this shows E ∗ ⊗ F ∗ can be naturally identified with the space of bilinear maps E × F → K. .  v= . . given by E∗ ⊗ F where Te∗ ⊗f (x) := e∗ . vn while a vector v ∗ is represented by a one-row matrix ∗ ∗ v ∗ = [v 1 . y ∈ F. vn ]· . uniquely defined by E∗ ⊗ F ∗ where Le∗ ⊗f ∗ . Using the functoriality of the tensor product and of the dualization construction one proves easily the following result. v = [v 1  v1  ∗  . . F ). e∗ ⊗ f ∗ −→ Le∗ ⊗f ∗ ∈ (E ⊗ F )∗ . vn ]. x ⊗ y = e∗ . x f. ∀x ∈ E.2. v = n ∗ i vi v.2 2 The finite dimensionality of E is absolutely necessary. In particular. A LINEAR ALGEBRA INTERLUDE while a vector v ∗ ∈ V ∗ admits a decomposition n 41 v∗ = i=1 ∗ i vi e. . e∗ ⊗ f −→ Te∗ ⊗f ∈ Hom(E. Then ∗ v ∗ .2. vn  where the · denotes the multiplication of matrices. . Proposition 2.8.2. . i=1 Classically. v∗. a vector v in V is represented by a one-column matrix  1  v  . (a) There exists a natural isomorphism E∗ ⊗ F ∗ ∼ = (E ⊗ F )∗ . 9.s). An element of Ts Example 2. 1) can be identified with a linear endomorphism of V . jβ ≤ dim V }. k ) can be identified with a k -linear map V × · · · × V → K. Example 2. and Einstein’s convention. ⊗ is an associative algebra.2. ⊗e .8.. 0 if i = j {ei1 ⊗ · · · ⊗ eir ⊗ ej1 ⊗ · · · ⊗ ejs / 1 ≤ iα . To achieve this pick a basis (ei ) of V .. The elements of T (V ) are called tensors. 1 T1 (V ) ∼ = End (V ). Using the bases (ei ) and (ej ). e j = δ j = r (V ) We then obtain a basis of Ts 1 if i = j . r is called tensor of type (r. 0) is called contravariant.e. For r. while a tensor of type (0.8 a tensor of type (1. s ≥ 0 set r Ts (V ) := V ⊗r ⊗ (V ∗ )⊗s . The tensor algebra of V is defined to be T (V ) := r. the adjunction identification can be described as the correspondence which j 1 associates to the tensor A = ai j ei ⊗ e ∈ T1 (V ). Prove the above proposition.. We use the term algebra since the tensor product induces bilinear maps r +r r r ⊗ : Ts × Ts → Ts +s .11. and the second time as subscript.12. +. and let (ei ) denote the dual basis in V ∗ uniquely defined by i ei .10. while a tensor of type (0.. r (V ) has a decomposition Any element T ∈ Ts j1 js 1 .2.2. 1 (V ) Using the adjunction morphism in Proposition 2. .. once as a superscript. Exercise 2.s r Ts (V ). we can identify the space T1 with the space End(V ) a linear isomorphisms. k A tensor of type (r. s) is called covariant. Let V be a vector space. i. NATURAL CONSTRUCTIONS ON MANIFOLDS Exercise 2.ir T = Tji1 .42 CHAPTER 2.2.2. According to Proposition 2.js ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · . Show that T (V ). where by definition V ⊗0 = (V ∗ )⊗0 = K. where we use Einstein convention to sum over indices which appear twice. the linear operator LA : V → V which j j maps the vector v = v ej to the vector LA v = ai j v ei .2. It is often useful to represent tensors using coordinates. 2.. We denote this action of σ ∈ Sr on an arbitrary element t ∈ T r (V ) by σt. (We refer to [34] for more general constructions. σ Ar (t) = Ar (σt) = (σ )Ar (t). v1 v2 ⊗ · · · vr ⊗ u2 ⊗ · · · ⊗ us . . Moreover. A LINEAR ALGEBRA INTERLUDE On the tensor algebra there is a natural contraction (or trace ) operation r −1 r tr : Ts → Ts −1 43 uniquely defined by tr (v1 ⊗ · · · ⊗ vr ⊗ u1 ⊗ · · · ⊗ us ) := u1 . These are special instances of the so called Schur functors.ir 2 . We set T r (V ) := T0 r objects. ∀t ∈ T r (V ).2.13. Ar (t) := 1 r! σt.. ∀v1 . Their proofs are left to the reader as exercises. . and we denote by S the group of permutations of r K = R..js = Tij2 . i. coincides with the usual trace on T1 = §2.e. σ ∈ Sr 1 r! σ ∈Sr 0 (σ )σt if r ≤ dim V . we denoted by (σ ) the signature of the permutation σ . C. if r > dim V Above. When r = 0 we set S0 := {1}. vr ∈ V.js .2.2. The following results are immediate. In this subsection we will describe two subspaces invariant under this action.. Every permutation σ ∈ Sr determines a linear map T r (V ) → T r (V ). The operators Ar and S r are projectors of T r (V ). 2 S2 r = S r . we see that the contraction 1 (V ) ∼ End (V ). .... In the coordinates determined by a basis (ei ) of V . uj ∈ V ∗ ... the contraction can be described as ii2 . uniquely determined by the correspondences v1 ⊗ · · · ⊗ vr −→ vσ(1) ⊗ · · · ⊗ vσ(r) . Ar = Ar . σ S r (t) = S r (σt) = S r (t). . . In particular.2 Symmetric and skew-symmetric tensors Let V be a vector space over r (V ). ∀vi ∈ V. S r (t) := and Ar : T r (V ) → T r (V ). Lemma 2. where again we use Einstein’s convention.ir (tr T )i j2 . Note that A0 = S 0 = K .) Define S r : T r (V ) → T r (V ). We will complete the proof of the proposition in two steps. A tensor T ∈ T r (V ) is called symmetric (respectively skew-symmetric ) if S r (T ) = T (respectively Ar (T ) = T ). and Λ• V := r≥0 Λr V. To prove the associativity of ∧1 consider the quotient algebra Q∗ = T ∗ /I∗ ... ⊗ vk ) = σ ∈Sk defined by (σ )vσ(1) ⊗ . and moreover π (α∧1 β ) = π (α) ∪ π (β ). (a) (Associativity) (α ∧ β ) ∧ γ = α ∧ (β ∧ γ ). (b) (Super-commutativity) ω r ∧ η s = (−1)rs η s ∧ ω r . where T ∗ is the associative algebra ( r≥0 T r (V ).2. The exterior product is the bilinear map ∧ : Λr V × Λs V → Λr+s V. and I∗ is the bilateral ideal generated by the set of squares {v ⊗ v/ v ∈ V }. η s ∈ Λs V. ∀ω r ∈ Λr V. The nonnegative integer r is called the degree of the (skew-)symmetric tensor. ∀t ∈ T r (V ). β γ ∈ Λ• V. ∧1 is an associative product. Definition 2. ω r ∧ η s := In particular.1) . ∀vi ∈ V.2.15.2) (2. ⊗). In particular. We first define a new product “∧1 ” by ω r ∧1 η s := Ar+s (ω ⊗ η ).. v1 ∧ · · · ∧ vk = k !Ak (v1 ⊗ . ω s ∈ Λs V. Step 1. (r + s)! Ar+s (ω ⊗ η ). We will prove that the map π : Λ• V → Q∗ is a linear isomorphism. The space of symmetric tensors (respectively skew-symmetric ones) of degree r will be denoted by S r V (and respectively Λr V ). NATURAL CONSTRUCTIONS ON MANIFOLDS Definition 2. The exterior product has the following properties.16.. The crucial observation is π (T ) = π (Ar (T )).2. r!s! Proposition 2. +.44 CHAPTER 2. Proof.2. ⊗ vσ(k) . Denote the (obviously associative) multiplication in Q∗ by ∪. Set S • V := r ≥0 S r V.2. The natural projection π : T ∗ → Q∗ induces a linear map π : Λ• V → Q∗ . ∀ω r ∈ Λr V. ∀α. which will turn out to be associative and will force ∧ to be associative as well.14. (2. Since (u + v )⊗2 ∈ I∗ . β = As (f1 ⊗ · · · ⊗ fs )..3) we obtain (2. any π (T ) can be alternatively described as π (ω ) for some ω ∈ Λ• V .2. then ω ∈ ker π = I∗ ∩ Λ• V so that ω = A∗ (ω ) = 0. If π (ω ) = 0 for some ω ∈ Λ• V . Consider α ∈ Λr V . The details are left to the reader.2).e. r!s!t! r!s!t! The associativity of ∧ is proved.2. Thus ∧1 is associative. β ∈ Λs V and γ ∈ Λt V . We have π (α∧1 β ) = π (Ar+s (Ar (e1 ⊗ · · · ⊗ er ) ⊗ As (f1 ⊗ · · · ⊗ fs ))) (2. Exercise 2.2) on monomials T = e1 ⊗ · · · ⊗ er .2. A LINEAR ALGEBRA INTERLUDE It suffices to check (2. Finish the proof of part (b) in the above proposition. i. . s. Step 2.2). The surjectivity of π follows immediately from (2. (Λr V ) ∧ (Λs V ) ⊂ Λr+s V.2.2. To prove (2. Note that Λr V = 0 for r > dim V (pigeonhole principle).2. Hence. for any σ ∈ Sr π (e1 ⊗ · · · ⊗ er ) = (σ )π (eσ(1) ⊗ · · · ⊗ eσ(r) ) 45 (2. The product ∧ is associative.2.2.2. The computation above shows that = e1 ∧ · · · ∧ ek = k !Ak (e1 ⊗ · · · ⊗ ek ).3) When we sum over σ ∈ Sr in (2.2). v ∈ V we deduce u ⊗ v = −v ⊗ u(mod I∗ ). We have (r + s)! (r + s + t)! (r + s)! α∧1 β ∧ γ = (α∧1 β )∧1 γ (α ∧ β ) ∧ γ = r!s! r!s! (r + s)!t! (r + s + t)! (r + s + t)! (α∧1 β )∧1 γ = α∧1 (β ∧1 γ ) = α ∧ (β ∧ γ ).2. The exterior algebra is a Z-graded algebra. ∀r. The space Λ• V is called the exterior algebra of V .17. To prove the injectivity of π note first that A∗ (I∗ ) = 0. It suffices to take ω = A∗ (T ). (b) The supercommutativity of ∧ follows from the supercommutativity of ∧1 (or ∪).2) def = π (Ar (e1 ⊗ · · · ⊗ er ) ⊗ As (f1 ⊗ · · · ⊗ fs )) = π (Ar (e1 ⊗ · · · ⊗ er )) ∪ π (As (f1 ⊗ · · · ⊗ fs )) = π (α) ∪ π (β ). ∀u.1) it suffices to consider only the special cases when α and β are monomials: α = Ar (e1 ⊗ · · · ⊗ er ). Indeed. ∧ is called the exterior product . ei ∈ V .2. To prove the latter one uses (2. 2. Let V be a vector space over K.18. The symmetric product “·” is also commutative. en is a basis of the vector space V then. It is often convenient to represent (skew-)symmetric tensors in coordinates. Let V be an n-dimensional K-vector space.. there exists an unique morphism of Kalgebras Φ : Λ• V → A such that the diagram below is commutative V O ιV M Λ• V A φ K Φ . The one dimensional vector space Λn V is called the determinant line of V .19 for the algebra S• V . Any linear map L : V → W induces a natural morphisms of algebras Λ• L : Λ• V → Λ• W. Proposition 2. The Λ• and the S • constructions are functorial in the following sense. Exercise 2. .21. i. β ∈ S s V. For any K-algebra A.. Prove Proposition 2. Formulate and prove the analogue of Proposition 2.2. ∀α ∈ S r V. ιV (v ) = v .22.19.2.2. ∀v ∈ V.19.2. S • L : S • V → S • W . and it is denoted by det V . There exists a natural injection ιV : V → Λ• V .2. NATURAL CONSTRUCTIONS ON MANIFOLDS Definition 2. This map enters crucially into the formulation of the following universality property. Exercise 2. Proposition 2. If e1 . and any linear map φ : V → A such that (φ(x)2 = 0.46 CHAPTER 2.2. for any 1 ≤ r ≤ n. Φ ◦ ιV = φ. such that ιV (v ) ∧ ιV (v ) = 0.. . the family {ei1 ∧ · · · ∧ eir / 1 ≤ i1 < · · · < ir ≤ n} is a basis for Λr V so that any degree r skew-symmetric tensor ω can be uniquely represented as ω= ω i1 ···ir ei1 ∧ · · · ∧ eir . 1≤i1 <···<ir ≤n Symmetric tensors can be represented in a similar way. The space of symmetric tensors S • V can be similarly given a structure of associative algebra with respect to the product α · β := S r+s (α ⊗ β ).e.20. 24. (a) Prove that if A is diagonalizable and its eigenvalues are a1 . . if n = dimK V . an . . the endomorphism Λn L can be identified with a scalar det L. − d log det( − zA) = dz ψr (A)z r−1 . . Suppose V is a complex n-dimensional vector space.23. and by ψr (A) the trace of the endomorphism S r (A) : S r V → S r V.2.2. . . ψr (A) = ar 1 + · · · + an . .2. . the determinant of the endomorphism L. and A is an endomorphism of V . then any linear endomorphism L : V → V defines an endomorphism Λn L : det V = Λn V → det V. if U −→ V −→ W are two linear maps. vr ) = (Lv1 ) · · · (Lvr ). Suppose V is a finite dimensional vector space and A : V → V is an endomorphism of V . Exercise 2. Moreover. We define σ0 (A) = 1. ψ0 (A) = dim V. Exercise 2. Prove the above proposition.2. then σr (A) = 1≤i1 <···<ir ≤n r ai1 · · · air . (b) Prove that for every sufficiently small z ∈ C we have the equalities det(V + zA) = j ≥0 σr (A)z r . r ≥1 . S r (BA) = (S r B )(S r A). . A B 47 In particular. A LINEAR ALGEBRA INTERLUDE uniquely defined by their actions on monomials Λ• L(v1 ∧ · · · ∧ vr ) = (Lv1 ) ∧ · · · ∧ (Lvr ). For every positive integer r we denote by σr (A) the trace of the induced endomorphism Λr A : Λr V → Λr V. and S • L(v1 . Definition 2. Since the vector space det V is 1-dimensional.25.2. then Λr (BA) = (Λr B )(Λr A). 2. The universality property of the tensor product implies the existence of a linear map Φ : Λ• V ⊗ Λ• W → Λ• (V ⊕ W ).2.2. uniquely determined by φ(v1 ∧ · · · ∧ vr . note that Λ• V ⊗ Λ• W is naturally a K-algebra by (ω ⊗ η ) ∗ (ω ⊗ η ) = (−1)deg η·deg ω (ω ∧ ω ) ⊗ (η ∧ η ). We want to mention a few general facts about Z-graded vector spaces.e. S • (V ⊕ W ) ∼ = S • V ⊗ S • W.4) = Λ• V ⊗ Λ• W. w) → ψ (v. The direct sum of two Z-graded vector spaces V and W is a Z-graded vector space with (V ⊕ W )n := Vn ⊕ Wn . . such that Φ ◦ ι = φ. To define the isomorphism in (2. NATURAL CONSTRUCTIONS ON MANIFOLDS The functors Λ• and S • have an exponential like behavior.48 CHAPTER 2. The uniqueness part in the universality property of the exterior algebra implies Φ ◦ Ψ = identity . w1 ∧ · · · ∧ ws ) = v1 ∧ · · · ∧ vr ∧ w1 ∧ · · · ∧ ws .. where ι is the inclusion of Λ• V × Λ• W in Λ• V ⊗ Λ• W . To construct the inverse of Φ. there exists a natural isomorphism Λ• (V ⊕ W ) ∼ (2. i. Note that Φ is also a morphism of K-algebras. For example. for any x ∈ V ⊕ W we have ψ (x) ∗ ψ (x) = 0. The spaces Λ• V and S • V are Z-graded vector spaces. (We will always assume that each Vn is finite dimensional.) The vectors in Vn are said to be homogeneous. the ring of polynomials K[x] is a K-graded vector space.2. One proves similarly that Ψ ◦ Φ = identity ..4). The universality property of the exterior algebra implies the existence of a unique morphism of K-algebras Ψ : Λ• (V ⊕ W ) → Λ• V ⊗ Λ• W. and one verifies easily that (Φ ◦ Ψ) ◦ ιV ⊕W = ιV ⊕W . vector spaces equipped with a direct sum decomposition V = n∈Z (2. w) = v ⊗ 1 + 1 ⊗ w ∈ Λ• V ⊗ Λ• W. The vector space V ⊕ W is naturally embedded in Λ• V ⊗ Λ• W via the map given by (v. Moreover. i. such that Ψ ◦ ιV ⊕W = ψ .4) consider the bilinear map φ : Λ• V × Λ• W → Λ• (V ⊕ W ).5) Vn .e. of degree n. and this concludes the proof of (2. The Poincar´ e series of K[x] is PK[x] (t) = 1 + t + t2 + · · · + tn−1 + · · · = 1 .30. if V has dimension n. while the second formula makes no sense (divergent series). then V ∼ = Kn so that PΛ• V (t) = (PΛ• K (t))n and PS • V (t) = (PS • K (t))n . and PS • K (t) = PK[x] (t) = 1 .28 we get χ(K[x]) = 1/2. 1−t Exercise 2. Let V be a K-vector space of dimension n.2.2. In particular.28. Prove the following statements are true (whenever they make sense). Example 2. 1−t . t−1 ]] by (dimK Vn )tn .27. Definition 2. The proposition follows using the equalities PΛ• K (t) = 1 + t. Let V be a Z-graded vector space.2.2. Remark 2.2. If we try to compute χ(K[x]) using the first formula in Definition 2.2. r+s=n To any Z-graded vector space V one can naturally associate a formal series PV (t) ∈ Z[[t. Proof. denoted by χ(V ).2.4) and (2. is defined by χ(V ) := PV (−1) = (−1)n dim Vn . From (2.2.2. (c) dim V = PV (1). PV (t) := n∈Z The series PV (t) is called the Poincar´ e series of V . Then PΛ• V (t) = (1 + t)n and PS • V (t) = 1 1−t n = 1 (n − 1)! d dt n−1 1 1−t .26. n∈Z whenever the sum on the right-hand side makes sense.2. The Euler characteristic of V. Proposition 2. (b) PV ⊗W (t) = PV (t) · PW (t).27 that for any vector spaces V and W we have PΛ• (V ⊕W ) (t) = PΛ• V (t) · PΛ• W (t) and PS • (V ⊕W ) (t) = PS • V (t) · PS • W (t).29.2. In particular. dim Λ• V = 2n and χ(Λ• V ) = 0. A LINEAR ALGEBRA INTERLUDE 49 The tensor product of two Z-graded vector spaces V and W is a Z-graded vector space with (V ⊗ W )n := Vr ⊗ Ws .5) we deduce using Exercise 2. Let V and W be two Z-graded vector spaces. (a) PV ⊕W (t) = PV (t) + PW (t). i.50 CHAPTER 2.33.e. Let V be a finite dimensional space. (b) A s-algebra over K is a Z2 -graded K-algebra.2.. •]s ≡ 0. They have parity (or chirality).. For any a ∈ Ai we denote its degree (mod 2) by |a|. a K-algebra A together with a direct sum decomposition A = A0 ⊕ A1 such that Ai · Aj ⊂ Ai+j (mod 2) . The elements in A0 are said to be even while the elements in A1 are said to be odd. E j ).2. and objects with different chiralities had to be treated differently. η j ∈ Aj by [ω i . The even endomorphisms have the form T00 0 0 T11 while the odd endomorphisms have the form 0 T01 T10 0 . We owe the “super” slang to the physicists.32. Example 2. NATURAL CONSTRUCTIONS ON MANIFOLDS §2. (a) A s-space is a Z2 -graded vector space. η j ]s := ω i η j − (−1)ij η j ω j . i. The elements in Ai are called homogeneous of degree i. In the quantum world many objects have a special feature not present in the Newtonian world. T10 T11 where Tji ∈ End (E i . Any linear endomorphism T ∈ End (E ) has a block decomposition T00 T01 T = . The exterior algebra Λ• V is naturally a s-algebra. a vector space V equipped with a direct sum decomposition V = V0 ⊕ V1 . . Example 2. the “super” slang adds the attribute super to most of the commonly used algebraic objects. From a strictly syntactic point of view.2. the prefix “s-” will abbreviate the word “super” Definition 2. . if the suppercommutator is trivial.31.e. [•. The even elements are gathered in Λeven V = r even Λr V. defined on homogeneous elements ω i ∈ Ai . We can use this block decomposition to describe a structure of s-algebra on End (E ).2. The “super” terminology provides an algebraic formalism which allows one to deal with the different parities on an equal basis. (c) The supercommutator in a s-algebra A = A0 ⊕ A1 is the bilinear map [•.3 The “super” slang The aim of this very brief section is to introduce the reader to the “super” terminology. . In this book. An s-algebra is called s-commutative. •]s : A × A → A. Let E = E 0 ⊕ E 1 be a s-space. [ . Often.. Any u∗ ∈ V ∗ defines an odd s-derivation of Λ• V denoted by iu∗ uniquely determined by its action on monomials.38. a hat indicates a missing entry.. for any x ∈ A.36.2.2.e. Exercise 2. y ∈ A.2. i. .2.34.32). ∧ v ˆi ∧ · · · ∧ vr . The multiplication in a (s-) Lie algebra is called the (s-)bracket.2. Example 2. Indeed. Definition 2. then A is called simply a Lie algebra.2. [x. y ∈ A. ]) be an s-algebra over K.2. for any homogeneous elements x.2. Definition 2. Rx is a s-derivation. For any x ∈ A we denote by Rx the right multiplication operator. Remark 2. The s-algebra Λ• V is s-commutative. not necessarily associative. assuming D is homogeneous (as an element of the s-algebra End (A)) then equality (2. Lx ]End( = LDx .2. Prove the statement in the above example.2. while for any z ∈ A we denoted by Lz the left multiplication operator a → z · a. A LINEAR ALGEBRA INTERLUDE while the odd elements are gathered in Λodd V = r odd 51 Λr V.35. r iu∗ (v0 ∧ v1 ∧ · · · ∧ vr ) = i=0 (−1)i u∗ . i. for all homogeneous elements x. if it is even (respectively odd ) as an element of the s-algebra End (A). a → [a. Let A = A0 ⊕ A1 be a s-algebra.2.e. A is called an s-Lie algebra if it is s-anticommutative. Let A = (A0 ⊕ A1 . ]s denotes the supercommutator in End (A) (with the s-structure defined in Example 2. The derivation iu∗ is called the interior derivation by u∗ or the contraction by u∗ . The relation (2. y ] + (−1)|x||y| [y. Let V be a vector space... An s-derivation is called even (respectively odd ). x] = 0. When A is purely even. one uses the alternate notation u∗ to denote this derivation. and ∀x ∈ A. A) [D. An s-derivation on A is a linear operator on D ∈ End (A) such that. A1 = {0}.37. As usual.6) where [ . +. x].6) is a super version of the usual Leibniz formula. s End(A) (2.6) becomes D(xy ) = (Dx)y + (−1)|x||D| x(Dy ). vi v0 ∧ v1 ∧ . y ∈ A. z ∈ A. z ∈ A.y]s . (2.2.39. x (c) ∀x ∈ A the bracket B : a → [a.43. 1.7) for all homogeneous elements x. x].8) Example 2. (a) For any D ∈ Ders (A). y ] = 0. Let E = E 0 ⊕ E 1 and F = F 0 ⊕ F 1 be two s-spaces. Notice in particular that any s-derivation of Λ• V is uniquely determined by its action on ∗ Λ1 V . D1 ∈ End (A) are also s-derivations. z ].2.2. (E ⊗ F ) := i+j ≡ (2) Ei ⊗ F j . (2.52 CHAPTER 2. and denote by Ders (A) the vector space of s-derivations of A. the s-commutator [D.2. Prove Proposition 2. Let E be a vector space (purely even). Moreover (A) [B x . B y ]End = B [x. (When ω = 1. ∀x. z ].40. Definition 2. x] + [[z.7). the fact that Rx is a s-derivation for all x ∈ A is equivalent with the super Jacobi identity [[y. uniquely determined by v ∗ × ω → Dv Dv ∗ ⊗ω ∗ ⊗ω . z ] + [[y.42. [[x. = 0 . where Dv ∗ ⊗ω is s-derivation defined by (v ) = v ∗ . which in this case is equivalent with the classical Jacobi identity. b] = ab − ba. Show that there exists a natural isomorphism of s-spaces V ∗ ⊗Λ• V ∼ = Ders (Λ• V ). Then A = End (E ) is a Lie algebra with bracket given by the usual commutator: [a. Let A = A0 ⊕ A1 be a s-algebra. [z.2. y. s Exercise 2. Proposition 2. Dv ⊗1 coincides with the internal derivation discussed in Example 2.2.40. then A is a Lie algebra over K if [ . Exercise 2. D ]s is again an s-derivation. y ]. v ω. To emphasize the super-nature of the tensor product we will use the symbol “⊗” instead of the usual “⊗”. ∀v ∈ V. When A is a purely even K-algebra. In down-to-earth terms.2. ∀x. End(A) (b) For any D. its homogeneous components D0 .) . x] = [[y.36. x]]. NATURAL CONSTRUCTIONS ON MANIFOLDS The above definition is highly condensed. y. x]s is a s-derivation called the bracket derivation determined by x. x].2.2. Their s-tensor product is the s-space E ⊗ F with the Z2 -grading. ] is anticommutative and satisfies (2. D ∈ Ders (A).2.41. z ] + (−1)|x||y| [y. This pairing is called the natural duality between a vector space and its dual. v → B (v. Let E = E0 ⊕ E1 be a finite dimensional s-space. y ]s . Observe that IBL = L. §2. Conversely. Definition 2. The natural pairing •. Then there exists a canonical s-trace trs on A uniquely defined by trs E = dim E0 − dim E1 .45. • ) ∈ W ∗ . A pairing B is called a duality if the adjunction map IB is an isomorphisms. Any pairing B : V × W → K defines a linear map IB : V → W ∗ . all vector spaces will be tacitly assumed finite dimensional. and we will use Einstein’s convention without mentioning it. Proposition 2.• = V ∗ : V ∗ → V ∗ . y ∈ A.2. and denote by A the s-algebra of endomorphisms of E .46. then we see that the space of s-traces is isomorphic with the dual of the quotient space A/[A. T00 T01 T10 T11 . Example 2. B (v. x. if T ∈ A has the block decomposition T = then trs T = tr T00 − tr T11 . A LINEAR ALGEBRA INTERLUDE 53 Let A = A0 ⊕ A1 be an s-algebra over K = R.2. If we denote by [A. A]s .44. In the sequel. y ]s ) = 0 ∀x.47. w ∈ W.2. w) = (Lv )(w). This section is devoted to those aspects of the atmosphere called duality which are relevant to differential geometry. Prove the above proposition. Exercise 2. One sees that I •. A pairing between two K-vector spaces V and W is a bilinear map B : V × W → K. y ∈ A . A supertrace on A is a K-linear map τ : A → K such that. any linear map L : V → W ∗ defines a pairing BL : V × W → K.2. C. called the adjunction morphism associated to the pairing. K will denote one of the fields R or C.2. A]s the linear subspace of A spanned by the supercommutators [x. τ ([x.4 Duality Duality is a subtle and fundamental concept which permeates all branches of mathematics.2. ∀v ∈ V. • : V ∗ × V → K is a duality. In fact. .2. This defines by restriction a pairing Λr B : Λr V × Λr W → K uniquely determined by Λr B (v1 ∧ · · · ∧ vr . In this case. w1 ⊗ w2 ) = B1 (v1 . Consider V a real vector space and ω : V × V → R a skew-symmetric bilinear form on V .48. then the operator L is called metric duality. (b) If (ei ) is a basis of V . the induced operator Iω : V → V ∗ is called symplectic duality.2. w1 ) · B2 (v2 . wj ))1≤i.49. Proposition 2. and set gij := (ei .54 CHAPTER 2.2.2.50. (a) The V has even dimension.•) : V → V ∗ . The notion of duality is compatible with the functorial constructions introduced so far. and in particular a natural isomorphism L := I(•.51.2.51 implies that given two spaces in duality B : V × W → K there is a naturally induced duality B ⊗n : V ⊗r × W ⊗r → K. ej ). This operator can be nicely described in coordinates as follows. j. 2) be two pairs of spaces in duality.2. then det(ωij )1≤i. When (•. w1 ∧ · · · ∧ wr ) := det (B (vi . w2 ). Let Bi : Vi × Wi → R (i = 1. NATURAL CONSTRUCTIONS ON MANIFOLDS Example 2. •) is positive definite. Let V be a finite dimensional real vector space. Prove the following. . The form ω is said to be symplectic if this pairing is a duality.51. •) : V × V → R defines a (self)duality. Pick a basis (ei ) of V .j ≤dim V = 0.52. Suppose that V is a real vector space. Example 2. and ωij := ω (ei . Let (ej ) denote the dual basis of V ∗ defined by j ej . and ω : V × V → R is a symplectic duality. Prove Proposition 2. Then there exists a natural duality B = B1 ⊗ B2 : (V1 ⊗ V2 ) × (W1 ⊗ W2 ) → R. ej ). Exercise 2. e i = δ i .2. Exercise 2.2. ∀i. Any symmetric nondegenerate quadratic form (•. uniquely determined by IB1 ⊗B2 = IB1 ⊗ IB2 ⇐⇒ B (v1 ⊗ v2 .j ≤r . The action of L is then Lei = gij ej . Proposition 2. 9) is a Gramm determinant. operation Ts s−1 .2. the natural duality • .js . If T ∈ Ts j1 js 1 . (2. ej ). An inner product on a real vector space V naturally induces an inner product on the tensor algebra T (V ).10) is classically referred to as lowering the indices..9) is symmetric and positive definite. wj ) )1≤i.2. • ) induces a self-duality ( • .ir T = Tji1 . r (V ) → T r−1 (V ) defined by This induces an operator L : Ts s+1 L(v1 ⊗ .js ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · ⊗ e .2.. . and in the exterior algebra Λ• V . (2.ir (LT )i jj1 . we have proved the following result... then ii2 .ir 2 . In a Euclidean vector space V the inner product induces the metric duality L : V → V ∗ .j ≤r .2.10) The operation defined in (2. • ). determined by (v1 ∧ · · · ∧ vr . . ⊗ vr ⊗ u1 ⊗ · · · ⊗ us ) = (v2 ⊗ · · · ⊗ vr ) ⊗ ((Lv1 ⊗ u1 ⊗ · · · ⊗ us ). Denote its inner product by ( • . In particular.53. • : Λ r V ∗ × Λ r V → R.2. Prove the above pairing is a duality.. The self-duality defined by ( • . The inverse of the metric duality L−1 : V ∗ → V induces a linear r (V ) → T r+1 (V ) called raising the indices. A duality B : V × W → K naturally induces a duality B † : V ∗ × W ∗ → K by 1 ∗ B † (v ∗ . w1 ∧ · · · ∧ wr ) := det ( (vi .jr = gij Tj1 . A LINEAR ALGEBRA INTERLUDE Exercise 2. where IB : V → W ∗ is the adjunction isomorphism induced by the duality B .. The reason for this nomenclature comes from the coordinate description of this operr (V ) is given by ation...9) The right hand side of (2. • : V ∗ × V → K induces a duality • . and in particular. Corollary 2. • ) : Λr V × Λr V → R....54. w∗ ) : = v ∗ . 55 This shows that we can regard the elements of Λr V ∗ as skew-symmetric r-linear forms V r → K.2. I− B w .2.2.. Now consider a (real ) Euclidean vector space V . and thus defines a natural isomorphism ∗ Λr V ∗ ∼ = (Λr V ) . Thus. where gij = (ei . the bilinear form in (2.2. 2.2. (c) Suppose or ∈ Or (V ). Equivalently. Let V be a real vector space. A volume form on V is a nontrivial linear form on the determinant line of V . or ).56 CHAPTER 2. We denote by Or (V ) the set of orientations of V . Then µ1 ∼ µ2 ⇐⇒ or µ1 = or µ2 . v )Λ• V . Definition 2. Let V be an Euclidean vector space. v ∈ V [ev . (a) An orientation on a vector space V is a continuous. we can identify the set of orientations with the set of equivalence classes of nontrivial volume forms. the basis is said to be negatively oriented. a choice of a volume form corresponds to a choice of a basis of det V . see Example 2. where V is a vector space. The associated orientation or ω is uniquely characterized by the condition or ω (ω ) = 1. iu ]s = ev iu + iu ev = (u. To every orientation or we can associate an equivalence class [µ]or of volume forms such that µ(ω )or (ω ) > 0. Observe that Or (V ) consists of precisely two elements. Show that for any u. . surjective map or : det V \ {0} → {±1}. Equivalently. ∀ω ∈ det V \ {0}. We define an equivalence relation on the space of nontrivial volume forms by declaring µ1 ∼ µ2 ⇐⇒ µ1 (ω )µ2 (ω ) > 0. Since det V is 1-dimensional.36). NATURAL CONSTRUCTIONS ON MANIFOLDS Exercise 2. a volume form on V is a nontrivial element of det V ∗ (n = dim V ). Thus.2. There is an equivalent way of looking at orientations.2.56. For any v ∈ V denote by ev (resp. note that any nontrivial volume form µ on V uniquely specifies an orientation or µ given by or µ (ω ) := sign µ(ω ). (b) A pair (V. Otherwise.57. ∀ω ∈ det V \ {0}. ∀ω ∈ det V \ {0}. and or is an orientation on V is called an oriented vector space. Definition 2. iv ) the linear endomorphism of Λ∗ V defined by ev ω = v ∧ ω (resp. µ : det V → R. A basis ω of det V is said to be positively oriented if or (ω ) > 0. to specify an orientation on V it suffices to specify a basis ω of det V . iv = ıv∗ where ıv∗ denotes the interior derivation defined by v ∗ ∈ V ∗ -the metric dual of v .55 (Cartan). To describe it. 58. . we have proved the following result. . Assume now that V is an Euclidean space. g ) canonically selects a volume form on V . to define an orientation on a vector space. and in particular. which determines a volume form µg = µor g . • ). call it ω . Definition 2. An orientation or on an Euclidean vector space (V. and the orientation given by e1 ∧ e2 . or 2 on the vector space V we define or 1 /or 2 ∈ {±1} to be or 1 /or 2 := or 1 (ω )or 2 (ω ). it determines a volume form µg defined by µg (λω ) = λ.2. it suffices to specify a total ordering of a given basis of the space. and we think of it as a basis of det V .. Show that. Given two orientations or 1 . it determines an orientation on V . Conversely. Let (V. If V = R2 with its standard metric. If we fix one of them. . The vector space det V has an induced Euclidean structure.60. Proposition 2. henceforth denoted by Detg = Detor g . g ) be an n-dimensional Euclidean vector space. . A LINEAR ALGEBRA INTERLUDE 57 To any basis {e1 . Denote the Euclidean inner product by g ( • . or 2 . or ) is said to be positively oriented if so is the associated basis of det V . We will say that or 1 /or 2 is the relative signature of the pair of orientations or 1 . ∀ω ∈ det V \ {0}. .59. . An ordered basis of an oriented vector space (V. • First.2. Note that a permutation of the indices 1. there exist exactly two length-one-vectors in det V ..2. .2. and or an orientation on V . .2. . en } of V one can associate a basis e1 ∧ · · · ∧ en of det V . we have Detor g (v1 ∧ · · · ∧ vn ) = or (v1 ∧ · · · ∧ vn ) (det g (vi . Thus. • Second. vj )). Thus. then we achieve two things. an orientation or ∈ Or (V ) uniquely selects a length-one-vector ω = ωor in det V . n changes the associated basis of det V by a factor equal to the signature of the permutation.. for any basis v1 . Exercise 2. vn of V . prove that | Detor g (v1 ∧ v2 )| is the area of the parallelogram spanned by v1 and v2 . Let ω ∈ Λ2 V . NATURAL CONSTRUCTIONS ON MANIFOLDS Definition 2.64. We define its pfaffian as P f (ω ) = P f or g (ω ) := exp ω = g 0 or 1 ∧n n! Detg (ω ) if dim V is odd . g ) is as in the above definition. Euclidean space.58 CHAPTER 2.. . ∀u. k! If (V. ej )ei ∧ ej . Let (V. Let A : V → V be a skew-symmetric endomorphism of an oriented Euclidean space V . We define the pfaffian of A by P f (A) := P f (ωA ). Euclidean space and denote by Detor g the associated volume form. The reader can check that ωA is independent of the choice of basis as above. Exercise 2.61. exp ω := k≥0 ωk . i. g. where e1 e2 denotes the standard basis. v ). g defined on homogeneous elements by ω= g 0 if deg ω < dim V . v ∈ V. ej )ei ∧ ej = 1 2 g (Aei .2. v ) = g (Au.2. Example 2. eN ) is a positively oriented orthonormal basis of V . If A= then ωA = θe1 ∧ e2 so that P f (A) = θ.62. or ) be an oriented. or ) is an oriented. g. 0 −θ θ 0 .2. . Notice that ωA (u. then we can define ωA ∈ Λ2 V by ωA = i<j g (Aei . where (V. or Detg ω if deg ω = dim V Definition 2. Prove that P f (A)2 = det A. and A : V → V is a skew-symmetric endomorphism of V .63.j where (e1 . if dim V = 2n where exp ω denotes the exponential in the (nilpotent) algebra Λ• V .. The Berezin integral or ( berezinian ) is the linear form : Λ• V → R. Let V = R2 denote the standard Euclidean space oriented by e1 ∧ e2 .. dim V = N .2. and an orientation on V ∗ .2. Aej ) is the (i. Definition 2. η n−r ∈ Λn−r V. Prove that P f or g (A) = (−1)n 2 n n! (σ )aσ(1)σ(2) · · · aσ(2n−1)σ(2n) σ ∈S2n = (−1)n σ ∈S2n (σ )aσ(1)σ(2) · · · aσ(2n−1)σ(2n) . •) the induced metric on Λn−r V ∗ .or : Λ V × Λ defined by 1 r n−r Ξ(ω r . ω r ∈ Λr V ∗ . and by (•. Let (V. ∀β ∈ Λ (2. j )-th entry in the matrix representing A in the basis (ei ). .11) .66. In the sequel we r (V ) → T r (V ∗ ) ∼ will continue to use the same notation Lg to denote the metric duality Ts = s r Ts (V ). ∀k.2. e2n . The Hodge ∗-operator is the adjunction isomorphism ∗ = IΞ : Λr V ∗ → Λn−r V ∗ induced by the Hodge duality. Prove that the Hodge pairing is a duality. and r is a nonnegative integer. The r-th Hodge pairing is the pairing r ∗ n−r Ξ = Ξr V → R. where aij = g (ei . g. and S2n denotes the set of permutations σ ∈ S2n satisfying σ (2k − 1) < min{σ (2k ).68. g g ω ∧η Exercise 2.65. The above definition obscures the meaning of the ∗-operator. g. Suppose (V. A LINEAR ALGEBRA INTERLUDE 59 Exercise 2. r ≤ dim V . σ (2k + 1)}. by definition. for every β ∈ Λn−s V ∗ the operator ∗ satisfies 1 −1 −1 n−r ∗ Detg (L− V . Let (V. real Euclidean vector space. . . or ) oriented Euclidean space. Then. η n−r ) := Detor L− .67. Consider A : V → V a skewsymmetric endomorphism and a positively oriented orthonormal frame e1 . • the standard pairing Λn−r V ∗ × Λn−r V → R. g α ∧ Lg β ) = ∗α. The metric duality Lg : V → V ∗ induces both a metric. β ).2.2. Let α ∈ Λr V ∗ so that ∗α ∈ Λn−r V ∗ . g.2. . oriented. Denote by •. or ) be an oriented Euclidean space of dimension 2n. We want to spend some time clarifying its significance.2. Definition 2. g. Lg β = (∗α. or ) be an n-dimensional.2. Let (V.72.2. and we will refer to 1-densities simply as densities. ∀u ∈ det V \ {0}.60 CHAPTER 2. Suppose (V. Euclidean space. V Example 2. Definition 2. e3 .2.2. ∀β ∈ Λn−k V ∗ . e2 . or ) is an oriented. and orientation determined by e1 ∧ e2 ∧ e3 .12) Equality (2.2. ∀β ∈ Λn−r V ∗ . Let V be a real vector space. ∗(e3 ∧ e1 ) = e2 .70. ∗1 = e1 ∧ e2 ∧ e3 .11) can be rewritten as α ∧ β.73. When r = 1 we set |Λ|V := |Λ|1 . ε) be an n-dimensional. The linear space of r-densities on V will be denoted by |Λ|r V . ∗(e2 ∧ e3 ) = e1 . and α ∧ ∗β = (α. ∀β ∈ Λ (2. Detg (∗1) = 1.2. The following result is left to the reader as an exercise. g . and ∗t is the Hodge operator corresponding to the metric gt and the same orientation.69. ∀α ∈ Λ V . g. and ∗t ω = tn−2p ∗ ω ∀ω ∈ Λp V ∗ . show that n or Detor tg = t Detg . Example 2. |ω |g . ∀λ = 0. ∗e3 = e1 ∧ e2 . real Euclidean space of dimension n. g. ∀α ∈ Λk V ∗ . β ) Detor V . Exercise 2. We conclude this subsection with a brief discussion of densities. NATURAL CONSTRUCTIONS ON MANIFOLDS Let ω denote the unit vector in det V defining the orientation. For every t > 0 Denote by gt the rescaled metric gt = t2 g .12) uniquely determines the action of ∗. Then ∗e1 = e2 ∧ e3 . ω = (∗α. β ). If ∗ is the Hodge operator corresponding to the metric g and orientation or . Then the associated Hodge ∗-operator satisfies ∗(∗ω ) = (−1)p(n−p) ω ∀ω ∈ Λp V ∗ . ∗e2 = e3 ∧ e1 . Thus r ∗ n−r ∗ α ∧ β = (∗α. Then (2. Any Euclidean metric g on V defines a canonical 1-density | Detg | ∈ |Λ|1 V which associated to each ω ∈ det V its length. oriented. For any r ≥ 0 we define an r-density to be a function f : det V → R such that f (λu) = |λ|r f (u). β ) ∗ 1. ∗(e1 ∧ e2 ) = e3 .71.2.2. Proposition 2. ∗(e1 ∧ e2 ∧ e3 ) = 1. . Let V be the standard Euclidean space R3 with standard basis e1 .2. A∗ w . where •. v ) > 0. A Hermitian metric defines a duality V × V → C. A LINEAR ALGEBRA INTERLUDE Observe that an orientation or ∈ Or (V ) defines a natural linear isomorphism ıor : det V ∗ → |Λ|V where ıor µ(ω ) = or (ω )µ(ω ).13) In particular. w ∈ Vc∗ .2. v ) ∈ Vc∗ .74. If V and W are complex Hermitian vector spaces.2. Its conjugate is the complex vector space V which coincides with V as a real vector space. We can rewrite the above fact as Av. •) is positive definite. Definition 2. an orientation in a Euclidean vector space canonically identifies |Λ|V with R. v → (·. ∀ω ∈ det V \ {0}. w = v. (v. ∀v ∈ V. but in which the multiplication by a scalar λ ∈ C is defined by λ · v := λv. • The bilinear from (•.2. A complex linear map W → Vc∗ is the same as a complex linear map W → Vc . ∗ . u).2.5 Some complex linear algebra In this subsection we want to briefly discuss some aspects specific to linear algebra over the field of complex numbers.2. The ∗ metric duality defines a complex linear isomorphism Vc ∼ = V so we can view the adjoint A∗ as a complex linear map W → V . ∀v ∈ V \ {0}. • For any u. If we forget the complex structure we obtain a real dual Vr∗ consisting of all real-linear maps V → R. •) : V × V → C satisfying the following properties. §2. The vector space V has a complex dual Vc∗ that can be identified with the space of complex linear maps V → C. then any complex linear map A : V → W induces a complex linear map A∗ : W → Vc∗ A∗ w := v → Av.e. v ) = (v. • denotes the natural duality between a vector space and its dual. det V ∗ µ → ıor µ ∈ |Λ|V . 61 (2. A Hermitian metric is a complex bilinear map (•. and hence it induces a complex linear isomorphism L : V → Vc∗ . v ∈ V we have (u. Let V be a complex vector space. i. .e. •) as an object over R. while ω is a real. V ∼ = (ker J + i). then the Hermitian metric decomposes as √ h = Re h − i ω. If we view (•. fj ) = 0 ∀i. Let V be an n-dimensional complex vector space and h a Hermitian metric on it. . NATURAL CONSTRUCTIONS ON MANIFOLDS Let h = (•.. f1 . n = dimC V . we denote by fj the vector iej . It is convenient to have a coordinate description of the abstract objects introduced above. which shows that ω = −Im h = i ei ∧ f i .. ej ) = δij . We will call an operator J as above a complex structure .. ω = − Im h. . ω is called the real 2-form associated to the Hermitian metric. . ej ) = δij = Re h(fi . fj ) = −Im h(ei . fj ) = 0. fj ) and Re h (ei . en . The operator J has no eigenvectors on V . VC = V ⊗ C. Denote by e1 . j. as an R-bilinear map V × V → C. Then Re h (ei .e.. i.2. i. ∀v ∈ V. Re h = i (ei ⊗ ei + f i ⊗ f i ). iej ) = δij . en of V . and h(ei . The multiplication √ by i = −1 defines a real linear operator which we denote by J . fn is an R-basis of V . i. . Pick an unitary basis e1 . Conversely. Obviously J satisfies J 2 = −V . Then the collection e1 . ω (ei .1 := ker(J + i) ∼ =C V . ej ) = ω (fi . f 1 .75. .0 := ker(J − i) ∼ =C V V 0. f n the dual R-basis in V ∗ . en . Prove that we have the following isomorphisms of complex vector spaces V ∼ = ker (J − i) Set V 1. a + bi ∈ C. The real part is an inner product on the real space V . . Also ω (ei .e. •) be a Hermitian metric on the complex vector space V .. . . and we have a splitting of VC as a direct sum of complex vector spaces (eigenspaces) VC = ker (J − i) ⊕ (ker J + i). i := −1. For each j . Exercise 2. if V is a real vector space then any real operator J : V → V as above defines a complex structure on V by (a + bi)v = av + bJv. and thus can be identified with an element of Λ2 R V . .62 CHAPTER 2. Any complex space V can be also thought of as a real vector space. has two eigenvalues ±i. . skew-symmetric ∗ bilinear form on V . Let V be a real vector space with a complex structure J on it. The natural extension of J to the complexification of V . . (2. .0 ⊗ Λ• V 0. ie1 .0 V )∗ c. . Jv ). v ). Then J ∗ ei = −f i . and. −J ) ∼ =C V . ∀u.q V )∗ c. J )∗ c = (Vr . More precisely. J ∗ ) ∼ =C (V. en .p V. We can define similarly Λp. . Λ0. .q V. and e1 . ien form a real basis of V . and the associated 2-form ω = −Im h are related by Moreover. and if we set 1 1 εi := √ (ei + if i ).1 V ∗ ∼ = (Λ0. 1)-form. more generally Λp. We can however complete this to a real basis. . then the above isomorphism can be reformulated as Λk VC ∼ = p+q =k Λp. Note that Λ1. then we have a natural isomorphism of complex vector spaces Vc∗ ∼ = (Vr∗ . en .1 . en be a basis of V over C. We obtain an isomorphism of Z-graded complex vector spaces Λ• VC ∼ = Λ• V 1. . J ).1 ∼ = V ⊕ V .q V ∗ as the Λp.1 .1 = spanC {ε ¯j }. ω (u. If h is a Hermitian metric on the complex vector space (V.0 V ∗ ∼ = (Λ1.2.2. v ) = ω (u. the vectors e1 . . . g (u. If we set Λp. and we have an isomorphism of complex vector spaces ∼ ∗ ∗ Vc∗ = (V. ω is a (1.q V := Λp V 1. n} the dual orthonormal basis in Vr∗ . To see this it suffices to pick an unitary basis (ei ) of V . This is not a real basis of V since dimR V = 2 dimC V .0 ⊕ V 0. and ω=i εi ∧ ε ¯i . J ).2. . v ) = g (Ju.14) Note that the complex structure J on V induces by duality a complex structure J ∗ on Vr∗ .q V ∗ ∼ = (Λp. . and construct as usual the associated real orthonormal basis {e1 . . A LINEAR ALGEBRA INTERLUDE 63 Thus VC ∼ = V 1.0 V ∗ = spanC {εi } Λ0. · · · . Let V be a complex vector space. ε ¯j := √ (ej − if j ). so that Λp.1 V )∗ c. f1 . f i . fn } (fi = Jei ). Denote by {ei .q -construction applied to the real vector space Vr∗ equipped with the complex structure J ∗ .0 ⊗C Λq V 0. 2 2 then Λ1.15) The Euclidean metric g = Re h.q V ∗ ∼ =C Λq.2. i = 1. (2. v ∈ V. . | det Z |2 = det Z .76. Clearly A is a closed subset of EndC (Cn ). ∀1 ≤ k ≤ n. k + iyk .16) We will prove that A = EndC (Cn ). We write Then j j j j ˆ zk = xj ˆj = iej . . Suppose (e1 . xk . The set A is nonempty since it contains all the diagonal matrices. and any complex linear automorphism T ∈ AutC (Cn ). j Then f1 ∧ if1 ∧ · · · ∧ fn ∧ ifn = | det Z |2 e1 ∧ ie1 ∧ · · · ∧ en ∧ ien . . . We want to show that det Z = | det Z |2 . ke Then. j fk = zk ej . for every Z ∈ EndC (Cn ). yk ∈ R. so it suffices to show that A is dense in EndC (Cn ). . ∀Z ∈ EndC (Cn ). where Z is the 2n × 2n real matrix Z= Re Z − Im Z Im Z Re Z = X −Y Y X . . .. j and X . Y denote the n × n real matrices with entries (xj k ) and respectively (yk ).e. . if we set = (−1)n(n−1)/2 ..2. ∧ en ∧ ien . en ) and (f1 .2. NATURAL CONSTRUCTIONS ON MANIFOLDS Proposition 2. Let A := Z ∈ EndC (Cn ). j and Z = (zk )1≤j. . . we deduce n f1 f1 ∧ if1 ∧ · · · ∧ fn ∧ ifn = = n (det Z )e1 ˆ ˆ ∧ · · · ∧ fn ∧ f 1 ∧ · · · ∧ fn ˆ ∧ · · · ∧ en ∧ e ˆ1 ∧ · · · ∧ e ˆn = (det Z )e1 ∧ ie1 ∧ . f k = ifk . and ˆ f k =− j n j yk ej + j xj ˆj . Proof. Observe that the correspondence EndC (Cn ) Z → Z ∈ EndR (R) is an endomorphism of R-algebras.64 CHAPTER 2. i. Then. (2.k is the complex matrix describing the transition from the basis e to the basis f . we have T ZT −1 = T Z T −1 . e fk = j j (xj k + iyk )ej = j xj k ej + j j yk e ˆj . fn ) are two complex bases of V .. ∧ en ∧ ien ∈ Λ2 R V . and let E → M be a rank r K-vector bundle over the smooth manifold M . Let E .3 Tensor fields §2. where {e1 . and gluing cocycles gαβ : Uαβ → GL(VE ).78. In particular.2. . Euclidean metric on V regarded as a real vector space. we will exclusively think of vector bundles in terms of gluing cocycles.. Definition 2.2. The bundle E is obtained by gluing these trivial pieces on the overlaps Uα ∩ Uβ using a collection of transition maps gαβ : Uα ∩ Uβ → GL(V ) satisfying the cocycle condition.16). and respectively hαβ : Uαβ → GL(VF ). A contains all the diagonalizable n × n complex matrices. The canonical orientation of a complex vector space V . The canonical orientation n ∗ c or c on V . R V → R. en } is any complex basis of V . n is the orientation defined by e1 ∧ ie1 ∧ . ∀T ∈ AutC (Cn ). TENSOR FIELDS Hence Z ∈ A ⇐⇒ T ZT −1 ∈ A. n = dimC V . 2. Suppose h is a Hermitian metric on the complex vector space V . 65 In other words. dimC V = n. .2. and a pffafian 1 ∗ 2 ∗ c P f h = P f or : Λ2 η → g (η n . Prove that the set of diagonalizable n × n complex matrices form a dense subset of the vector space EndC (Cn ). then so will any of its conjugates. if a complex matrix Z satisfies (2. Let K denote one of the fields R or C. .3. . given by a (common) open cover (Uα ).. and more specifically.77.2. and these form a dense subset of EndC (Cn ) (see Exercise 2. n! and we conclude P f h ω = 1. and the metric g define a volume form Detor ∈ Λ2 g R Vr .1 Operations with vector bundles We now return to geometry. In the sequel. a collection of gluing maps as above satisfying the cocycle condition uniquely defines a vector bundle. Exercise 2. Then g = Re h defines is real. then c Detg = Detor = g 1 n ω .2. where V is an r-dimensional vector space over the field K.77). F be two vector bundles over the smooth manifold M with standard fibers VE and respectively VF . to vector bundles.3. . we can find an open cover (Uα ) of M such that each restriction E |Uα is trivial: E |Uα ∼ = V × Uα . Conversely. Detg ). Λ Vr h n! If ω = − Im h is real 2-form associated with the Hermitian metric h. According to the definition of a vector bundle. E ∗ .3. φ : M → M . for any m ∈ M . Any bundle maps S : E → E and T : F → F . Let E . The reader can check easily that these vector bundles are independent of the choices of transition maps used to characterize E and F (use Exercise 2. a morphism S † : (E )∗ → E ∗ . E ⊗ F . It is called the determinant line bundle of E . Let E → M be a K-vector bundle over M .1. both covering the same diffeomorphism φ : M → M . We define the tensor bundles of M r r Ts (M ) : =Ts (T M ) = (T M )⊗r ⊗ (T ∗ M )⊗s . and respectively Λr E . Λr gαβ : Uαβ → GL(Λr VE ). The direct sum E ⊕ F is also called the Whitney sum of vector bundles.3.3. the bundle Λr E has rank 1. induce bundle morphisms S⊕T :E⊕F →E ⊕T . The bundle E ∗ is called the dual of the vector bundle E .66 Then the collections CHAPTER 2. Note that we have a natural identification E∗ ⊗ F ∼ = Hom (E. E ∼ = T M . S⊗T :E⊗F →E ⊗F .2. Prove the assertion above. covering φ.) These above constructions are natural in the following sense. satisfy the cocycle condition. (Observe that a vector space can be thought of as a vector bundle over a point. where † denotes the transpose of a linear map.3. if r = rankK E .2 Tensor fields We now specialize the previous considerations to the special situation when E is the tangent bundle of M . A metric on E is a section h of E ∗ ⊗K E ∗ (E = E if K = R) such that.3. NATURAL CONSTRUCTIONS ON MANIFOLDS gαβ ⊕ hαβ : Uαβ → GL(VE ⊕ VF ). covering φ−1 etc. and it is denoted by det E . §2. so that we get an induced map (S −1 )† ⊗ T : E ∗ ⊗ F → (E )∗ ⊗ F .32). and therefore define vector bundles which we denote by E ⊕ F . In particular.1. Consider bundle isomorphisms S : E → E and T : F → F covering the same diffeomorphism of the base. All the functorial constructions on vector spaces discussed in the previous section have a vector bundle correspondent. Exercise 2. The cotangent bundle is then T ∗ M := (T M )∗ . Definition 2. Then (S −1 )† : E ∗ → (E )∗ is a bundle isomorphism covering φ. E and F be vector bundles over a smooth manifold M . Let E and F be vector bundles over the same smooth manifold M . . Example 2. h(m) defines a metric on Em (Euclidean if K = R or Hermitian if K = C). gαβ ⊗ hαβ : Uαβ → GL(VE ⊗ VF ). F ). † −1 ∗ (gαβ ) : Uαβ → GL( VE ). F . 1 ≤ j1 .. . then a tensor field can be viewed as a smooth selection of a tensor in each of the tangent spaces. (a) A tensor field of type (r.. it is a symmetric (0. ..ir T = Tji1 . .2... any tensor T ∈ Ts 1 .. ir ≤ n. j. .. More precisely.5. ⊗ ir ⊗ dxj1 ⊗ .. (c) A Riemannian metric on a manifold M is a metric on the tangent bundle. they define a framing of the restriction of T M to U . s) over the open set U ⊂ M is a section r (M ) over U .. . gij = gji .3.. It is often very useful to have a local description of these objects. i 1 ∂x ∂x In the above equality we have used Einstein’s convention.e. . In particular. 1 ≤ i1 . TENSOR FIELDS 67 Definition 2. n. such that for every x ∈ M . the bilinear map gx : Tx M × Tx M → R defines an Euclidean metric on Tx M .j gij dxi ⊗ dxj .. a Riemann metric defines a smoothly varying procedure of measuring lengths of vectors in tangent spaces. In particular. ∀i. . We can form a dual framing of T ∗ M |U .. They satisfy the duality conditions dxi . ∂xi1 ∂x r (M ) has a local description Hence. ⊗ dxjs . . then the vector fields ∂ ∂ ( ∂x 1 . ∂xn ) trivialize T M |U .. using the 1-forms dxi .ir dxi1 ∧ · · · ∧ dxir . while a Riemann metric g has the local description g= i. i = 1. . .js ∂ ∂ ⊗ .. . i. of Ts (b) A degree r differential form (r-form for brevity) is a section of Λr (T ∗ M ). ⊗ ir ⊗ dxj1 ⊗ . js ≤ n .. . ⊗ dxjs . ∂xj ∂ ∂ ⊗ . If (x1 . If we view the tangent bundle as a smooth family of vector spaces.. .4. We set Ω• (M ) := r≥0 Ωr (M ). xn ) are local coordinates on an open set U ⊂ M .3.3. r (T M ) is given by A basis in Ts x ∂ i = δj . 2)-tensor g ... . Example 2. an r-form ω has the local description ω= 1≤i1 <···<ir ≤n ωi1 . .. . The space of (smooth) r-forms over M is denoted by Ωr (M ). . naturally defines a C ∞ (M )-multilinear map s S: 1 Vect (M ) → C ∞ (M ). Yk ) = Sp f∗ Y1 . k ) tensor field on N . In particular. (Dp f )−1 Yk . which we regard as a C ∞ (M )-multilinear map S : Vect (M ) × · · · × Vect (M ) → C ∞ (M ). Conversely. (X1 . (f∗ )−1 Yk = Sp (Dp f )−1 Y1 . . . . NATURAL CONSTRUCTIONS ON MANIFOLDS Remark 2. . i. .3. Suppose f : M → N is a diffeomorphism. (a) A covariant tensor field.6. any such map uniquely defines a (0.. for any Y1 . Any diffeomorphism f : M → N induces bundle isomorphisms Df : T M → T N and (Df −1 )† : T ∗ M → T ∗ N covering f . . k )-tensor field on M . Let q ∈ N . Thus. . . In the sequel we will always regard the differential of a smooth function f as a 1-form and to indicate this we will use the notation df (instead of the usual Df ). Xs ) → p → Sp X1 (p). and S is a (0.3. k Then f∗ S is a (0.e. Notice that the wedge product in the exterior algebras induces an associative product in Ω• (M ) which we continue to denote by ∧. .3. . (b) Let f ∈ C ∞ (M ). . we get a smooth C ∞ (M )-linear map df : Vect (M ) → C ∞ (M ) defined by df (X )m := Df (X )f (m) ∈ Tf (m) R ∼ = R. Then. . a diffeomorphism f induces a linear map r r f∗ : Ts (M ) → Ts (N ). we have −1 (f∗ S )q (Y1 .1) called the push-forward map. . an r-form η can be identified with a skew-symmetric C ∞ (M )-multilinear map r η: 1 Vect (M ) → C ∞ (M ). Its Fr´ echet derivative Df : T M → T R ∼ = R × R is naturally a 1-form. Yk ∈ Tq N . . Xs (p) ∈ C ∞ (M ). .68 CHAPTER 2. s)-tensor field S . . .. and set p := f −1 (q ). s)-tensor field. . . .. Example 2. Indeed. . In particular. . . . (2.7. the group of diffeomorphisms of M acts naturally (and linearly) on the space of tensor fields on M . a (0. ∂x etc. .3. Example 2. then f ∗ S is the covariant tensor field defined by (f ∗ S )p (X1 (p). . For simplicity. r = (x2 + y 2 )1/2 . . . . p ∈ M . ∀X1 . Dp f (Xs ) . . Explicitly. We have F ∗ dx = d(r cos θ) = cos θdr − rsinθdθ. Xs ∈ Vect (M ). x ≥ 0}.8. where f∗ is the push-forward map defined in (2. θ) → (x = r cos θ. F ∗ (dx ∧ dy ) = d(r cos θ) ∧ d(rsinθ) = (cos θdr − rsinθdθ) ∧ (sinθdr + r cos θdθ) = r(cos2 θ + sin2 θ)dr ∧ dθ = rdr ∧ dθ. The pullback of g by F is the symmetric (0. To compute F∗ dr we need to express r as a function of x and y . . ∂x instead of ∂ ∂ ∂r .1). . 2π ) → R2 . we will write ∂r . To compute F∗ ∂r and F∗ ∂θ we use the chain rule which implies F∗ ∂r = ∂y 1 ∂x ∂x + ∂y = cos θ∂x + sin θ∂y = (x∂x + y∂y ) ∂r ∂r r = (x2 1 (x∂x + y∂y ). The map F defines the usual polar coordinates. any smooth map f : M → N defines a linear map 0 0 f ∗ : Ts (N ) → Ts (M ). called the pullback by f . if S is such a tensor defined by a C ∞ (M )-multilinear map S : (Vect (N ))s → C ∞ (N ). Consider the map F∗ : (0. (r. F ∗ dy = d(r sin θ) = sinθdr + r cos θdθ.3. 2)–tensor g = dx2 + dy 2 . . Xs (p)) := Sf (p) Dp f (X1 ). y = r sin θ). and then we have F∗ dr = (F −1 )∗ dr = d(x2 + y 2 )1/2 = x(x2 + y 2 )−1/2 dx + y (x2 + y 2 )−1/2 dy. + y 2 )1/2 The Euclidean metric is described over U by the symmetric (0. Note that when f is a diffeomorphism we have −1 † f ∗ = (f∗ ). . ∞) × (0. . . TENSOR FIELDS 69 For covariant tensor fields a more general result is true. It is a diffeomorphism onto the open subset U := R2 \ {(x. 2)–tensor F ∗ (dx2 + dy 2 ) = d(r cos θ) 2 + d(r sin θ) 2 = (cos θdr − rsinθdθ)2 + (sinθdr + r cos θdθ)2 = dr2 + r2 dθ2 .2. More precisely. 0).3. . 10.. a Riemann metric defines a duality L : Vect (M ) → Ω1 (M ) etc. If we choose local coordinates (xi ) on an open set U associated tensor bundles Ts then.. Proposition 2. they arise naturally in geometry.ir tr Tji0 .3.. In local coordinates the contraction has the form 0 .. ∂xi1 ∂x If we denote by (g ij ) the inverse of the matrix (gij ).. while a tensor field T of type (r.9. for every point p ∈ U .ir T = Tji1 . ⊗ dxjs .. In particular.ir = {Tij }... . ∀ωj ∈ Ω1 (M ).js ∂ ∂ ⊗ .kr 1 .js ii1 . Let f : M → N be a smooth map.. as explained above. NATURAL CONSTRUCTIONS ON MANIFOLDS All the operations discussed in the previous section have natural extensions to tensor fields.3 Fiber bundles We consider it is useful at this point to bring up the notion of fiber bundle. and they impose themselves as worth studying....j gij dxi ⊗ dxj . Prove the above proposition. s .. the length r (M ) is the number |T (p)| defined by of T (p) ∈ Ts p g k1 .3. On the other hand.. s) can be described as 1 ...ir |T (p)|g = gi1 k1 .. There are several reasons to do this. there exists a contraction operator r+1 r tr : Ts +1 (M ) → Ts (M ) defined by tr (X0 ⊗ · · · ⊗ Xr ) ⊗ (ω0 ⊗ · · · ⊗ ωs ) = ω0 (X0 )(X1 ⊗ · · · Xr ⊗ ω1 ⊗ · · · ⊗ ωs ). On one hand. .. ⊗ ir ⊗ dxj1 ⊗ .70 CHAPTER 2. There exists a tensor multiplication. Exercise 2. 1 . The exterior product defines an exterior product on the space of smooth differential forms ∧ : Ω• (M ) × Ω• (M ) → Ω• (M )..3. gir kr g j1 1 · · · g js s Tji1 . The pullback by f defines a morphism of associative algebras f ∗ : Ω• (N ) → Ω• (M )..js Let us observe that a Riemann metric g on a manifold M induces metrics in all the r (M )... then. the metric g can be described as g= i. The space Ω• (M ) is then an associative algebra with respect to the operations + and ∧. . they provide a very elegant and concise language to describe many phenomena in geometry.js T 1 . ∀Xi ∈ Vect (M )... where in the above equalities we have used Einstein’s convention. §2. Then the counterclockwise rotations about the z -axis define a smooth left action of S 1 on S 2 .3. Tg = M . (a) Let M be a smooth manifold. then for every ϕ ∈ R mod 2π ∼ = S 1 we define Tϕ : S 2 → S 2 by Tϕ (r. z ). More formally. such that T1 ≡ M and Tg (Th m) = Tgh m (respectively Tg (Th m) = Thg m) ∀g. we deduce that ∀g ∈ G the map Tg is a diffeomorphism of M . This is an example of trivial fiber bundle. x2 + y 2 + z 2 = 1 }. (g. m) → Tg m. In particular. y = r sin θ. This is a very loose description. TENSOR FIELDS 71 We have already met examples of fiber bundles when we discussed vector bundles. (b) Let G act on M . These were “smooth families of vector spaces”. Example 2. and G a Lie group. Definition 2.3. The action is called effective if. In general. It is convenient to regard this as a family of manifolds (Fb )b∈B . g ∈ G} is called the orbit of the action through m. h ∈ G. θ. and X is called the total space. z = 0. The action is called free if ∀g ∈ G. and ∀m ∈ M Tg m = m. For any m ∈ M the set G · m = {Tg m. The resulting map T : S 1 × S 2 → S 2 . but it offers a first glimpse at the notion about to be discussed. The gluing may encode a symmetry of the fiber. and we would like to spend some time explaining what do we mean by symmetry. . We say the group G acts on M from the left (respectively right).3. ∀g ∈ G. if there exists a smooth map Φ : G × M → M. p) → Tϕ (p) defines a left action of S 1 on S 2 encoding the rotational symmetry of S 2 about the z -axis. (θ + ϕ) mod 2π. z ) = (r. It is useful to think of a Lie group action on a manifold as encoding a symmetry of that manifold. a fiber bundle is obtained by gluing a bunch of trivial ones according to a prescribed rule. z ). if we use cylindrical coordinates (r. yz ) ∈ R3 . θ. Consider the unit sphere S 1 = (x. m ∈ M. The model situation is that of direct product X = F × B . A fiber bundle wants to be a smooth family of copies of the same manifold.12. (ϕ. F is called the standard (model) fiber. The manifold B is called the base.11. x = r cos θ. where B and F are smooth manifolds.2. 72 CHAPTER 2. (c) a smooth manifold B called the base . then the bundle (E. For any g ∈ G denote by Lg (resp.3. F.. the tautological linear action of SO(n) on Rn defines a linear representation of SO(n). ∀(f. B ).14. G×F (g. If G is a Lie group. In this way we get the tautological left (resp. Rg ) the left (resp right) translation by g . b) = (Tαβ (b)f.13. (b) a smooth manifold F called the standard fiber . where Uαβ = −1 ◦ ψ . A linear representation of G on a vector space V is a left action of G on V such that each Tg is a linear map. F.e. b) ∈ F × Uα . F × Uα     Ψα M π−1(Uα) π Uα We can form the transition (gluing) maps Ψαβ : F × Uαβ → F × Uαβ . . and diffeomorphisms Ψα : F × Uα → π −1 (Uα ) such that π ◦ Ψα (f. (f) There exists an effective left action of the Lie group G on F .3. A smooth fiber bundle is an object composed of the following: (a) a smooth manifold E called the total space . We will denote this fiber bundle by (E. Let G be a Lie group. right) action of G on itself. these maps can be written Uα ∩ Uβ .. b) = b. π.e. The group G is called the symmetry group of the bundle. NATURAL CONSTRUCTIONS ON MANIFOLDS Example 2. defined by Ψαβ = ψα β as ψαβ (f. Definition 2. i. π.15. (d) a surjective submersion π : E → B called the natural projection . B ) is called a G-fiber bundle if it satisfies the following additional conditions. the diagram below is commutative. Example 2. According to (e).3. an open cover (Uα ) of the base B . Let G be a Lie group. b). x) → g · x = Tg x ∈ F. (e) a collection of local trivializations. i. where Tαβ (b) is a diffeomorphism of F depending smoothly upon b ∈ Uαβ . For example. One says V is a G-module. such that.2. and any f ∈ F . s(b) ∈ π −1 (b). and for any trivializing cover (Uα ) (as in Definition 2.3.15) there exists a smooth map gα : Uα → G such that 1 Ψ− α T Ψα (f. if we are given a cover (Uα )α∈A of the base B . We need to describe when two such choices are equivalent. ∀ b ∈ B . x). TENSOR FIELDS (g) There exist smooth maps gαβ : Uαβ → G satisfying the cocycle condition gαα = 1 ∈ G.. and a collection of transition maps gαβ : Uα ∩ Uβ → G satisfying the cocycle condition. Indeed. ∀α. B.18. b) ∈ F × Uβ . A rank r vector bundle (over K = R. b ). Definition 2. and such that Tαβ (b) = Tgαβ (b) . F. (a) A fiber bundle is an object defined by conditions (a)-(d) and (f) in the above definition. i.. f. we have 1 Φ− α Ψi (f. then we can get a bundle by gluing the trivial pieces Uα × F along the overlaps.16. gγα = gγβ · gβα .3. Two open covers (Uα ) and (Vi ). A G-automorphism of this bundle is a diffeomorphism T : E → E such that π ◦ T = π . (One can think the structure group is the group of diffeomorphisms of the standard fiber). K) acts on Kr in the natural way.17. for any x ∈ Uα ∩ Vi . π . 73 The choice of an open cover (Uα ) in the above definition is a source of arbitrariness since there is no natural prescription on how to perform this choice. K)-fiber bundle with standard fiber Kr . together with the collections of local trivializations Φα : F × Uα → π −1 (Uα ) and Ψi : F × Vi → π −1 (Vi ) are said to be equivalent if. a collection of gluing data determines a G-fiber bundle. b) ∈ F × Uα is identified with the element (gβα (b) · f. if b ∈ Uα ∩ Uβ . x) = (Tαi (x)f. G). π.e.3. for all α. then the element (f. A G-bundle structure is defined by an equivalence class of trivializing covers. T maps fibers to fibers. there exists a smooth map Tαi : Uα ∩ Vi → G. Example 2. Let E → B be a G-fiber bundle. ∀b. b) = ( gα (b)f.3. More precisely. We will denote a G-fiber bundle by (E. Definition 2. π (b) A section of a fiber bundle E → B is a smooth map s : B → E such that π ◦ s = B . γ. C) is a GL(r. and where the group GL(r. i.3. β. As in the case of vector bundles. i.e. (a) P is a principal G-bundle. Xn ) → (m. where G acts on itself by left translations. g ) → p · g. Exercise 2. u) · g. . Xi ∈ Tm M and span(X1 . right action of G on G.. Hint: A matrix T = (Tji ) ∈ GL(n. .. u ∈ Uα . Prove the following are equivalent. Let P be a fiber bundle with fiber a Lie group G. (b) Show F (M ) is a principal GL(n. . i. Exercise 2. Consider a trivializing cover (Uα )α∈A . . ∀g. (Alternative definition of a principal bundle). Let M n be a smooth manifold. Denote by F (M ) the set of frames on M . u) = Ψα (h. 1 Xi .19. X1 .. V.3.. (T −1 )i )n Xi ).22..e. V ∈ U. (Associated fiber bundles). (T Example 2. (p. f ) → τ (g )f.3.21.. .e.3. (b) There exists a free. A principal G-bundle is a G-fiber bundle with fiber G. W ∈ U) gU V (x)gV W (x)gW U (x) = 1 ∈ G. Xn ) → m is a submersion. such that ∀x ∈ U ∩ V ∩ W (U. such that Ψα (hg.. NATURAL CONSTRUCTIONS ON MANIFOLDS Example 2. Let G be a Lie group. X1 . The bundle F (M ) is called the frame bundle of the manifold M . h ∈ G. Let π : P → G be a principal Gbundle... Assume G acts (on the left) on a smooth manifold F τ : G × F → F. .. . and denote by gαβ : Uα ∩ Uβ → G a collection of gluing maps determined by this cover. m ∈ M. X1 . F (M ) = {(m. (The frame bundle of a manifold). (a) Prove that F (M ) can be naturally organized as a smooth manifold such that the natural projection p : F (M ) → M .3. and there exists a trivializing cover Ψα : G × Uα → π −1 (Uα ) . (Prove this!) It is called the bundle associated to P via τ and is denoted by P ×τ F . Xn ) = Tm M } . a principal G-bundle over a smooth manifold M can be described by an open cover U of M and a G-cocycle.74 CHAPTER 2. such that its orbits coincide with the fibers of the bundle P ... .. This new bundle is independent of the various choices made (cover (Uα ) and transition maps gαβ )..20. Equivalently. The collection ταβ = τ (gαβ ) : Uαβ → Diffeo (F ) satisfies the cocycle condition and can be used (exactly as we did for vector bundles) to define a G-fiber bundle with fiber F . i. (g. R)-bundle. . a collection of smooth maps gU V : U ∩ V → G U. (m. P × G → P. Xn ). K) acts on the right on F (M ) by −1 i (m. The space of orbits of this action is naturally identified with the complex projective space CPn−1 .25.2. . (p is the obvious projection)..23.3. Any metric h on E (euclidian or Hermitian) defines a submanifold S (E ) ⊂ E by S (E ) = {v ∈ E . The bundle S (E ) is usually called the sphere bundle of E . |v |h = 1}. (b) Prove that the tautological line bundle over CPn−1 is associated to the Hopf bundle via the natural action of S 1 on C1 . . Prove that S (E ) is a fibration over M with standard fiber a sphere of dimension rank E − 1. eiθ zn ). Prove that the tangent bundle of a manifold M n is associated to F (M ) via the natural action of GL(n. |z0 |2 + · · · + |zn |2 = 1 then we detect an S 1 -action on S 2n−1 given by eiθ · (z1 . (The Hopf bundle) If we identify the unit odd dimensional sphere S 2n−1 with the submanifold (z1 .. TENSOR FIELDS 75 Exercise 2..3. Exercise 2.. zn ) ∈ Cn . R) on Rn . . zn ) = (eiθ z1 . .3.. . Let E be a vector bundle over the smooth manifold M ... Exercise 2.24. (a) Prove that p : S 2n−1 → CPn−1 is a principal S 1 bundle called Hopf bundle.. Describe one collection of transition maps.3. 3. such that (a) Φ0 (m) = m. (s + t. When N = R × M . ∞) defines a balanced open Nf := (t. 76 . We will discuss how one can operate with the various objects we have introduced so far. t ∈ R.1 The Lie derivative §3. m) → Φt (m). Note that any continuous function f : M → (0. where N is a balanced neighborhood of {0} × M in R × M . A neighborhood N of {0} × M in R × M is called balanced if. Φs (m)) ∈ N. m). (t. ∞] such that ( R × {m} ) ∩ N = (−r. (t. Definition 3. ∀m ∈ M . there exists r ∈ (0. We strongly recommend [4] for more details. we will introduce several derivations of the various algebras of tensor fields. m ∈ M such that (s. Φ is called a flow. ∀m ∈ M . m). m) ∈ R × M . and excellent examples. In particular. A local flow is a smooth map Φ : N → M . +) on M . (b) Φt (Φs (m)) = Φt+s (m) for all s.1 Flows on manifolds The notion of flow should be familiar to anyone who has had a course in ordinary differential equations. and we will also present the inverse operation of integration. r) × {m}. |t| < f (m) . The conditions (a) and (b) above show that a flow is nothing but a left action of the additive (Lie) group (R.Chapter 3 Calculus on Manifolds This chapter describes the “kitchen” of differential geometry.1.1. In this section we only want to describe some classical analytic facts in a geometric light.1. The map X : {Local flows on M } → Vect (M ). Let M be a smooth n-dimensional manifold. ε) × K → M. then Φ1 = Φ2 on N1 ∩ N2 .e.3. Exercise 3. there exists a unique integral curve for X .2) are two local flows such that XΦ1 = XΦ2 ..1. xn (t) . Example 3. Its generator is the vector field XA on Rn defined by XA (u) = d |t=0 etA u = Au. for any precompact open subset K ⊂ U .1. Proof. X (p) is the tangent vector to the smooth path t → Φt (p) at t = 0. as a consequence of the smooth dependence upon initial data we deduce that the map ΦK : NK = (−ε. for all x ∈ K . . . . 43]) show that.. . is a surjection.1. Let Φ : N → M be a local flow on M . dt i. b) → M such that γ ˙ (t) = X (γ (t)).5. . THE LIE DERIVATIVE n Example 3. Surjectivity. ..2) Moreover.1. Classical existence results (see e. and X = X i ∂x i . Show that XΦ is a smooth vector field.3. (3. In local coordinates (xi ) over on open subset U ⊂ M this condition can be rewritten as x ˙ i (t) = X i x1 (t).6.g. ∀p ∈ M.. γx : (−ε.1. Φ → XΦ . Definition 3. xn (t)).. . dt Proposition 3. [4. It generates a flow Φt A on R by ∞ 77 Φt Ax =e x= k=0 tA tk k A k! x.. is smooth.2. The above equality is a system of ordinary differential equations. (x.1. there exists ε > 0 such that.1) ∂ where γ (t) = (x1 (t). An integral curve for X is a smooth curve γ : (a.1.. n. (3.4. Let X be a vector field on M .1. t) → γx (t). Consider the flow etA on Rn generated by an n × n matrix A. Moreover. ∀i = 1. ε) → M satisfying γx (0) = x. if Φi : Ni → M (i=1. This path is called the flow line through p. Let A be a n × n real matrix. The infinitesimal generator of Φ is the vector field X on M defined by X (p) = XΦ (p) := d |t=0 Φt (p). 3) . The uniqueness of solutions of initial value problems for ordinary differential equations implies that Φ satisfies all the conditions in the definition of a local flow.2 The Lie derivative Let X be a vector field on the smooth n-dimensional manifold M and denote by Φ = ΦX the local flow it generates. §3. εα ) × Kα → M solving the initial value problem (3. θ). the map Φt is a diffeomorphism of M and so it induces a push-forward map on the space of tensor fields. we assume Φ is actually a flow so its domain is R × M . y = r sin θ. and then describe it in cylindrical coordinates (z.78 CHAPTER 3. z ) ∈ R3 . Exercise 3. Moreover. Φ = Φα on Nα .1.7. For each t ∈ R. For every point p ∈ S 2 we denote by X (p) ∈ Tp R3 . 1) onto Tp S 2 . the orthogonal projection of the vector k = (0. (3.1-2). (Φ1 : N1 → M ) ≺ (Φ2 : N2 → M ) if and only if N1 ⊂ N2 . Consider the unit sphere S 2 = (x. and as above. y. and it is denoted by ΦX .1. For simplicity. X is the infinitesimal generator of Φ. x2 + y 2 + z 2 = 1 . If S is a tensor field on M we define its Lie derivative along the direction given by X as LX Sm := − lim 1 (Φt ∗ S )m − Sm t→0 t ∀m ∈ M. The local flow situation is conceptually identical.1. Define N := α∈A Nα . but notationally more complicated. The second part of the proposition follows from the uniqueness in initial value problems. precompact. where x = r cos θ. (b) Describe explicitly the flow generated by X . and set Φ : N → M . Tautologically. (a) Prove that p → X (p) is a smooth vector field on S 2 . we deduce Φα = Φα on Nα ∩ Nβ . The family of local flows on M with the same infinitesimal generator X ∈ Vect(M ) is naturally ordered according to their domains. CALCULUS ON MANIFOLDS Now we can cover M by open. we get smooth maps Φα : Nα = (−εα . local coordinate neighborhoods (Kα )α∈A . This family has a unique maximal element which is called the local flow generated by X . 0.1. r = (x2 + y 2 )1/2 . by uniqueness. and we let him keep track of the the sizes of the tensor S carried by the flow at the point p.1. Der (C ∞ (M )) ∼ = Vect (M ). i.9. To show that the limit exists. then one sees that LX S is a tensor of the same type as S . •. THE LIE DERIVATIVE 79 Intuitively. Let X. In particular the Lie bracket induces a Lie algebra structure on Vect (M ). coincides with the commutator of the two derivations of C ∞ (M ) defined by X and Y i. Hence. The Lie derivatives measures the rate of change in these sizes. . Let Φt = Φt X be the local flow generated by X .3) exists. X p . t→0 t dt Exercise 3. The vector field [X. Here is the intuition behind this very suggestive terminology. ∂xi ∂xj Arnold refers to the Lie derivative LX as the “fisherman’s derivative”. X ) → α(X ) ∈ C ∞ (M ). LX is a derivation of C ∞ (M ). C ∞ (T ∗ M ) × C ∞ (T M ) (α. ∀ f ∈ C ∞ (M ). Y ] = LX Y is called the Lie bracket of X and Y . viewed as a derivation of C ∞ (M ). i. Proof. Assume for simplicity that it is defined for all t. • denotes the natural duality between T ∗ M and T M . Lemma 3. In particular.1.e. Above. • : C ∞ (T ∗ M ) × C ∞ (T M ) → C ∞ (M ).10. Y ]f. we will provide more explicit descriptions of this operation. We will work in local coordinates (xi ) near a point m ∈ M so that X = Xi 1 ∂ ∂ and Y = Y j . Y ∈ Vect (M ). For any X ∈ Vect (M ) and f ∈ C ∞ (M ) we have Xf := LX f = df. Proof. Lemma 3. Prove that any derivation of the algebra C ∞ (M ) is of the form LX for some X ∈ Vect(M ). We place an observer (fisherman) at a fixed point p ∈ M . Then the Lie derivative of Y along X is a new vector field LX Y which.e. The map Φt acts on C ∞ (M ) by the pullback of its inverse.e. X = df (X ). LX Y f = [X.1. LX S measures how fast is the flow Φ changing1 the tensor S . •.1. −t ∗ Φt ∗ = (Φ ) . for point p ∈ M we have 1 d LX f (p) = lim (f (p) − f (Φ−t p)) = − |t=0 f (Φ−t p) = df.1.8. If the limit in (3.3. . This concludes the proof of the lemma. ∂xk (3. ∂xi : Tγ (0) M → Tγ (−t) M is the linearization of the map j Yγj(−t) = Ym − tX i i 2 (xi ) → xi 0 − tX + O (t ) . ∂ ] = 0. using the geometric series ( − A)−1 =  + A + A2 + · · · . CALCULUS ON MANIFOLDS We first describe the commutator [X. we deduce that the differential −t −1 Φt : Tγ (−t) M → Tγ (0) M.j (3.. the basic vectors ∂x commute as derivations.1. and t Note that Φ− ∗ ∂Y j + O(t2 ).1. so that the commutator of the two derivations is the derivation defined by the vector field [X.1. i ∂xj i So far we have not proved the vector field in (3. i. ∗ = (Φ∗ ) has the matrix form Φt ∗ =+t Using (3.4) is independent of coordinates.e.7) in (3. Y ]f = (X i = X iY j ∂ ∂f ∂ ∂f )(Y j j ) − (Y j )(X i i ) ∂xi ∂x ∂xj ∂x − X iY j i ∂2f j ∂X ∂f + Y ∂xi ∂xj ∂xj ∂xi j ∂2f i ∂Y ∂f + X ∂xi ∂xj ∂xi ∂xj .6) In particular.80 CHAPTER 3.4) ∂ ∂ Note in particular that [ ∂x . (3.1.7) = t Xi k ∂Y k j ∂X − Y ∂xi ∂xj + O(t2 ). We will achieve this by identifying it with the intrinsically defined vector field LX Y . Y ] = Xi k ∂Y k j ∂X − Y ∂xi ∂xj ∂ . then [X.1. If f ∈ C ∞ (M ). where A is a matrix of operator norm strictly less than 1.1.5) so it has a matrix representation t Φ− ∗ =−t ∂X i ∂xj + O(t2 ). Y ].5) we deduce k Ym − Φt ∗ YΦ−t m k ∂X i ∂xj + O(t2 ). i.1.j (3. Then i 2 Φ−t m = γ (−t) = γ (0) − γ ˙ (0)t + O(t2 ) = xi 0 − tX + O (t ) . Set γ (t) = Φt m so that we have a parametrization γ (t) = (xi (t)) with x ˙ i = X i . i. 3) we deduce that LX (S ⊗ T ) exists. We have Φt ∗ ω = (Φ ) ω . Fix a point p ∈ M . for any p ∈ M . we have X · ω (Y ) = We deduce that X · ω (Y ) − (LX ω )Y = i. Y ]p ).e.1. We can now completely describe the Lie derivative on the algebra of tensor fields.j j ωi (0)X0 ∂Y i .9) Since any tensor field is locally a linear combination of tensor monomials of the form X1 ⊗ · · · ⊗ Xr ⊗ ω1 ⊗ · · · ⊗ ωs . and any smooth tensor field S .11.1. and any vector fields X. Denote by Φt the local flow generated by X . ∂xi i = X i (p). Y ]). and any Y ∈ Vect (M ). X = i Xi ∂ .j ωi (0) One the other hand. and LX (S ⊗ T ) = LX S ⊗ T + S ⊗ LX T. we have −t (Φt ∗ ω )p (Yp ) = ωΦ−t p Φ∗ Yp . ωj ∈ Ω1 (M ). (3. we deduce that the Lie derivative LX S exists for every X ∈ Vect (M ). Y = ∂xi Yi i ∂ .. (3. −t ∗ Proof. . then using (3. ∂xj ∂Y i ∂X i j − Y ∂xj ∂xj 0 = ωp ([X. ∂xj 0 ∂X i j Y .1. X0 0 Using (3. i.j j ωi (0) X0 d |t=0 dt ωi (t)Y i (t) = i i ω ˙ i (0)Y0i + i. T are two tensor fields on M such that both LX S and LX T exist. ∂xj 0 Hence −(LX ω )Y = d |t=0 (Φt ∗ ω )p (Yp ) = − dt ω ˙ i (0)Y0i − i i. We set ωi (t) = ωi (γ (t)). Denote by γ (t) the path t → Φt (p).1.1.1. For any differential form ω ∈ Ω1 (M ). Xi ∈ Vect (M ). Then ω= i ω i dxi .3. Observe that if S. Y ∈ V ect(M ) we have (LX ω )(Y ) = LX (ω (Y )) − ω ([X.6) we deduce (Φt ∗ ω )p (Yp ) = i ωi (−t) · Y0i − t j ∂X i j Y + O(t2 ) . THE LIE DERIVATIVE 81 Lemma 3.8) where X · ω (Y ) denotes the (Lie) derivative of the function ω (Y ) along the vector field X . and Y i = Y i (p). and choose local coordinates (xi ) near p. 8). Y ]. LX is a natural operation. for any diffeomorphism φ : M → N we have φ∗ ◦ LX = Lφ∗ X ◦ φ∗ . ∀f ∈ C (M ). LY ]Z = L[X. r+1 r (c) LX commutes with the contraction tr : Ts +1 (M ) → Ts (M ). [LX . As for part (c). Corollary 3. The commutator [LX .1.Y ] on C ∞ (M ). Let X be a vector field on the smooth manifold M . The naturality of LX is another way of phrasing the coordinate independence of this operation.Y ] .e.9).1. LY ] is a derivation (as a commutator of derivations). Proof. Exercise 3. Moreover.. ∀Z ∈ Vect (M ).14.9). X2 ]. S2 ]). [S1 . LY ] = L[X. The reader can check easily that the general case of property (c) follow from this observation coupled with the product rule (3. X = Xf . φ∗ (LX ) = Lφ∗ X . (b) LX Y = [X. LY ] = L[X. S2 )] = D([X1 . as derivations of the algebra of tensor fields on M . since the contraction commutes with both LX and LY it obviously commutes with LX LY − LY LX . (3. and ω ∈ Ω1 (M ). so that [LX . We leave the reader to fill in the routine details.12. [D(X1 .. the equality LX tr T = tr LX T is equivalent to LX (ω (Y )) = (LX ω )(Y ) + ω (LX (Y ).82 CHAPTER 3.Y ] on Vect (M ). By Lemma 3. Properties (a) and (b) were proved above.1. a simple computation shows that [LX .e. S1 ).1. For any X. Finally.1. its restriction to C ∞ (M ) ⊕ Vect (M ) ⊕ Ω1 (M ) uniquely determines the action on the entire algebra of tensor fields which is generated by the above subspace. where Y ∈ Vect (M ). S ) = LX + S is well defined and is a linear isomorphism. The fact that LX is a derivation follows from (3. D(X2 .10) which is precisely (3. Also. this says that Vect (M ) ∗ (M ) is a Lie subalgebra of the Lie algebra of derivations. Since LX is a derivation of the algebra of tensor fields. ∀X ∈ Vect (M ).10. The corollary is proved. derivative LX is the unique derivation of T∗ ∞ (a) LX f = df. as a space of derivations of T∗ Proof. Moreover. ∀X. i. . CALCULUS ON MANIFOLDS Proposition 3. Then the Lie ∗ (M ) with the following properties.13. Prove that the map ∗ D : Vect (M ) ⊕ End (T M ) → Der(T∗ (M )) given by D(X. LY ] = L[X.1. in its simplest form. Y ∈ Vect (M ) we have [LX .1. i. In particular.1. Y ∈ Vect (M ). when T = Y ⊗ ω .Y ] Z. 3 Examples ∂ Example 3. LX (ω ∧ η ) = (LX ω ) ∧ η + ω ∧ (LX η ). Let M be a smooth manifold.16. Proof. Prove that if p ∈ M .1. ∀s.1.2. X (p) = 0 = p ∈ M .1. LX ω = In particular. Y ] = 0.e. ∀ω. Y (p) = 0 . Let ω = ωi dxi be a 1-form on Rn . The statement in the proposition follows immediately from the straightforward observation that the Lie derivative commutes with this operator (which is a projector). We leave the reader to fill in the details. THE LIE DERIVATIVE ∗ with the remarkable property LX is a derivation of T∗ 83 LX (Ω∗ (M )) ⊂ Ω∗ (M ). The wedge product makes Ω∗ (M ) a s-algebra.1. if X = ∂xi = j δ ij ∂xj . ∂X i ∂ ∂xk ∂xi . and it is natural to ask whether LX is a s-derivation with respect to this product. the two flows commute if and only if [X. If X = X j ∂x j is a vector field on n k R then LX ω = (LX ω )k dx is defined by (LX ω )k = (LX ω )( Hence ∂ ∂ ∂ ) = Xω ( ) − ω (LX ) = X · ωk + ω ∂xk ∂xk ∂xk Xj ∂ωk ∂X j + ω j ∂xj ∂xk dxk . .17. Ψ : R × M → M are two smooth flows on M with infinitesimal generators X and respectively Y .1. As in Subsection 2. Proposition 3. §3. and suppose that Φ. Exercise 3. ∂xi If X is the radial vector field X = ix i∂ .15. t ∈ R. then n LX ω = L∂xi ω = k=1 ∂ωk k dx . denote by A the anti-symmetrization operator A : (T ∗ M )⊗k → Ωk (M ). xi then LX ω = k (X · ωk + ωk )dxk .3. i. The Lie derivative along a vector field X is an even s-derivation of Ω∗ (M ). We say that the two flows commute if Φt ◦ Ψs = Ψs ◦ Ψt . η ∈ Ω∗ (M ).2. where dv is the volume form on R3 . the local flow generated by X preserves the form dv . Using the computation in the previous example we get LX (dx) = dF := so that LX (dv ) = ∂F ∂F ∂F dx + dy + dz. Y ] = [(Lg )∗ X. LX (dy ) = dG. Hence LG is a Lie subalgebra of Vect (G). A tensor field T on G is called left (respectively right ) invariant if for any g ∈ G (Lg )∗ T = T (respectively (Rg )∗ T = T ). X → exp(X ) called the exponential map of the group G. Y ∈ LG . [X. . dv = dx ∧ dy ∧ dz . and the right translation (Rg ) on G. We will get a better understanding of this statement once we learn integration on manifolds. Φt X = Rexp (tX ) . later in this chapter.1.19. (The exponential map of a Lie group). LX (dz ) = dH ∂x ∂y ∂z ∂F ∂G ∂H + + ∂x ∂y ∂z dv = (div X )dv. Since LX is an even s-derivation of Ω∗ (M ). Rg (h) = h · g. Any X ∈ LG defines a local flow Φt X on G that is is defined for all t ∈ R. so that ∀X. (Lg )∗ Y ]. Lg (h) = g · hy. We thus get a map exp : T1 G ∼ = LG → G. Consider a Lie group G. In particular. Fact 1. Φt X (g ) = g · exp (tX ).e.18.1. we deduce that if div X = 0. In other wors. Fact 3. dim LG = dim G. CALCULUS ON MANIFOLDS ∂ ∂ ∂ Example 3. i. We want to compute LX dv . Y ] ∈ LG . Consider a smooth vector field X = F ∂x + G ∂y + H ∂z on R3 . the left invariance implies that the restriction map LG → T1 G. Fact 2.. It is called called the Lie algebra of the group G.84 CHAPTER 3. ( Exercise ) Set X exp(tX ) : =Φt X (1). Any element g ∈ G defines two diffeomorphisms of G: the left (Lg ). We will often find it convenient to identify the Lie algebra of G with the tangent space at 1. The set of left invariant vector fields on G is denoted by LG . X → X1 is an isomorphism (Exercise ). Indeed. Example 3. The naturality of the Lie bracket implies (Lg )∗ [X. we deduce LX (dx ∧ dy ∧ dz ) = (LX dx) ∧ dy ∧ dz + dx ∧ (LX dy ) ∧ dz + dx ∧ dy ∧ (LX dz ). ∀h ∈ G. Φt is a flow. K). C ). (d) Prove that we have the following isomorphisms of real Lie algebras. C). dt We can write (g · exp(tX )) = Lg exp(tX ) so that d d |t=0 (Lg exp(tX )) = (Lg )∗ |t=0 exp(tX ) dt dt = (Lg )∗ X = Xg . show that u(n). Exercise 3. Y ∈ T1 G.3. K). This means the tangent space T1 G can be identified with a linear space of matrices. (left invariance).1. K) of all n × n matrices with the bracket given by the commutator of two matrices. C) are real Lie subalgebras of gl(n..21. ∼ ∼ LU (n) ∼ u( n ). . su(n) ⊂ gl(n. then the Lie algebra of G is the Lie algebra gl(n. and denote by exp(tX ) and exp(tY ) the 1-parameter groups with they generate.1. Y ]alg (temporarily) denotes the commutator of the two matrices X and Y . where the bracket [X. Show that at 1 ∈ G [X. (a) Show that gs. THE LIE DERIVATIVE Indeed. Prove the statements left as exercises in the example above.2) is a Lie subalgebra with respect to the commutator [ · . i. C).20. · ]. the traditional equality exp(tX ) exp(tY ) = exp(t(X + Y )) no longer holds. Y ]geom the Lie bracket of X and Y viewed as left invariant vector fields on G. (c) Show that o(n) ⊂ gl(n. and set g (s. k! Exercise 3. Y ) + · · · . one has the Campbell-Hausdorff formula exp(tX ) · exp(tY ) = exp td1 (X. while su(n. LO(n) ∼ = o(n). Y ) + t3 d3 (X. t) = exp(sX ) exp(tY ) exp(−sX ) exp(−tY ). In general.t = 1 + [X. Instead.1. Let G be a matrix Lie group.1. L su( n ) and L sl( n. and for any X ∈ LG we have (Exercise ) exp (X ) = eX = k ≥0 1 k X .e.2. Let X. (b) Denote (temporarily) by [X. it suffices to check that d |t=0 (g · exp (tX )) = Xg .22. = SU (n) = SL(n.C) = Remark 3. Similarly. Y ]alg st + O((s2 + t2 )3/2 ) as s. 85 The reason for the notation exp (tX ) is that when G = GL(n. in a non-commutative matrix Lie group G. R) (defined in Section 1. t → 0. Y ]geom . a Lie subgroup of some general linear group GL(N. Y ]alg = [X. C) is even a complex Lie subalgebra of gl(n. Y ) + t2 d2 (X. Y ]] + [Y. . d2 (X. More precisely. an odd derivation iX . The above s-derivations by no means exhaust the space of s-derivations of Ω• (M ). Xr−1 ) = ω (X. ∀X1 .2.2. Y ]. . Y ) = [X. Proposition 3. 1 d1 (X. . (b) The super-commutator of LX and iY as s-derivations of Ω∗ (M ) is given by [LX . The fact that iX is an odd s-derivation is equivalent to iX (ω ∧ η ) = (iX ω ) ∧ η + (−1)deg ω ω ∧ (iX η ). The proof uses the fact that the Lie derivative commutes with the contraction operator. This section is devoted precisely to these new features. 12 For more details we refer to [41. . . X ]]) etc. 2 1 ([X. df coincides with the differential of f . via the contraction map. Prove that the interior derivation along a vector field is a s-derivation. Xr−1 ). The vector field X also defines. . Y ) = 3. ∀ω ∈ Ωr (M ). or the contraction by X . The Lie derivative along a vector field X defines an even derivation in Ω• (M ). Xr−1 ∈ Vect (M ). and it is left to the reader as an exercise.3. its space of derivations has special features.1. Exercise 3. iX ω is the (r − 1)-form determined by (iX ω )(X1 . Y ) = X + Y. η ∈ Ω∗ (M ).2. [X. iY ]s = LX iY − iY LX = i[X. . CALCULUS ON MANIFOLDS where dk are homogeneous polynomials of degree k in X .86 CHAPTER 3. ∀ω. In fact we have the following fundamental result. called the interior derivation along X .2. d3 (X. .2.1 The exterior derivative The super-algebra of exterior forms on a smooth manifold M has additional structure. The dk ’s are usually known as Dynkin polynomials. and Y with respect to the multiplication between X and Y given by their bracket. (a) For any smooth function f ∈ Ω0 (M ). X1 . . Proposition 3.2 Derivations of Ω∗ (M ) §3. iX ω : =tr (X ⊗ ω ). .Y ] . . . . [Y. Often the contraction by X is denoted by X . There exists an odd s-derivation d on the s-algebra of differential forms Ω• ( · ) uniquely characterized by the following conditions. 84]. (a) [iX . iY ]s = iX iY + iY iX = 0. For example. and in particular. while over V we have ω= 1≤j1 <···<jr ≤n ωj1 .ir . Then.. and d(dxi ) = 0. y n ) the local coordinates along V ..jr dy j1 ∧ · · · ∧ dy jr ∂ ωj1 .. = 1≤i1 <···<ir ≤n (3. Since d is an s-derivation.. and for any form ω on M . over U dω = 1≤i1 <···<ir ≤n (dωi1 . ∧ dxir 1≤i1 <···<ir ≤n dω = 1≤i1 <···<ir ≤n ∂ωi1 . if U . so that on the overlap U ∩ V we can describe the y ’s as functions of the x’s. DERIVATIONS OF Ω∗ (M ) (b) d2 = 0. and this completes the proof of the uniqueness of d. Over U we have ω= ωi1 .. ... we have dφ∗ ω = φ∗ dω (⇐⇒ [φ∗ ..ir dxi1 ∧ . To prove that this is a well defined operation we must show that. i... ∧ dxir ).. Uniqueness. .. 87 (c) d is natural.jr are skew-symmetric.. xn )... For each coordinate neighborhood U we define dω |U as in (3. d] = 0). ∀σ ∈ Sr ωiσ(1) .... Denote by (x1 .. the form dω is uniquely determined on any coordinate neighborhood. The derivation d is called the exterior derivative....1). Let U be a local coordinate chart on M n with local coordinates (x1 .. over U .. Consider an r-form ω . i.e... . .. and by (y 1 ..ir ) ∧ (dxi1 ∧ · · · ∧ dxir ) ∂ωi1 .e. xn ) the local coordinates on U .2. Proof...jr j dy (dy j1 ∧ · · · ∧ dy jr ). for any smooth function φ : N → M .ir dxi1 ∧ . we deduce that.ir i dx ∂xi ∧ (dxi1 ∧ · · · ∧ dxir ). V are two coordinate neighborhoods.iσ(r) = (σ )ωi1 . Existence. then dω |U = dω |V on U ∩ V. .1) Thus. ∧ dxir ... .ir i dx ∂xi ∧ (dxi1 ∧ .2.2.ir and ωj1 . ..3.. ∂y j dω = 1≤j1 <···<jr ≤n The components ωi1 ... any r-form ω can be described as ω= 1≤i1 <···<ir ≤n ωi1 . .......ir = ∂xi r k=1 ∂y j1 ∂y jr · · · ωj ... Let θ = f dxi1 ∧ · · · ∧ dxir and ω = gdxj1 ∧ · · · ∧ dxjs ...j .. .2... ∧ dxir ∂xi 1≤i1 <···<ir ≤n r = i k=1 ∂y j1 ∂ 2 xjk ∂y jr · · · · · · ωj ......jr i dx ∂xi ∧ ∂y j1 i1 dx ∂xi1 ∧ ··· ∧ ∂y jr ir dx ∂xir = (dωj1 .jr j dy ∧ dy j1 ∧ · · · ∧ dy jr .jr ∂y j1 ∂y jr ∂y j1 · · · · · · · · · ω + j ... Consequently on U ∩ V ∂ωi1 .ir i dx ∧ dxi1 ∧ .. jr according to Einstein’s convention.jr i1 ir dx ∧ dx · · · ∧ dx = · · · ∧ dxi ∧ dxi1 · · · dxir ∂xi ∂xi1 ∂xir ∂xi = ∂ ωj1 .ir i ∂y j1 ∂y jr ∂ ωj1 . We deduce ∂ωi1 ... ∧ dxir ∂xi1 ∂xi ∂xik ∂xir 1 r r + i k=1 ∂y j1 ∂y jr ∂ ωj1 .. ∧ dxir . ∂xi1 ∂xir ∂xi ∂2 ∂2 = ..2) vanishes. To prove that d is an odd s-derivation it suffices to work in local coordinates and show that the (super)product rule on monomials. We have thus constructed a well defined linear map d : Ω• (M ) → Ω•+1 (M )..2.. Then d(θ ∧ ω ) = d(f gdxI ∧ dxJ ) = d(f g ) ∧ dxI ∧ dxJ .. where in the above equality we also sum over the indices j1 ...jr ) ∧ dy j1 ∧ · · · ∧ dy jr = ∂ ωj1 .j r 1 i i i i i ∂x 1 ∂x ∂x k ∂x r ∂x 1 ∂xir ∂xi . ∂xi ∂xik ∂xik ∂xi (3. ∂y j This proves dω |U = dω |V over U ∩ V .2) Notice that while dxi ∧ dxik = −dxik ∧ dxi so that the first term in the right hand side of (3..ir = Hence ∂ωi1 ... CALCULUS ON MANIFOLDS and similarly for the ω s.88 CHAPTER 3. ∂xi1 ∂xir 1 r ∂ 2 y jk ∂y jr ∂ ωj1 . Since ω |U = ω |V over U ∩ V we deduce ωi1 .jr i · · · dx ∧ dxi1 ∧ .j dxi ∧ dxi1 ∧ . We set for simplicity dxI := dxi1 ∧ · · · ∧ dxir and dxJ := dxj1 ∧ · · · ∧ dxjs .. X Above... (−1)i+j ω ([Xi . (b) (Cartan’s homotopy formula) [d. the usual chain rule gives dN (φ∗ ωI ) = φ∗ (dM ωI ). . The proposition is proved. X0 . We check this on monomials f dxI as above... We now prove d2 = 0. d2 (f dxI ) = d(df ∧ dxI ) = (d2 f ) ∧ dxI . ∂xi ∂xj 89 The desired conclusion follows from the identities ∂f 2 ∂f 2 = and dxi ∧ dxj = −dxj ∧ dxi .. We have d2 f = ∂f 2 dxi ∧ dxj . Xr ) = i=0 ˆ i . i j ∂x ∂x ∂xj ∂xi Finally. Proposition 3... ]s denotes the super-commutator in the s-algebra of real endomorphisms of Ω• (M ).. dN (dφI ) = 0. For functions.2. In particular. (c) [d.. . d]s = 2d2 = 0. .3) + 0≤i<j ≤r ˆ i . An immediate consequence of the homotopy formula is the following invariant description of the exterior derivative: r (dω )(X0 .2..2. .. Xr )..4. . We put all the above together and we deduce dN (φ∗ ω ) = φ∗ (dM ω I ) ∧ dφI = φ∗ (dM ω I ) ∧ φ∗ dxI = φ∗ (dM ω ). iX ]s = diX + iX d = LX ∀X ∈ Vect (M ). X (3. it suffices to show d2 f = 0 for all smooth functions f .. the hat indicates that the corresponding entry is missing. DERIVATIONS OF Ω∗ (M ) = (df · g + f · dg ) ∧ dxI ∧ dxJ = df ∧ dxI ∧ dxJ + (−1)r (f ∧ dxI ) ∧ (dg ∧ dxJ ) = dθ ∧ ω + (−1)deg θ θ ∧ dω. Here I runs through all ordered multi-indices 1 ≤ i1 < · · · < ir ≤ dim M .. Xj ]. The exterior derivative satisfies the following relations. LX ]s = dLX − LX d = 0. We have dN (φ∗ ω ) = I dN (φ∗ ωI ) ∧ φ∗ (dxI ) + φ∗ ω I ∧ d(φ∗ dxI ) . ∀X ∈ Vect (M ). and [ . .. X ˆ j . (a) [d. let φ be a smooth map N → M and ω = I ωI dxI be an r-form on M .3.. In terms of local coordinates (xi ) the map φ looks like a collection of n functions φi ∈ C ∞ (N ) and we get φ∗ (dxI ) = dφI = dN φi1 ∧ · · · ∧ dN φir .. Xr )) (−1)i Xi (ω (X0 . . . Thus. X1 . X ). Part (c) of the proposition is proved in a similar way.5) (3. We prove it in two special case r = 1 and r = 2.8) The general case in (3. It is a local s-derivation. D be two local s-derivations of Ω• (M ) which have the same parity.. Y ) −ω ([X.2. If D = D on Ω0 (M ) ⊕ Ω1 (M ). Z ) − Y ω (X. then D = D on Ω• (M ). Z ]). We deduce from the homotopy formula (dω )(X. Consider a 2-form ω and three vector fields X .2. Z ) − Zω (X. The homotopy formula is now a consequence of the following technical result left to the reader as an exercise. Y. D is an even s-derivation of Ω∗ (M ). they are either both even or both odd.3) in the case r = 1. Y. we deduce Xω (Y ) = LX (ω (Y )) = (LX ω )(Y ) + ω ([X. Y ].4) and we get (dω )(X.2. Z ]. Y ]) − (dω (X ))(Y ) = Xω (Y ) − Y ω (X ) − ω ([X. Hence dω (X.6) (3. Y ) − ω (X. X ) = X (ω (Y. Y ]). Z ) = Xω (Y. Z )) − ω ([X. Since LX commutes with contractions we deduce (LX ω )(Y. (3. Y. Z ]. Z ) + ω ([X. if ω ∈ Ω∗ (M ) vanishes on some open set U then Dω vanishes on that open set as well.6) we deduce (dω )(X. [X..2.2.2.3) can be proved by induction. Y ) = (iX dω )(Y ) = (LX ω )(Y ) − (dω (X ))(Y ). Z ) − ω (Y. X ).5. We apply now (3. The reader can check easily by direct computation that Dω = LX ω . CALCULUS ON MANIFOLDS Proof.2. Y ) = Xω (Y ) − ω ([X. [X. [Y.3) is a simple consequence of the homotopy formula.2.2. Z ) − ω (Y.7) (3. Z ) + Zω (X. X ) = Y (iX ω (Z )) − Z (iX ω (Y )) − (iX ω )([Y.2.7) in (3. Lemma 3.e.4) . ∀ω ∈ Ω0 (M ) ⊕ Ω1 (M ).2.5) into (3. Y and Z .3) for r = 1 to the 1-form iX ω . Y ) − ω ([Y. (3. Equality (3. The proof of the proposition is complete.e.2.2. We deduce from the homotopy formula dω (X. Let D.2. To prove the homotopy formula set D := [d. The case r = 1. Y ]. Z ) = (LX − diX )ω (Y. Y ]). since LX commutes with the contraction operator.90 CHAPTER 3. Z ). i. Z ]) − d(iX ω )(Y. We substitute (3. Z ]). Y ∈ Vect (M ). iX ]s = diX + iX d. Let ω be an 1-form and let X. i. On the other hand. We get d(iX ω )(Y. Z ) = X (ω (Y. Z ]) = Y ω (X. Z )) − ω ([X. This proves (3. The case r = 2. If we use (3. Z ) = (iX dω )(Y. Y ]. Xj ].8...2. Y ∈ LG we have (see (3.2. For X. . Xr ) = 0≤i<j ≤r ˆ i . 1 ∗ Indeed.. Y ]).1. Since ω .2. Then dω = dP ∧ dx + dQ ∧ dy + dR ∧ dz ∂R ∂Q ∂P ∂R ∂Q ∂P − dx ∧ dy + − dy ∧ dz + − ∂x ∂y ∂y ∂z ∂z ∂x so that dω looks very much like a curl. X1 . the scalars ω (X ) and ω (Y ) are constants.2.. ∗ ∼ ∗ ∼ 1 In particular.. X (3..3) imply that for all X0 . Let G be a connected Lie group. This looks very much like a divergence. Example 3.. ω = P dx + Qdy + Rdz .3) in the general case. . Xr ). (c) Let ω = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy ∈ Ω2 (R3 ).3) ) dω (X. DERIVATIONS OF Ω∗ (M ) Exercise 3... .2. Finish the proof of (3. X1 . Xr ∈ LG we have dω (X0 . In Example 3. (The exterior derivative in R3 ). X and Y are left invariant. Y ) = Xω (Y ) − Y ω (X ) − ω ([X. Exercise 3. §3. If we identify LG = T1 G.2 Examples 91 Example 3.9. (a) Let f ∈ C ∞ (R3 ). The exterior derivative of a form in Ω∗ lef t can be described only in terms of the algebraic structure of LG . Set Ωr lef t (G) = left invariant r -forms on G.. then we get a natural isomorphism ∼ r ∗ Ωr lef t (G) = Λ LG . Then = dω = ∂P ∂Q ∂R + + ∂x ∂y ∂z dx ∧ dy ∧ dz. ∂x ∂y ∂z The differential df looks like the gradient of f . dz ∧ dx.9) . (b) Let ω ∈ Ω1 (R3 ). Y ]). Then df = ∂f ∂f ∂f dx + dy + dz. Prove Lemma 3.6. L∗ G = Ωlef t (G).. let ω ∈ L∗ G = Ωlef t (G). X ˆ j .. Y ) = −ω ([X.11 we defined the Lie algebra LG of G as the space of left invariant vector fields on G.2.2... More generally.3.2.5. then the same arguments applied to (3.7. Thus.2. if ω ∈ Ωr lef t . the first two terms in the above equality vanish so that dω (X. .2.2.. . (−1)i+j ω ([Xi . u) → ∇X u. ∀f ∈ C ∞ (M ). Technically. For such an arbitrary E . ∀f ∈ C ∞ (M ) : ∇f X u = f ∇X u. and ∀u ∈ C ∞ (E ). we expect that after “rescaling” the direction X the derivative along X should only rescale by the same factor. Hence it has to be an operator ∇ : Vect (M ) × C ∞ (E ) → C ∞ (E ). one would like to have a procedure of measuring the rate of change of a section u of E along a direction described by a vector field X . such that.1. . However. is C ∞ (M )-linear so that it defines a bundle map ∇u ∈ Hom (T M. E ) ∼ = C ∞ (T ∗ M ⊗ E ). For tensor fields. we encounter a problem which was not present in the case of tensor bundles. (b) If we think of the usual directional derivative. The conditions (a) and (b) can be rephrased as follows: for any u ∈ C ∞ (E ). Summarizing. the local flow generated by the vector field X on M no longer induces bundle homomorphisms. and very often one is interested in measuring the “oscillations” of sections of vector bundles. C).e. the tensor bundles are not the only vector bundles arising in geometry. the map ∇u : Vect (M ) → C ∞ (E ). CALCULUS ON MANIFOLDS Connections on vector bundles §3.3 CHAPTER 3. i. To obtain something that looks like a derivation we need to formulate clearly what properties we should expect from such an operation.3. Definition 3. the transport along a flow was a method of comparing objects in different fibers which otherwise are abstract linear spaces with no natural relationship between them. Namely.92 3. we can formulate the following definition. (a) It should measure how fast is a given section changing along a direction given by a vector field X . ∀f ∈ C ∞ (M ). (X. Hence we require ∇X (f u) = (Xf )u + f ∇X u. we have ∇(f u) = df ⊗ u + f ∇u.3. A covariant derivative (or linear connection ) on E is a K-linear map ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ). u ∈ C ∞ (E ). it has to satisfy a sort of (Leibniz) product rule. Let E be a K-vector bundle over the smooth manifold M (K = R. X → ∇X u. (c) Since ∇ is to be a derivation.1 Covariant derivatives We learned several methods of differentiating tensor objects on manifolds.. The only product that exists on an abstract vector bundle is the multiplication of a section with a smooth function. pick a smooth partition of unity (µβ ) subordinated to this cover.. The space C ∞ (KM ) of smooth sections coincides with the space C ∞ (M. Now define ∇ := α. Denote such a connection by ∇α . The space A(E ) of linear connections on E is an affine space modeled on Ω1 (End (E )). Notation. Kr ) → C ∞ (M. Conversely. it admits at least one connection. T ∗ M ⊗ E )) ∼ = C ∞ (T ∗ M ⊗ E ∗ ⊗ E ) ∼ = Ω1 (E ∗ ⊗ E ) ∼ = Ω1 (EndE ). is a linear connection.3. Let Kr M = K × M be the rank r trivial vector bundle over M . To check that A (E ) is an affine space. . the trivial one. We will refer to these sections as differential k -forms with coefficients in the vector bundle F. A ∈ C ∞ (Hom (E. To see this. as in the above example. Thus. dfr ). Remark 3.3. Proposition 3. For any vector bundle F over M we set Ωk (F ) : =C ∞ (Λk T ∗ M ⊗ F ). ∀u ∈ C ∞ (E ). consider two connections ∇0 and ∇1 . ∇1 be two connections on a vector bundle E → M . CONNECTIONS ON VECTOR BUNDLES 93 ∼ r Example 3. given ∇0 ∈ A(E ) and A ∈ Ω1 (EndE ) one can verify that the operator ∇A = ∇0 + A : C ∞ (E ) → Ω1 (E ).2.β µβ ∇α . Kr ) of Kr -valued smooth functions on M .. We first prove that A (E ) is not empty. satisfying A(f u) = f A(u). We can define ∇0 : C ∞ (M.. Next. One checks easily that ∇ is a connection. This is called the trivial connection. Proof.3.3. Then for any α ∈ C ∞ (M ) the map ∇ = α∇1 + (1 − α)∇0 : C ∞ (E ) → C ∞ (T ∗ ⊗ E ) is again a connection. Since E |Uα is trivial. This concludes the proof of the proposition.4. . T ∗ M ⊗ Kr ) ∇0 (f1 .3. .3.. Let E be a vector bundle.. Their difference A = ∇1 − ∇0 is an operator A : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ). Let ∇0 . One checks easily that ∇ is a connection so that A(E ) is nonempty.. fr ) = (df1 . choose an open cover {Uα } of M such that E |Uα is trivial ∀α. α = 1. (3. Consider the sections eα = φ(δα ). CALCULUS ON MANIFOLDS The tensorial operations on vector bundles extend naturally to vector bundles with connections. K). • : C ∞ (E1 ) × C ∞ (E1 ) → C ∞ (M ) ∗ and E . er (x)) describes a basis of the moving fiber Ex . X ∈ Vect (M ). is the pairing induced by the natural duality between the fibers of E1 1 any connection ∇E on a vector bundle E induces a connection ∇End(E ) on End (E ) ∼ = E ∗ ⊗ E by (∇End(E ) T )(u) = ∇E (T u) − T (∇E u) = [∇E . ∀u ∈ C (E1 ).3. u + v. .1) ∀T ∈ End (E ).2) It is convenient to view Γβ iα as an r × r -matrix valued 1-form. The guiding principle behind this fact is the product formula. for any section uα eα of E |U we have α i ∇u = duα ⊗ eα + Γβ iα u dx ⊗ eβ . ∗ has a natural connection ∇∗ defined by the identity The dual bundle E1 1 ∞ ∞ ∗ X v. It is often useful to have a local description of a covariant derivative. . u ∈ C ∞ (E ). ∇E1 ⊗E2 (u1 ⊗ u2 ) = (∇1 u1 ) ⊗ u2 + u1 ⊗ ∇2 u2 . whence the terminology moving frame... As x moves in U . K).. Let E → M be a K-vector bundle of rank r over the smooth manifold M . and we write this as i Γβ iα = dx ⊗ Γi .. ∇X u .3. if Ei (i = 1.. the covariant derivative is completely described by its action on a moving frame. A section u ∈ C ∞ (E |U ) can be written as a linear combination u = uα eα uα ∈ C ∞ (U. the collection (e1 (x). A moving frame 2 is a bundle isomorphism r φ : Kr U = K × M → E |U . In particular. u = ∇∗ X v. Pick a coordinate neighborhood U such E |U is trivial. 2) are two bundles with connections ∇i . Thus. then E1 ⊗ E2 has a naturally induced connection ∇E1 ⊗E2 uniquely determined by the product rule. if ∇ is a covariant derivative in E . . 2 A moving frame is what physicists call a choice of local gauge. T ]u. Thus. Γiα ∈ C (U. More precisely. (3. Then ∇eα ∈ 1 Ω (E |U ) so that we can write β i ∞ ∇eα = Γβ iα dx ⊗ eβ . pick local coordinates (xi ) over U . v ∈ C (E1 ). where δα are the natural basic sections of the trivial bundle Kr U . Hence. we have ∇u = duα ⊗ eα + uα ∇eα . where ∗ •.. To get a more concrete description.94 CHAPTER 3. This can be obtained using Cartan’s moving frame method. r. 3. .e. With respect to the local frame (eα ) the section σ has a decomposition σ = u α eα . i. or using the frame f . and we can rewrite (3.4) in the above equality we get ˆ u.3. ∇fα = Γ α Consider a section σ of E |U .3. Γ The above relation is the transition rule relating two local gauge descriptions of the same connection. K).2) as ∇u = du + Γu. Let Γ ˆ β fβ . the derivative ∇σ is identified with the Kr -valued 1-form du + Γu. (3.4): ˆ u).. ˆ denote the connection 1-form corresponding to the new moving frame. while in the second case it is identified with ˆ u.3. The two decompositions are related by u = gu ˆ. du + Γu = g (du ˆ + Γˆ Using (3.3. A moving frame allows us to identify sections of E |U with Kr -valued functions on U . du ˆ + Γˆ These two identifications are related by the same rule as in (3.3) A natural question arises: how does the connection 1-form changes with the change of the local gauge? Let f = (fα ) be another moving frame of E |U . while with respect to (fβ ) it has a decomposition σ=u ˆβ fβ . (dg )ˆ u + gdu ˆ + Γg u ˆ = gdu ˆ + g Γˆ Hence ˆ = g −1 dg + g −1 Γg.3. The correspondence eα → fα defines an automorphism of E |U . we can identify the E -valued 1-form ∇σ with a Kr -valued 1-form in two ways: either using the frame e. The map g is called the local gauge transformation relating e to f . Using the local frame e we can identify this correspondence with a smooth map g : U → GL(r.4) Now. In the first case. (3.3. CONNECTIONS ON VECTOR BUNDLES 95 The form Γ = dxi ⊗ Γi is called the connection 1-form associated to the choice of local gauge. 96 . (3. The standard basis in K . Exercise 3. transition maps gβα ◦ f and 1-forms f ∗ Γα . The above arguments can be reversed producing the following global result. ∼ ∼ = = More concretely. CHAPTER 3. Then any collection of matrix valued 1-forms Γα ∈ Ω1 (End Kr Uα ) satisfying −1 −1 −1 −1 Γβ = (gαβ dgαβ ) + gαβ Γα gαβ = −(dgβα )gβα + gβα Γα gβα over Uα ∩ Uβ . Suppose we are given the following data. We can use the local description in Proposition 3. These are like an object and its image in a mirror. if tα : Eα −→ Kr × Uα (respectively tβ : Eβ −→ Kr × Uβ ) is a trivialization of a bundle E over an open set Uα (respectively Uβ ).6.i .A word of warning. induces two local moving frames on E : −1 1 eα. Proposition 3. these data define a connection f ∗ ∇ on f ∗ E described by the open cover f −1 (Uα ). On the overlap Uα ∩ Uβ these two frames are related by the local gauge transformation −1 eβ.5). • A rank r K-vector bundle E → M defined by the open cover (Uα ). • A smooth map f : N → M .3.3.3.5. and transition maps gβα : Uα ∩ Uβ → GL(Kr ). Then. This is precisely the opposite way the two trivializations are identified.5) uniquely defines a covariant derivative on E .5 to define the notion of pullback of a connection. Prove the above proposition.i = t− α (δi ) and eβ.i = gβα eα. then the transition 1 r map “from α to β ” over Uα ∩ Uβ is gβα = tβ ◦ t− α . and (Uα ) a trivializing cover with transition maps gαβ : Uα ∩ Uβ → GL(r.3. This connection is independent of the various choices and it is called the pullback of ∇ by f . Let E → M be a rank r smooth vector bundle. • A connection ∇ on E defined by the 1-forms Γα ∈ Ω1 (End Kr Uα ) satisfying the gluing conditions (3.i = tβ (δi ). CALCULUS ON MANIFOLDS The identification {moving f rames} ∼ = {local trivialization} should be treated carefully.i = gαβ eα. denoted by (δi ). and there is a great chance of confusing the right hand with the left hand. K).3. ωβ = zαβ §3.6) becomes (using Einstein’s convention) duα β = 0 + Γα tβ u dt . 1]. The connection 1-form corresponding to this moving frame will be denoted by Γ ∈ Ω1 (End (Kr )). the space of connections on L. define Tt u0 as the value at t of the solution of the initial value problem ∇ d u(t) = 0 dt . let us make the simplifying assumption that γ (t) lies entirely in some coordinate neighborhood U with coordinates (x1 . ut should satisfy d = γ. C).3. xn )... u(0) = u0 (3. We will see in this subsection that such a procedure is all we need to define covariant derivatives.6) The equation (3. the main reason we could not construct natural derivations on the space of sections of a vector bundle was the lack of a canonical procedure of identifying fibers at different points. To see this.7) α u (0) = uα 0 .3. and any t ∈ [0.3. More exactly. the only way we know how to derivate sections is via ∇. . such that E |U is trivial. (3.3. A connection on L is simply a collection of C-valued 1-forms ω α on Uα related on overlaps by dzαβ + ω α = d log zαβ + ω α .3. if u0 ∈ Eγ (0) then the path t → ut = Tt u0 ∈ Eγ (t) should be thought of as a “constant” path. where dt dt The above equation suggests a way of defining Tt . The rigorous way of stating this “constancy” is via derivations: a quantity is “constant” if its derivatives are identically 0. Now. Let E → M be a rank r K-vector bundle and ∇ a covariant derivative on E . 1] → M we will define a linear isomorphism Tγ : Eγ (0) → Eγ (1) called the parallel transport along γ . End (L) ∼ = L∗ ⊗ L − 1 is trivial since it can be defined by transition maps (zαβ ) ⊗ zαβ = 1. ˙ ∇ d ut = 0. This is always happening.. CONNECTIONS ON VECTOR BUNDLES 97 Example 3.3.. One should think of this Tt as identifying different fibers. it offers a simple way of identifying different fibers.3. A (L) is an affine space modelled by the linear space of complex valued 1-forms. Let L → M be a complex line bundle over the smooth manifold M . (Complex line bundles). More precisely. Thus.2 Parallel transport As we have already pointed out.e. Let {Uα } be a trivializing cover with transition maps zαβ : Uα ∩ Uβ → C∗ = GL(1. For any smooth path γ : [0. Denote by (eα )1≤α≤r a local moving frame trivializing E |U so that u = uα eα . The bundle of endomorphisms of L. In particular. For any u0 ∈ Eγ (0) .7. at least on every small portion of γ .6) is a system of linear ordinary differential equations in disguise. i. Equation (3. we will construct an entire family of linear isomorphisms Tt : Eγ (0) → Eγ (t) .3. we will show that once a covariant derivative is chosen. η ∈ Ωs (E ) d∇ (ω ∧ η ) = dω ∧ η + (−1)r ω ∧ d∇ η. if γ (t) is an integral curve for X . then. t very small. .3. xn (t) .3. the contraction −iX Γ = −Γ(X ) ∈ End (E ) describes the infinitesimal parallel transport along the direction prescribed by the vector field X .10) with Γ0 = (iX Γ) |γ (0) . given a local moving frame for E in a neighborhood of γ (0).3. . Proposition 3. (a) d∇ |Ω0 (E ) = ∇. in the non-canonical identification of nearby fibers via a local moving frame. dt More explicitly. and Tt denotes the parallel transport along γ from Eγ (0) to Eγ (t) . Tt is identified with a t-dependent matrix which has a Taylor expansion of the form Tt =  − Γ0 t + O(t2 ). then Γt is the endomorphism given by Γ t eβ = x ˙ i Γα iβ eα . The system (3. (3.3 The curvature of a connection Consider a rank r smooth K-vector bundle E → M over the smooth manifold M .3. §3.3. We deduce u ˙ (0) = −Γt u0 . (3. . .7) can be rewritten as    duα dt + Γα ˙ i uβ = 0 iβ x uα (0) = uα 0 . (b) ∀ω ∈ Ωr (M ). The connection ∇ has a natural extension to an operator d∇ : Ωr (E ) → Ωr+1 (E ) uniquely defined by the requirements. if the path γ (t) is given by the smooth map Γt = t → γ (t) = x1 (t). .9) This gives a geometric interpretation for the connection 1-form Γ: for any vector field X . (3. and let ∇ : Ω0 (E ) → Ω1 (E ) be a covariant derivative on E .98 where CHAPTER 3.8) This is obviously a system of linear ordinary differential equations whose solutions exist for any t.3. CALCULUS ON MANIFOLDS d Γ=γ ˙ Γ ∈ Ω0 (End (Kr )) = End (Kr ). In more intuitive terms.8. In the former case we have d2 = 0. The sections of this bundle are smooth Kr -valued functions on M .3.3. Still. This coincides with the usual differential d : Ω0 (M ) ⊗ K → Ω1 (M ) ⊗ K. which is a consequence of the commutativity ∂ . B ∈ End (E ). the operator (d∇ )2 can be identified with a section of HomK (E. For any connection ∇ on a smooth vector bundle E → M .3. Example 3. Definition 3.11) so it has to coincide with the operator described above using a given partition of unity.the trivial connection. and any other connection differs from d by a Mr (K)-valued 1-form on M . Any operator with the properties (a) and (b) acts on monomials as in (3. For any smooth function f ∈ C ∞ (M ). Example 3. The 0 extension d∇ is the usual exterior derivative.3. u ∈ Ω0 (E ) set d∇ (ω ⊗ u) = dω ⊗ u + (−1)r ω ∇u. 99 (3. the equality [ ∂x i ∂xj (d∇ )2 = 0 does not hold in general.11) Using a partition of unity one shows that any η ∈ Ωr (E ) is a locally finite combination of monomials as above so the above definition induces an operator Ωr (E ) → Ωr+1 (E ). CONNECTIONS ON VECTOR BUNDLES Outline of the proof Existence. and it is usually denoted by F (∇). Consider the trivial bundle Kr M .3. something very interesting happens. Lemma 3. Kr ). We compute (d∇ )2 (f ω ) = d∇ (df ∧ ω + f d∇ ω ) = −df ∧ d∇ ω + df ∧ d∇ ω + f (d∇ )2 ω = f (d∇ )2 ω.11. Thus. In the second case. The ∧-operation above is defined for any vector bundle E as the bilinear map Ωr (End (E )) × Ωs (End (E )) → Ωr+s (End (E )).10. The trivial bundle KM has a natural connection ∇0 .3. As a map Ω0 (E ) → Ω2 (E ). then the curvature of the connection d + A is the 2-form F (A) defined by F (A)u = (d + A)2 u = (dA + A ∧ A)u. uniquely determined by (ω r ⊗ A) ∧ (η s ⊗ B ) = ω r ∧ η s ⊗ AB. We let the reader check that this extension satisfies conditions (a) and (b) above. where (xi ) are local coordinates on M . Uniqueness.3. If A is such a form. The exterior derivative d defines the trivial connection on Kr M. (d∇ )2 (f ω ) = f {(d∇ )2 )ω }.3. Λ2 T ∗ M ⊗R E ) ∼ = E ∗ ⊗ Λ2 T ∗ M ⊗R E ∼ = Λ2 T ∗ M ⊗R EndK (E ). Proof. ∀u ∈ C ∞ (M. and an arbitrary d∇ . Hence (d∇ )2 is a bundle morphism Λr T ∗ M ⊗ E → Λr+2 T ∗ M ⊗ E . There is a major difference between the usual exterior derivative d. For ω ∈ Ωr (M ). (d∇ )2 is an EndK (E )-valued 2-form. and any ω ∈ Ωr (E ) we have . ∂ ] = 0.9. the object (d∇ )2 ∈ Ω2 (EndK (E )) is called the curvature of ∇.12. A. 3. Y ) is an endomorphism of E . iZ d∇ + d∇ iZ = ∇Z .12) (3. Y ∈ Vect (M ) the quantity F (X.14) F (X. Y )u = iY iX (d∇ )2 u = iY (iX d∇ )∇u = iY (∇X − d∇ iX )∇u = (iY ∇X )∇u − (iY d∇ )∇X u = (∇X iY − i[X. In the second half of this book we will discuss in more detail this fact. ∇j = ∇ ∂ .14) Fij = −Fji := F ( ∂ ∂ . Let E → M be a smooth vector bundle on M and ∇ a connection on it. and we set ∇i := ∇ ∂ .100 CHAPTER 3. u ∈ Ω0 (E ). CALCULUS ON MANIFOLDS We conclude this subsection with an alternate description of the curvature which hopefully will shed some light on its analytical significance. the endomorphism Fij measures the extent to which the partial derivatives ∇i .12)-(3.3. . ∀ω ∈ Ωr (M ). ∇X iY − iY ∇X = i[X.3. ∇Y ] − ∇[X. ∂xi ∂xj (3.13. we denote by iZ : Ωr (E ) → Ωr−1 (E ) the C ∞ (M ) − linear operator defined by iZ (ω ⊗ u) = (iZ ω ) ⊗ u.3.3. A connection ∇ such that F (∇) = 0 is called flat. A natural question arises: given an arbitrary vector bundle E → M do there exist flat connections on E ? If E is trivial then the answer is obviously positive. ∇Z satisfy the usual super-commutation identities. Y ) = [∇X . iZ . For any u ∈ Ω0 (E ) we compute using (3. Definition 3. For any vector field Z .16) Thus. then we deduce ∂xi ∂xj (3. Denote its curvature by F = F (∇) ∈ Ω2 (End (E )). Hence F (X. ∇j fail to commute. (3.Y ] . Maybe an intelligent choice will restore the classical commutativity of partial derivatives so we should concentrate from the very beginning to covariant derivatives ∇ such that F (∇) = 0.3.Y ] )∇u − ∇Y ∇X u = (∇X ∇Y − ∇Y ∇X − ∇[X.3. ) = [∇i .15) ∂ ∂ and Y = ∂x . The covariant derivative ∇Z extends naturally to elements of Ωr (E ) by ∇Z (ω ⊗ u) := (LZ ω ) ⊗ u + ω ⊗ ∇Z u.Y ] . This is in sharp contrast with the classical calculus and an analytically oriented reader may object to this by saying we were careless when we picked the connection. ∇j ]. iX iY + iY iX = 0. In general. The operators d∇ . the answer is negative.13) (3. For any X. where (xi ) are local coordinates If in the above formula we take X = ∂x i j on M . In the remaining part of this section we will give a different description of this endomorphism.3.Y ] )u. and this has to do with the global structure of the bundle. .. Using (3. 0). p2 = (s. Let E → M be a rank r smooth K-vector bundle.1.. CONNECTIONS ON VECTOR BUNDLES 101 §3. .. Throughout this subsection we will use Einstein’s convention.. 0. and ∇ a connection on it. 0). t.3. .. It is defined by the equalities (∇i := ∇ ∂ ∂xi ) (3. Denote by s the parallel transport (using the connection ∇) from (x1 . p1 = (s. .. .. . where p0 = (0.. . Define T2 x2 instead of x1 . and maybe this will explain this choice of terminology. β ≤ r..3. 0..3. ∂ Γj ∂ Γi − j + Γi Γj − Γj Γi i ∂x ∂x . .4 Holonomy The reader may ask a very legitimate question: why have we chosen to name curvature. x2 ) described in Figure 3. Consider local coordinates (x1 . In this subsection we describe the geometric meaning of curvature. xn ) on an open subset U ⊂ M such that E |U is trivial. . . . .. 0) lies in U .. .3.18) Though the above equation looks very complicated it will be the clue to understanding the geometric significance of curvature. The connection 1-form associated to this moving frame is i Γ = Γi dxi = (Γα iβ )dx . Pick a moving frame (e1 .17) ∇i eβ = Γα iβ eα . 1 ≤ α. . . . . .. Look at the parallelogram Ps. 0). the deviation from commutativity of a given connection. . . p3 = (0. x2 .. .1: Parallel transport along a coordinate parallelogram. 0).3. Assume for simplicity that the point of coordinates (0. t. . xn ) to (x1 + s.t in the “plane” (x1 .3. er ) of E over U .. . xn ) T1 t in a similar way using the coordinate along the curve τ → (x1 + τ... p 3 p 2 p 0 p 1 Figure 3. . x2 . (3. xn ).16) we compute Fij eβ = (∇i ∇j − ∇j ∇i )eβ = ∇i (Γj eβ ) − ∇j (Γi eβ ) = so that Fij = ∂ Γα jβ ∂xi − ∂ Γα jβ ∂xj γ α α eα + Γ γ jβ Γiγ − Γiβ Γjγ eα . 0. t) = T1 u0 = u0 − sΓ1 (p0 )u0 + C1 s2 + O(s3 ). CALCULUS ON MANIFOLDS We now perform the counterclockwise parallel transport along the boundary of Ps.3.102 CHAPTER 3. where E0 is the fiber of E over p0 Set F12 := ∂ ∂ F ( ∂x 1 ∂x2 ) |(0.t = T2 T1 T2 T1 .20) +C1 s2 + C2 t2 + O(3). The curvature is an infinitesimal measure of this deviation. t) = T2 T1 u = T2 u1 = u1 − tΓ2 (p1 )u1 + C2 t2 + O(t3 ) = u0 − sΓ1 (p0 )u0 − tΓ2 (p1 )(u0 − sΓ1 (p0 )u0 ) + C1 s2 + C2 t2 + O(3) =  − sΓ1 (p0 ) − tΓ2 (p1 ) + tsΓ2 (p1 )Γ1 (p0 ) u0 + C1 s2 + C2 t2 + O(3). In the sequel the letter C (eventually indexed) will denote constants. t) =  − tΓ2 + st ∂ Γ1 ∂ Γ2 − + Γ2 Γ1 − Γ1 Γ2 ∂x2 ∂x1 |p0 u0 .19) C1 is a constant vector in E0 whose exact form is not relevant to our computations.. F12 is an endomorphism of E0 . Proposition 3. ∂x1 ∂ Γ2 − Γ2 Γ1 ∂x1 |p0 u0 (3. t → 0. The Γ-term in the right-hand-side of the above equality can be approximated as Γ1 (p2 ) = Γ1 (p0 ) + s ∂ Γ1 ∂ Γ1 (p0 ) + t 2 (p0 ) + O(2).14.. 1 ∂x ∂x Using u2 described as in (3.. The outcome is a linear map Ts.. For any u ∈ E0 we have F12 u = − ∂2 Ts. (3. Proof. Similarly.t can be described as −t −s t s Ts.t u. we have −s t s −s u3 = u3 (s.20) we get after an elementary computation u3 = u3 (s.3. t s t u2 = u2 (s. ∂s∂t We see that the parallel transport of an element u ∈ E0 along a closed path may not return it to itself.t . O(k ) denotes an error ≤ C (s2 + t2 )k/2 as s.3.3. The parallel transport along ∂Ps. t) = T1 T2 T1 u0 = T1 u2 = u2 + sΓ1 (p2 )u2 + C3 s2 + O(3).t : E0 → E0 . s :E →E The parallel transport T1 0 p1 can be approximated using (3.9) s u1 = u1 (s. Now use the approximation Γ2 (p1 ) = Γ2 (p0 ) + s to deduce u2 =  − sΓ1 − tΓ2 − st ∂ Γ2 (p0 ) + O(2).3.0) . We see that the curvature measures the holonomy along infinitesimal parallelograms.0 ≡.√s dTs. Remark 3. Alternatively. the last equality states that the curvature is the the “amount of holonomy per unit of area”. CONNECTIONS ON VECTOR BUNDLES +C4 s2 + C5 t2 + O(3). Let E → M be a vector bundle with a connection ∇.16. The curvature provides an infinitesimal measure of that twist.3. We can do so along paths starting at a fixed point. t) = u0 + st ∂ Γ2 (p0 ) + C7 t2 + O(3). using the parallel transport. The same goes for C9 .3.3.15. A connection can be viewed as an analytic way of trivializing a bundle. Definition 3. Ts.t ds Loosely speaking.3. . the constant C8 is the second order correction in the Taylor expansion of s → Ts.21) we get u4 (s. The result in the above proposition is usually formulated in terms of holonomy. Exercise 3. u4 = u4 (s. Hint: Use the parallel transport defined by a connection on the vector bundle E to produce a bundle isomorphism E → E0 × Rn . Thus we have −F12 = dT√s. Prove that any vector bundle E over the Euclidean space Rn is trivializable.t =  − stF12 + O(3). dareaPs. t) = T2 103 (3.t = .e.3. Clearly ∂ 2 u4 ∂s∂t = −F12 (p0 )u0 as claimed.3. ∂x2 ∂ Γ1 ∂ Γ2 − + Γ2 Γ1 − Γ1 Γ2 |p0 u0 ∂x2 ∂x1 +C8 s2 + C9 t2 + O(3) = u0 − stF12 (p0 )u0 + C8 s2 + C9 t2 + O(3). If we had kept track of the various constants in the above computation we would have arrived at the conclusion that C8 = C9 = 0 i. but using different paths ending at the same point we may wind up with trivializations which differ by a twist. The holonomy of ∇ along a closed path γ is the parallel transport along γ . so it has to be 0. where E0 is the fiber of E over the origin.3. Finally.21) with Γ2 (p3 ) = Γ2 (p0 ) + t Using (3. we have −t = u3 + tΓ2 (p3 )u3 + C6 t2 + O(3).17. ∇Z ]] u. Hence.3.3. Y )u) − F (Z. [X. Z we have iX D = ∇X − DiX .[X. Y ]) = ∇Z . Let K be the trivial line bundle over a smooth manifold M . Then DF (∇) = 0. the Bianchi identity states that (d∇ )3 is 0.12)–(3. Recall that ∇ induces a connection in any tensor bundle constructed from E . In particular. where d is the trivial connection.Z ] iX − ∇Z iY iX )F = (i[X. Proof. and ω is a K-valued 1-form on M .Z ] u − [∇X . .) Let E → M and ∇ as above.18. Proposition 3.14). The Bianchi identity now follows from the classical Jacobi identity for commutators. ∇[X. The Bianchi identity describes one remarkable algebraic feature of the curvature.Y ]] .Z ] iX + i[Z. Z ) = iZ iY iX DF = iZ iY (∇X − DiX )F = iZ (∇X iY − i[X.X ] iY )F − (∇X iY iZ + ∇Y iZ iX + ∇Z iX iY )F. (The Bianchi identity. Roughly speaking. Y . The Bianchi identity is in this case precisely the equality d2 ω = 0. Y. For any vector fields X .5 The Bianchi identities Consider a smooth K-vector bundle E → M equipped with a connection ∇. We will use the identities (3. This extends to an “exterior derivative” D : Ωp (End (E )) → Ωp+1 (End (E )). [∇Y . Y )∇X u = [∇X .3. F (Z. We compute immediately i[X. (DF )(X. Example 3.Y ] − ∇[Z.3.3.19. Any connection on K has the form ∇ω = d + ω . We have seen that the associated exterior derivative d∇ : Ωp (E ) → Ωp+1 (E ) does not satisfy the usual (d∇ )2 = 0.Y ] iZ F = F (Z. CALCULUS ON MANIFOLDS §3. and the curvature is to blame for this. ∇[Y. it induces a connection in E ∗ ⊗ E ∼ = End (E ) which we continue to denote by ∇.Z ] iY − iZ i[X.Y ] iZ + i[Y. The curvature of this connection is F (ω ) = dω.Y ] )F − iZ (∇Y − DiY )iX F = (∇X iZ iY − i[X. Also for any u ∈ Ω0 (E ) we have (∇X iY iZ F )u = ∇X (F (Z.104 CHAPTER 3. Y )] u = ∇X .Y ] )F − (∇Y iZ iX − i[Y. 3. A connection on T M is said to be symmetric if T = 0. i. For tangent bundles it happens that. Y ) = [∇X . Definition 3. f Y ) = f T (X. so that T (•. the torsion has the description k T (∂i . and it is thus i very natural to work with this frame. they ∂ automatically define a moving frame of the tangent bundle.. In the remaining of this subsection we will try to sketch the geometrical meaning of torsion.3. where p is a point in Tx M .3. and e is a basis of the . Then ∀f ∈ C ∞ (M ) T (f X. To find such an interpretation. Recall that. Y ] ∈ Vect (M ). For X. The proof of this lemma is left to the reader as an exercise. Y ∈ Vect (M ) consider T (X. The tangent space Tx M can be coordinatized using affine frames. and then a local moving frame for the vector bundle. a 2-form whose coefficients are vector fields on M . but instead we will treat them as points.e. It will be convenient to regard Tx M as an affine space modeled by Rn . n = dim M . The coefficients Γk ij are usually known as the Christoffel symbols of the connection. this is not the only tensor naturally associated to ∇. ∂j ) = (Γk ij − Γji )∂k . However. The tensor T is called the torsion of the connection ∇. ∇Y ] − ∇[X. CONNECTIONS ON VECTOR BUNDLES 105 §3.20.3. These are pairs (p. we have to look at the finer structure of the tangent space at a point x ∈ M . In terms of Christoffel symbols.21. Y ) = ∇X Y − ∇Y X − [X. •) is a tensor T ∈ Ω2 (T M ). For an arbitrary vector bundle there is no correlation between these two choices. ∂i = ∂x . let ∇ be a connection on T M . Lemma 3. the tangent bundles have some peculiar features which enrich the structure of a connection. With the above notations we set ∇i ∂j = Γk ij ∂k (∇i = ∇∂i ). we have to first choose local coordinates on the manifolds.Y ] ∈ End (T M ).6 Connections on tangent bundles The tangent bundles are very special cases of vector bundles so the general theory of connections and parallel transport is applicable in this situation as well. Y ). We guess by now the reader is wondering how the mathematicians came up with this object called torsion. e). Y ) = T (X. when looking for a local description for a connection on a vector bundle.3. Still. we will no longer think of the elements of Tx M as vectors. once local coordinates (xi ) are chosen. As usual we construct the curvature tensor F (X. Hence. Thus. 106 CHAPTER 3. We write T = S + i If now (x ) are local coordinates on M . Exercise 3. (∂i )). followed by a translation v .. CALCULUS ON MANIFOLDS underlying vector space. The frame (∂i ) suffers a parallel transport measured as usual by  − tiX Γ + O(t2 ). both modelled by Rn . then they define an affine frame Ax at each x ∈ M : (Ax = (0. Consider a smooth vector bundle E → M over the smooth manifold M . We assume that both E and T M are equipped with connections and moreover the ˆ the induced connection on Λ2 T ∗ M ⊗ connection on T M is torsionless. the “amount of drift per unit of area”.14 one can show the torsion measures the translation component of the holonomy along an infinitesimal parallelogram. Since we will not need this fact we will not include a proof of it. The total affine transport will be ˆ tX + O(t2 ). Denote by ∇ End (E ). Z ∈ Vect (M ) ˆ X F (Y. An affine isomorphism T : A → B can then be described using these coordinates as n T : Rn x → Ry x → y = Sx + v. Y ) = 0. an affine map is described by a “rotation” S . Exercise 3. where p is thought of as the origin. As in Proposition 3. B are two affine spaces.e. then describing T This is equivalent to describing the infinitesimal affine transport at a given point x0 ∈ M along a direction given by a vector X = X i ∂i ∈ Tx0 M . we will determine Tt by imposing the initial condition T0 = . (∂i )). and (p. close to x0 then its coordinates satisfy i i 2 xi t = x0 + tX + O (t ). Consider the vector valued 1-form ω ∈ Ω1 (T M ) defined by ω (X ) = X ∀X ∈ Vect (M ).22. where v is a vector in Rn . and a smooth path γ : I → M . f ) are affine frames of A and respectively B . If xt is a point along the integral curve of X . then d∇ ω = T ∇ . This shows the origin x0 of Ax0 “drifts” by tX + O(t2 ). Tt = ( − tiX Γ)+ The holonomy of ∇ along a closed path will be an affine transformation and as such it has two components: a “rotation”” and a translation. Suppose that A. Z ) + ∇ ˆ Y F (Z. and by (y j ) the coordinates in B induced by the frame (q . f ). e) . Denote by (xi ) the coordinates in A induced by the frame (p. X ) + ∇ ˆ Z F (X.23. Given a connection ∇ on T M . Prove that ∀X.3. A frame allows one to identify Tx M with Rn . ∇ . In fact. and ˙t . and S is an invertible n × n real matrix. Thus. The affine frame of Tx0 M is Ax0 = (0. we will construct a family of affine isomorphisms Tt : Tγ (0) → Tγ (t) called the affine transport of ∇ along γ .3. Show that if ∇ is a linear connection on T M .3. i. (p. e). This vector measures the “drift” ˆx of the origin. where T ∇ denotes the torsion of ∇. Y. namely the notion of 1-density on a manifold. The bundle of r-densities on M is r |Λ|r M : =|Λ| (T M ). It helps to have local descriptions of densities. ∞ (|Λ| ) its Denote by C ∞ (|Λ|M ) the space of smooth sections of |Λ|M .4. Let E → M be a rank k .1 Integration of 1-densities We spent a lot of time learning to differentiate geometrical objects but. The densities on a manifold resemble in many respects the differential forms of maximal degree. smooth real vector bundle over a manifold M defined by an open cover (Uα ) and transition maps gβα : Uαβ → GL(k. The fiber at p ∈ M of this bundle consists of r-densities on Ep (see Subsection 2. To this aim.. Denote the local coordinates on Uα by (xi α ). and by C0 M subspace consisting of compactly supported densities. Classically. just as in classical calculus. A change in coordinates should be thought of as a change in the measuring units. pick an open cover of M consisting of coordinate neighborhoods (Uα ). We call |Λ|M the density bundle of M. and in this subsection we will describe the differential geometric analogue of a measure. Let M be a smooth manifold and r ≥ 0.. ∂xn of Tp M spans an α α infinitesimal parallelepiped and µα (p) is its “volume”.3. R) satisfying the cocycle condition.4.1. When r = 1 we will use the notation |Λ|M = |Λ|1 M . where n is the dimension of M . ∂ ∂ It may help to think that for each point p ∈ Uα the basis ∂x 1 .4. integration requires a background measure.2. . A density is a map µ : C ∞ (det T M ) → C ∞ (M ). For any r ∈ R we can form the real line bundle |Λ|r (E ) defined by the same open cover and transition maps tβα := | det gβα |−r = | det gαβ |r : Uαβ → R>0 → GL(1. Denote by det T M = Λdim M T M the determinant line bundle of T M . INTEGRATION ON MANIFOLDS 3.j ≤n . . The gluing rules describe how the numerical value of the volume changes from one choice of units to another. A 1-density on M is then a collection of functions µα ∈ C ∞ (Uα ) related by −1 µα = δαβ µβ .4 Integration on manifolds 107 §3..4). Definition 3. the story is only half complete without the reverse operation. Set δαβ = | det Tαβ |. R). integration. This choice of a cover produces a trivializing cover of T M with transition maps Tαβ = ∂xi α ∂xj β 1≤i. ij )1≤i. Then the restriction of |dVg | to Uα has the description |dVg | = |gα | |dxα |.. i. for all smooth functions f : M → R. and all e ∈ C ∞ (det T M ). (c) Suppose g is a Riemann metric on the smooth manifold M . then. Then any top degree form ω ∈ Ωm (M ) defines a density |ω | on M which associates to each section e ∈ C ∞ (det T M ) the smooth function x → |ωx ( e(x) )| Observe that |ω | = | − ω |. so this map Ωm (M ) → C ∞ (|Λ|M ) is not linear. xn ). Example 3. then on each Uα we have top degree forms m dxα := dx1 α ∧ · · · ∧ dxα . any smooth map φ : M → N between manifolds of the same dimension induces a pullback transformation φ∗ : C ∞ (|Λ|N ) → C ∞ (|Λ|M ).. The volume density defined by g is the density denoted by |dVg | which associates to each e ∈ C ∞ (det T M ) the pointwise length x → |e(x)|g .2. en the canonical basis.j We denote by |gα | the determinant of the symmetric matrix gα = (gα. . in particular.. where f is some smooth function on Rn .. This extends to a trivialization of T Rn and. If φ : Rn → Rn is a smooth map. Denote by e1 . φ∗ (|dvn |) = det ∂φi ∂xj · |dvn |. .108 CHAPTER 3. The reader should think of |dvn | as the standard Lebesgue measure on Rn . .. (b) Suppose M is a smooth manifold of dimension m.ij dxi g= α ⊗ dxα . . .. (a) Consider the special case M = Rn . In this case. ∧ en ) = 1.j ≤m . φn = φn (x1 .. any smooth density on Rn takes the form µ = f |dvn |. CALCULUS ON MANIFOLDS such that µ(f e) = |f |µ(e). viewed as a collection of n smooth functions φ1 = φ1 (x1 .. In the coordinates (xi α ) the metric g can be described as j gα.. .4. to which we associate the density |dxα |. It has a nowhere vanishing section |dvn | defined by |dvn |(e1 ∧ . If (Uα . .. In particular. . the bundle of densities comes with a natural trivialization. xn ).. described by (φ∗ µ)(e) = µ (det φ∗ ) · e = | det φ∗ | µ(e). ∀e ∈ C ∞ (det T M ). (xi α ) ) is an atlas of M . and any µ ∈ C0 M φ∗ µ = M N µ. for any smooth manifolds M . (a) M is invariant under diffeomorphisms. any diffeomorphism φ : M → N .4. and can be written as αµ = µα |dxα |. and we denote by |dxα | (n = dim M ) the density on Uα defined by |dxα | ∂ ∂ ∧ .. Proof.3. More precisely. (ii) For each Uα we pick a collection of local coordinates (xi α ). we have µ= M ∞ (Rn ) we have (c) For any ρ ∈ C0 U µ. N of the ∞ (|Λ| ). Rn where in the right-hand-side stands the Lebesgue integral of the compactly supported function ρ. we have same dimension n. for any open set U ⊂ M . and any µ ∈ C0 M supp µ ⊂ U .e. ∞ (|Λ| ) with (b) M is a local operation. Proposition 3.. ∞ (M ) such that ∀α ∈ A the support supp α lies (i) A smooth partition of unity A ⊂ C0 entirely in some precompact coordinate neighborhood Uα . and such that the cover (Uα ) is locally finite.. i. where µα is some smooth function compactly supported on Uα .e. We set αµ := µα |dxα |..3. i. INTEGRATION ON MANIFOLDS 109 The importance of densities comes from the fact that they are exactly the objects that can be integrated. ∧ n 1 ∂xα ∂xα = 1. M is called the integral on M . Uα Rn . the product αµ is a density supported in Uα .4. smooth function. Under this identification |dxi α | corresponds n to the Lebesgue measure |dvn | on R . we have the following abstract result. There exists a natural way to associate to each smooth manifold M a linear map M ∞ : C0 (|Λ|M ) → R uniquely defined by the following conditions. and µα is a compactly supported. Rn ρ|dvn | = ρ(x)dx. To establish the existence of an integral we associate to each manifold M a collection of data as follows. For any µ ∈ C ∞ (|Λ|). The local coordinates allow us to interpret µα as a function on Rn . We have Let µ ∈ C0 M αµ = Uα β Uα ⊂Rn βαµ = β Uαβ αβµ. ∞ (M ) two partitions of unity • Independence of the partition of unity. We can view αµ as a compactly supported function on Rn .4. (α.110 Finally. B ⊂ C0 on M . define CHAPTER 3. (3.1) over α and (3.4. The above sum contains only finitely many nonzero terms since supp µ is compact. Fix the partition of unity A. These determine two densities |dxi | and collection of local coordinates (yα α α j ∞ (|Λ| ) we have respectively |dyα |. CALCULUS ON MANIFOLDS A µ= M M µ = def α∈A Uα αµ. and thus it intersects only finitely many of the Uα s which form a locally finite cover. Let A. µα ∈ C0 (Uα ) are related by µy α = det The equality Rn ∂xi α j ∂yα µx α. (3. β ) ∈ A × B ∞ ⊂ C0 (M ). We will prove A A∗B B = M M = M .4. Form the partition of unity A ∗ B := αβ . . To prove property (a) we will first prove that the integral defined as above is indepen∞ (M ). • Independence of coordinates. and consider a new i ) on each U .2) over β we get the desired conclusion. y ∞ where µx α . Note that supp αβ ⊂ Uαβ = Uα ∩ Uβ . µx α |dxα | = Rn µy α |dyα | is the classical change in variables formula for the Lebesgue integral.4.1) Similarly βµ = Uβ α Uαβ αβµ. For each µ ∈ C0 M y αµ = αµx α |dxα | = αµα |dyα |. ∞ (|Λ| ).2) Summing (3. We will show that A B = M M . and the local coordinates dent of the various choices: the partition of unity A ⊂ C0 i (xα )α∈A . This explains the “big surprise” produced by the famous counter-example due to M¨ obius in the first half of the 19th century. we have to require that the background manifold is oriented. For any ∞ partition of unity A ⊂ C0 (M ) with associated cover (Vα )α∈A we can form a new partition of unity A ∗ B of U with associated cover Vαβ = Vα ∩ Vβ . More precisely. For any density µ on M supported on U we have µ= M α Vα α∈A forms a partition of unity on M .4. More precisely.2 Orientability and integration of differential forms Under some mild restrictions on the manifold. the oriented manifolds are the “2-sided manifolds”.3. i. the calculus with densities can be replaced with the richer calculus with differential forms. one can distinguish between an “inside face” and an “outside face” of the manifold. A smooth manifold M is said to be orientable if the determinant line bundle det T M (or equivalently det T ∗ M ) is trivializable. Property (a) now αµ = α∈A β ∈B Vαβ αβµ = αβ ∈A∗B Vαβ αβµ = U µ. The uniqueness of the integral is immediate.2. To prove property (b) on the local character of the integral. We see that det T ∗ M is trivializable if and only if it admits a nowhere vanishing section. The 2-sidedness can be formulated rigorously as follows. Roughly speaking. for M = Rn . we can assume that all the local coordinates chosen are Cartesian. and using the independence of the integral on partitions of unity. Definition 3. (Think of a 2-sphere in R3 (a soccer ball) which is naturally a “2-faced” surface. consider a partition of unity ∞ (N ). and any µ ∈ C0 N have (φ∗ α)φ∗ µ = φ−1 (U α) αµ. §3. and then ∞ (U ) subordinated to the open cover (V ) choose a partition of unity B ⊂ C0 β β ∈B . We use this partition of unity to compute integrals over U .) The 2-sidedness feature is such a frequent occurrence in the real world that for many years it was taken for granted. From the classical change in variables formula we deduce that. INTEGRATION ON MANIFOLDS 111 To prove property (a) for a diffeomorphism φ : M → N .4.4. for any A ⊂ C0 ∞ (|Λ| ) we coordinate neighborhood Uα containing the support of α ∈ A.e. pick U ⊂ M . . We say that two volume forms ω1 and ω2 are equivalent if there exists f ∈ C ∞ (M ) such that ω2 = ef ω1 . Uα The collection φ∗ α = α ◦ φ follows by summing over α the above equality.. and we leave the reader to fill in the details. Property (c) is clear since. He produced a 1-sided surface nowadays known as the M¨ obius band using paper and glue. he glued the opposite sides of a paper rectangle attaching arrow to arrow as in Figure 3. The mild restrictions referred to above have a global nature. Such a section is called a volume form on M .4. and an equivalence class of volume forms will be called an orientation of the manifold. then for every p ∈ M the map Or (M ) or → or p ∈ Or (Tp M ) is a bijection. then there exists a nowhere vanishing smooth function ρ : M → R such that ω = ρω . ∀q ∈ M .p . then ∗ M is a nontrivial volume form on T M . Proposition 3. We deduce sign ρ(q ) = (q ). and thus constant. Proof.p on Tp M . If or is given by the volume form ω and or is given by the volume form ω . The function M q → (q ) = or q /or q ∈ {±1} is continuous. ∀q ∈ M. orientable smooth manifold M .p = or ω2 . A pair (orientable manifold. In particular. If M is a connected. Let us observe that if M is orientable. If the orientation or on M is defined by a volume form ω . then for every point p ∈ M we have a natural map Or (M ) → Or (Tp M ). It is clear that if ω1 and ω2 are equivalent volume form then or ω1 . Suppose or and or are two orientations on M such that or p = or p .4.112 CHAPTER 3.2: The Mobius band. for any volume form ω . We denote by Or (M ) the set of orientations on the smooth manifold M .p = −or ω. Or (M ) or → or p ∈ Or (Tp M ). orientation) is called an oriented manifold. This is indeed an equivalence relation. (q ) = (p) = 1. which canonically defines an orientation ωp ∈ det Tp p or ω. This shows that the two forms ω and ω are equivalent and thus or = or . and thus Or (M ) = ∅. defined as follows. This map is clearly a surjection because or −ω.5. .p . CALCULUS ON MANIFOLDS Figure 3. let ν denote the normal line bundle. Hence all spheres are orientable. We deduce det T M is trivial since det T M ∼ = det(Rk ⊕ T M ) ⊗ (det Rk )∗ . Indeed.4. a choice of an orientation of one of its tangent spaces uniquely determines an orientation of the manifold. In the remaining part of this section we will describe several simple ways to detect orientability. . In particular. We see this is just a special instance of the more general question we addressed in Chapter 2: how can one decide whether a given vector bundle is trivial or not. Suppose M is a manifold such that the Whitney sum Rk M ⊕ T M is trivial.7. If the tangent bundle of a manifold M is trivial. Both det Rk and det(Rk ⊕ T M ) are trivial. (see Figure 3. it is often convenient to decide the orientability issue using ad-hoc arguments.3: The normal line bundle to the round sphere. The orientability question can be given a very satisfactory answer using topological techniques. Example 3. Indeed. The line bundle ν has a remarkable feature: ν ⊕ T S n = Rn+1 . Then M is orientable. However. How can one decide whether a given manifold is orientable or not.6.4.3).3. This is clearly a trivial line bundle since it has a tautological nowhere vanishing section p → p ∈ νp . we have det(Rk ⊕ T M ) = det Rk ⊗ det T M.4. A natural question arises. This trick works for example when M ∼ = S n . then clearly T M is orientable. The last proposition shows that on a connected. all Lie groups are orientable. orientable manifold. n The fiber of ν at a point p ∈ S is the 1-dimensional space spanned by the position vector of p as a point in Rn . INTEGRATION ON MANIFOLDS 113 Figure 3. Example 3. 114 CHAPTER 3. CALCULUS ON MANIFOLDS . We can think of p as the “outer” normal to the round sphere. The canonical orientation on Rn is the orientation defined by the volume form dx1 ∧· · ·∧ dxn . ∆αβ = det = j µα ∂xβ n 1 n A permutation ϕ of the variables x1 α . Then n ω |Uα = µα dx1 α ∧ · · · ∧ dxα . . 1. Lemma 3.. βωϕ(β ) . there exists a map ϕ : B → A such that supp β ⊂ Uϕ(β ) ∀β ∈ B. The unit sphere S n ⊂ Rn+1 is orientable. To this aim we will use the relation Rn+1 ∼ = νp ⊕ Tp S n . 2.3) implies that on an overlap Uα1 ∩ · · · ∩ Uαm the forms ωα1 . An element ω ∈ det Tp S n defines the canonical orientation if p ∧ ω ∈ det Rn+1 defines the canonical orientation of Rn+1 .Important convention.3) Proof.e. ωαm differ by a positive multiplicative factor. The form ω is nowhere vanishing since condition (3. A smooth manifold M is orientable if and only if there exists an open n cover (Uα )α∈A . We assume that there exists an open cover with the properties in the lemma. i.... where the smooth functions µα are nowhere vanishing. In the sequel we will exclusively deal with its canonical orientation.. .8... When n = 1 it coincides with the counterclockwise orientation of the unit circle S 1 . by p we denoted the position vector of p as a point inside the Euclidean space Rn+1 . xα ) on Uα such that det ∂xi α ∂xj β > 0 on Uα ∩ Uβ . ∞ (M ) subordinated to the cover (U ) Consider a partition of unity B ⊂ C0 α α∈A . and we will prove that det T ∗ M is trivial by proving that there exists a volume form.. This will insure the positivity condition ∆αβ > 0.. Define ω := β n where for all α ∈ A we define ωα := dx1 α ∧ · · · ∧ dxα . where x1 . and on the overlaps they satisfy the gluing condition µβ ∂xi α . (xi α )). (3.4.4. Conversely.. Above. let ω be a volume form on M and consider an atlas (Uα . We call this orientation outer normal first. xα will change dxα ∧ · · · ∧ dxα by a factor (ϕ) so we can always arrange these variables in such an order so that µα > 0. xn are the canonical Cartesian coordinates.. .. and local coordinates (x1 α . To describe this orientation it suffices to describe a positively oriented basis of det Tp M for some p ∈ S n .4. . .. 8 we deduce immediately the following consequence.11. INTEGRATION ON MANIFOLDS The lemma is proved.13).76 we deduce the following result. Using Lemma 3. Using connected sums and products we can now produce many examples of manifolds. an orientation or on a smooth manifold M defines an isomorphism ıor : C ∞ (det T ∗ M ) → C ∞ (|Λ|M ).2. ΘM := Θ(T M ).4. linear isomorphism ıor : det V ∗ → |Λ|V .4. The statement in Lemma 3. The reader can check immediately that the product of two orientable manifolds is again an orientable manifold. and gluing cocycle βα := sign det gβα : Uαβ → R∗ = GL(1.4. 115 We can rephrase the result in the above lemma in a more conceptual way using the notion of orientation bundle. By now the reader may ask where does orientability interact with integration. .4. and gluing cocycle gβα : Uαβ → GL(r. Any complex manifold is orientable. R).4.2. From Lemma 3. We define orientation bundle ΘM of a smooth manifold M as the orientation bundle associated to the tangent bundle of M .12. In particular. The answer lies in Subsection 2.10.8 and Proposition 2.4.4. the connected sums of g tori is an orientable manifold.4 where we showed that an orientation or on a vector space V induces a canonical.9. the complex Grassmannians Grk (Cn ) are orientable. Suppose E → M is a real vector bundle of rank r on the smooth manifold M described by the open cover (Uα )α∈A . The connected sum of two orientable manifolds is an orientable manifold.4. Exercise 3. Supply the details of the proof of Proposition 3. Proposition 3.8 can now be rephrased as follows.12. Similarly. A smooth manifold M is orientable if and only if the orientation bundle ΘM is trivializable. The orientation bundle associated to E is the real line bundle Θ(E ) → M described by the open cover (Uα )α∈A .3. see (2. Prove the above result. For any compactly supported differential form ω on M of maximal degree we define its integral by ω := M M ıor ω.4.2. R). Proposition 3. Corollary 3. Exercise 3.13. In particular.4. Consider the 2-form on R3 . (3. ρα ∈ C ∞ (Uα ).j ≤M > 0. 0. ϕ.4. which is precisely ıor ω .4. θ).4: Spherical coordinates. ω = xdy ∧ dz . and due to the condition (3. n = dim M. . Thus they glue together to a density on M . (xi α ) such that det ∂xi β ∂xj α 1≤. Example 3.4) the collection of desities µα = ρα |dxα | ∈ C ∞ (Uα . and let S 2 denote the unit sphere. CALCULUS ON MANIFOLDS ϕ O θ r P y x Figure 3. where S 2 has the canonical orientation. We ought to pause and explain in more detail the isomorphism ıor : C ∞ (det T ∗ M ) → ∞ C (|Λ|M ).4. 0) we have ∂r = ∂x = p.14. We want to emphasize that this definition depends on the choice of orientation.  z = r cos ϕ At the point p = (1.4) and on on each coordinate patch Uα the orientation is given by the top degree from n dxα = dx1 α ∧ · · · ∧ dxα .i. Since M is oriented we can choose a coordinate atlas Uα . To compute this integral we will use spherical coordinates (r. These are defined by (see Figure 3.4. We want to compute S 2 ω |S 2 . ∂θ = ∂y ∂ϕ = −∂z .116 z CHAPTER 3.)   x = r sin ϕ cos θ y = r sin ϕ sin θ . | |Λ| |M ) satisfy µα = µβ on the overlap Uαβ . The differential form is described by a collection of forms ωα = ρα dxα . (3. and we deduce ω= S2 [0. Thus ∆ ≡ 1 which establishes the right invariance of dVG . (Invariant integration on compact Lie groups).3. and in particular. connected Lie group. On S 2 we have r ≡ 1 and dr ≡ 0 so that xdy ∧ dz |S 2 = sin ϕ cos θ (cos θ sin ϕdθ + sin θ cos ϕdϕ)) ∧ (− sin ϕ)dϕ = sin3 ϕ cos2 θdϕ ∧ dθ. The standard orientation associates to this form de density sin3 ϕ cos2 θ|dϕdθ|. and in particular. If there exists x ∈ G such that ∆(x) = 1. INTEGRATION ON MANIFOLDS 117 so that the standard orientation on S 2 is given by dϕ ∧ dθ. .π ]×[0. To prove this. We can extend ω by left translations to a left-invariant volume form on G which we continue to denote by ω .5) (assuming a fixed orientation on G). 3 As we will see in the next subsection the above equality is no accident. G → (R \ {0}.4. or ∆(x−1 ) > 1.2π ] sin3 ϕ cos2 θ|dϕdθ| = 0 π sin3 ϕdϕ · 0 2π cos2 θdθ 4π = volume of the unit ball B 3 ⊂ R3 . Let G be a compact.4. Consider a positively oriented volume element ω ∈ det L∗ G . This defines an orientation. Fix once and for all an orientation on the Lie algebra LG . = Example 3. We claim dVG is also right invariant. Since G is connected ∆(G) ⊂ R+ .5) The differential form dVG is the unique left-invariant n-form (n = dim G) on G satisfying (3. so it has The quantity ∆(h) is independent of h because Rh G ∗ ∗ R∗ we deduce to be a scalar multiple of dVG . by integration. the set ∆(G) is bounded. and since G is compact. ∗ dV is a left invariant form. Since (Rh1 h2 ) = (Rh2 Rh1 )∗ = Rh h2 1 ∆(h1 h2 ) = ∆(h1 )∆(h2 ) Hence h → ∆h is a smooth morphism ∀h1 . Set dVG = 1 c ω so that G dVG = 1. we get a positive scalar c= G ω. consider the modular function G h → ∆(h) ∈ R defined by ∗ Rh (dVG ) = ∆(h)dVG . h2 ∈ G.15.4. we would deduce the set (∆(xn ))n∈Z is unbounded. ·).4. then either ∆(x) > 1. 118 CHAPTER 3. CALCULUS ON MANIFOLDS The invariant measure dVG provides a very simple way of producing invariant objects on G. More precisely, if T is tensor field on G, then for each x ∈ G define Tx = ((Lg )∗ T )x dVG (g ). G Then x → T x defines a smooth tensor field on G. We claim that T is left invariant. Indeed, for any h ∈ G we have (Lh )∗ T = u=hg G (Lh )∗ ((Lg )∗ T )dVG (g ) = G ((Lhg )∗ T )dVG (g ) = G (Lu )∗ T L∗ h−1 dVG (u) = T ( L∗ h−1 dVG = dVG ). If we average once more on the right we get a tensor G x→ G (Rg )∗ T x dVG , which is both left and right invariant. Exercise 3.4.16. Let G be a Lie group. For any X ∈ LG denote by ad(X ) the linear map LG → LG defined by LG Y → [X, Y ] ∈ LG . (a) If ω denotes a left invariant volume form prove that ∀X ∈ LG LX ω = tr ad(X )ω. (b) Prove that if G is a compact Lie group, then tr ad(X ) = 0, for any X ∈ LG . §3.4.3 Stokes’ formula The Stokes’ formula is the higher dimensional version of the fundamental theorem of calculus (Leibniz-Newton formula) b df = f (b) − f (a), a where f : [a, b] → R is a smooth function and df = f (t)dt. In fact, the higher dimensional formula will follow from the simplest 1-dimensional situation. We will spend most of the time finding the correct formulation of the general version, and this requires the concept of manifold with boundary. The standard example is the lower half-space 1 n n 1 Hn − = { (x , ..., x ) ∈ R ; x ≤ 0 }. Definition 3.4.17. A smooth manifold with boundary of dimension n is a topological space with the following properties. (a) There exists a smooth n-dimensional manifold M that contains M as a closed subset. (b) The interior of M , denoted by M 0 , is non empty. 3.4. INTEGRATION ON MANIFOLDS 119 (c) For each point p ∈ ∂M := M \ M 0 , there exist smooth local coordinates (x1 , ..., xn ) defined on an open neighborhood N of p in M such that (c1 ) M 0 ∩ N = q ∈ N; x1 (q ) < 0, . (c2 ) ∂M ∩ N = { x1 = 0 }. The set ∂M is called the boundary of M . A manifold with boundary M is called orientable if its interior M 0 is orientable. Example 3.4.18. (a) A closed interval I = [a, b] is a smooth 1-dimensional manifold with boundary ∂I = {a, b}. We can take M = R. (b) The closed unit ball B 3 ⊂ R3 is an orientable manifold with boundary ∂B 3 = S 2 . We can take M = R3 . (c) Suppose X is a smooth manifold, and f : X → R is a smooth function such that 0 ∈ R is a regular value of f , i.e., f (x) = 0 =⇒ df (x) = 0. Define M := x ∈ X ; f (x) ≤ 0 . A simple application of the implicit function theorem shows that the pair (X, M ) defines a manifold with boundary. Note that examples (a) and (b) are special cases of this construction. In the case (a) we take X = R, and f (x) = (x−a)(x−b), while in the case (b) we take X = R3 and f (x, y, z ) = (x2 +y 2 +z 2 )−1. Definition 3.4.19. Two manifolds with boundary M1 ⊂ M1 , and M2 ⊂ M2 are said to be diffeomorphic if, for every i = 1, 2 there exists an open neighborhood Ui of Mi in Mi , and a diffeomorphism F : U1 → U2 such that F (M1 ) = M2 . Exercise 3.4.20. Prove that any manifold with boundary is diffeomorphic to a manifold with boundary constructed via the process described in Example 3.4.18(c). Proposition 3.4.21. Let M be a smooth manifold with boundary. Then its boundary ∂M is also a smooth manifold of dimension dim ∂M = dim M − 1. Moreover, if M is orientable, then so is its boundary. The proof is left to the reader as an exercise. Important convention. Let M be an orientable manifold with boundary. There is a (non-canonical ) way to associate to an orientation on M 0 an orientation on the boundary ∂M . This will be the only way in which we will orient boundaries throughout this book. If we do not pay attention to this convention then our results may be off by a sign. We now proceed to described this induced orientation on ∂M . For any p ∈ ∂M choose local coordinates (x1 , ..., xn ) as in Definition 3.4.17. Then the induced orientation of Tp ∂M is defined by dx2 ∧ · · · ∧ dxn ∈ det Tp ∂M, = ±1, 120 where CHAPTER 3. CALCULUS ON MANIFOLDS is chosen so that for x1 < 0, i.e. inside M , the form dx1 ∧ dx2 ∧ · · · ∧ dxn is positively oriented. The differential dx1 is usually called an outer conormal since x1 increases as we go towards the exterior of M . −dx1 is then the inner conormal for analogous reasons. The rule by which we get the induced orientation on the boundary can be rephrased as {outer conormal} ∧ {induced orientation on boundary} = {orientation in the interior}. We may call this rule “outer (co)normal first” for obvious reasons. Example 3.4.22. The canonical orientation on S n ⊂ Rn+1 coincides with the induced orientation of S n+1 as the boundary of the unit ball B n+1 . Exercise 3.4.23. Consider the hyperplane Hi ⊂ Rn defined by the equation {xi = 0}. i Prove that the induced orientation of Hi as the boundary of the half-space Hn,i + = {x ≥ 0} is given by the (n − 1)-form (−1)i dx1 ∧ · · · ∧ dxi ∧ · · · dxn where, as usual, the hat indicates a missing term. Theorem 3.4.24 (Stokes formula). Let M be an oriented n-dimensional manifold with boundary ∂M and ω ∈ Ωn−1 (M ) a compactly supported form. Then dω = M0 ∂M ω. In the above formula d denotes the exterior derivative, and ∂M has the induced orientation. Proof. Via partitions of unity the verification is reduced to the following two situations. Case 1. The (n − 1)-form ω is compactly supported in Rn . We have to show dω = 0. Rn It suffices to consider only the special case ω = f (x)dx2 ∧ · · · ∧ dxn , where f (x) is a compactly supported smooth function. The general case is a linear combination of these special situations. We compute dω = Rn Rn ∂f dx1 ∧ · · · ∧ dxn = ∂x1 Rn−1 R ∂f dx1 dx2 ∧ · · · ∧ dxn = 0, ∂x1 since R ∂f dx1 = f (∞, x2 , ..., xn ) − f (−∞, x2 , ..., xn ), ∂x1 and f has compact support. 3.4. INTEGRATION ON MANIFOLDS Case 2. The (n − 1)-form ω is compactly supported in Hn − . Let ω= i 121 fi (x)dx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn . Then dω = i (−1)i+1 ∂f ∂xi dx1 ∧ · · · ∧ dxn . One verifies as in Case 1 that Hn − ∂f 1 dx ∧ · · · ∧ dxn = 0 for i = 1. ∂xi For i = 1 we have Hn + ∂f dx1 ∧ · · · ∧ dxn = ∂x1 0 Rn−1 −∞ ∂f dx1 dx2 ∧ · · · ∧ dxn ∂x1 = Rn−1 f (0, x2 , ..., xn ) − f (−∞, x2 , ..., xn ) dx2 ∧ · · · ∧ dxn = Rn−1 f (0, x2 , ..., xn )dx2 ∧ · · · ∧ dxn = ∂ Hn − ω. The last equality follows from the fact that the induced orientation on ∂ Hn − is given by dx2 ∧ · · · ∧ dxn . This concludes the proof of the Stokes formula. Remark 3.4.25. Stokes formula illustrates an interesting global phenomenon. It shows that the integral M dω is independent of the behavior of ω inside M . It only depends on the behavior of ω on the boundary. Example 3.4.26. xdy ∧ dz = S2 B3 dx ∧ dy ∧ dz = vol (B 3 ) = 4π . 3 Remark 3.4.27. The above considerations extend easily to more singular situations. For example, when M is the cube [0, 1]n its topological boundary is no longer a smooth manifold. However, its singularities are inessential as far as integration is concerned. The Stokes formula continues to hold dω = In ∂I ω ∀ω ∈ Ωn−1 (I n ). The boundary is smooth outside a set of measure zero and is given the induced orientation: “ outer (co)normal first”. The above equality can be used to give an explanation for the terminology “exterior derivative” we use to call d. Indeed if ω ∈ Ωn−1 (Rn ) and Ih = [0, h] then we deduce dω |x=0 = lim h−n ω. (3.4.6) h→0 n ∂Ih When n = 1 this is the usual definition of the derivative. 122 CHAPTER 3. CALCULUS ON MANIFOLDS Example 3.4.28. We now have sufficient technical background to describe an example of vector bundle which admits no flat connections, thus answering the question raised at the end of Section 3.3.3. Consider the complex line bundle Ln → S 2 constructed in Example 2.1.36. Recall that Ln is described by the open cover S 2 = U0 ∪ U1 , U0 = S 2 \ {South pole}, U1 = S 2 \ {North pole}, and gluing cocycle g10 : U0 ∩ U1 → C∗ , g10 (z ) = z −n = g01 (z )−1 , where we identified the overlap U0 ∩ U1 with the punctured complex line C∗ . A connection on Ln is a collection of two complex valued forms ω0 ∈ Ω1 (U1 ) ⊗ C, ω1 ∈ Ω1 (U1 ) ⊗ C, satisfying a gluing relation on the overlap (see Example 3.3.7) ω1 = − If the connection is flat, then dω0 = 0 on U0 and dω1 = 0 on U1 . Let E + be the Equator equipped with the induced orientation as the boundary of the northern hemisphere, and E − the equator with the opposite orientation, as the boundary of the southern hemisphere. The orientation of E + coincides with the orientation given by the form dθ, where z = exp(iθ). We deduce from the Stokes formula (which works for complex valued forms as well) that E+ dg10 dz + ω0 = n + ω0 . g10 z ω0 = 0 E+ ω1 = − E− ω1 = 0. On the other hand, over the Equator, we have ω1 − ω0 = n from which we deduce 0= E+ dz = nidθ, z ω0 − ω1 = ni dθ = 2nπ i !!! E+ Thus there exist no flat connections on the line bundle Ln , n = 0, and at fault is the gluing cocycle defining L. In a future chapter we will quantify the measure in which the gluing data obstruct the existence of flat connections. 3.4. INTEGRATION ON MANIFOLDS 123 §3.4.4 Representations and characters of compact Lie groups The invariant integration on compact Lie groups is a very powerful tool with many uses. Undoubtedly, one of the most spectacular application is Hermann Weyl’s computation of the characters of representations of compact semi-simple Lie groups. The invariant integration occupies a central place in his solution to this problem. We devote this subsection to the description of the most elementary aspects of the representation theory of compact Lie groups. Let G be a Lie group. Recall that a (linear) representation of G is a left action on a (finite dimensional) vector space V G×V →V (g, v ) → T (g )v ∈ V, such that the map T (g ) is linear for any g . One also says that V has a structure of G-module. If V is a real (respectively complex) vector space, then it is said to be a real (respectively complex) G-module. Example 3.4.29. Let V = Cn . Then G = GL(n, C) acts linearly on V in the tautological manner. Moreover V ∗ , V ⊗k , Λm V and S V are complex G-modules. Example 3.4.30. Suppose G is a Lie group with Lie algebra LG − T1 G. For every g ∈ G, the conjugation C g : G → G, h → C g (h) = ghg −1 , sends the identity 1 ∈ G to itself. We denote by Adg : LG → L the differential of h → C g (h) at h = 1. The operator Adg is a linear isomorphism of LG , and the resulting map G → Aut(LG ), g → Adg , is called adjoint representation of G. For example, if G = SO(n), then LG = so(n), and the adjoint representation Ad : SO(n) → Aut(so(n)), can be given the more explicit description so(n) X −→ gXg −1 ∈ so(n), ∀g ∈ SO(n). Ad(g ) Exercise 3.4.31. Let G be a Lie group, with Lie algebra LG . Prove that for any X, Y ∈ LG we have d |t=0 Adexp(tX ) (Y ) = ad(X )Y = [X, Y ]. dt Definition 3.4.32. A morphism of G-modules V1 and V2 is a linear map L : V1 → V2 such that, for any g ∈ G, the diagram below is commutative, i.e., T2 (g )L = LT1 (g ). V1 T1 (g ) L M V2 M V2 K V1 K T2. (g ) L The space of morphisms of G-modules is denoted by HomG (V1 , V2 ). The collection of isomorphisms classes of complex G-modules is denoted by G − M od. 124 CHAPTER 3. CALCULUS ON MANIFOLDS If V is a G-module, then an invariant subspace (or submodule) is a subspace U ⊂ V such that T (g )(U ) ⊂ U , ∀g ∈ G. A G-module is said to be irreducible if it has no invariant subspaces other than {0} and V itself. Proposition 3.4.33. The direct sum “⊕”, and the tensor product “⊗” define a structure of semi-ring with 1 on G − M od. 0 is represented by the null representation {0}, while 1 is represented by the trivial module G → Aut (C), g → 1. The proof of this proposition is left to the reader. ∗ Example 3.4.34. Let Ti : G → Aut (Ui ) (i = 1, 2) be two complex G-modules. Then U1 ∗ − 1 † ∗ is a G-module given by (g, u ) → T1 (g ) u . Hence Hom (U1 , U2 ) is also a G-module. Explicitly, the action of g ∈ G is given by (g, L) −→ T2 (g )LT1 (g −1 ), ∀L ∈ Hom(U1 , U2 ). We see that HomG (U1 , U2 ) can be identified with the linear subspace in Hom (U1 , U2 ) consisting of the linear maps U1 → U2 unchanged by the above action of G. Proposition 3.4.35 (Weyl’s unitary trick). Let G be a compact Lie group, and V a complex G-module. Then there exists a Hermitian metric h on V which is G-invariant, i.e., h(gv1 , gv2 ) = h(v1 , v2 ), ∀v1 , v2 ∈ V . Proof. Let h be an arbitrary Hermitian metric on V . Define its G-average by h(u, v ) := G h(gu, gv )dVG (g ), where dVG (g ) denotes the normalized bi-invariant measure on G. One can now check easily that h is G-invariant. In the sequel, G will always denote a compact Lie group. Proposition 3.4.36. Let V be a complex G-module and h a G-invariant Hermitian metric. If U is an invariant subspace of V then so is U ⊥ , where “⊥” denotes the orthogonal complement with respect to h. Proof. Since h is G-invariant it follows that, ∀g ∈ G, the operator T (g ) is unitary, T (g )T ∗ (g ) = V . Hence, T ∗ (g ) = T −1 (g ) = T (g −1 ), ∀g ∈ G. If x ∈ U ⊥ , then for all u ∈ U , and ∀g ∈ G h(T (g )x, u) = h((x, T ∗ (g )u) = h(x, T (g −1 )u) = 0. Thus T (g )x ∈ U ⊥ , so that U ⊥ is G-invariant. Corollary 3.4.37. Every G-module V can be decomposed as a direct sum of irreducible ones. 3.4. INTEGRATION ON MANIFOLDS 125 If we denoted by Irr (G) the collection of isomorphism classes of irreducible G-modules, then we can rephrase the above corollary by saying that Irr (G) generates the semigroup (G − M od, ⊕). To gain a little more insight we need to use the following remarkable trick due to Isaac Schur. Lemma 3.4.38 (Schur lemma). Let V1 , V2 be two irreducible complex G-modules. Then dimC HomG (V1 , V2 ) = 1 if V1 0 if V1 ∼ = V2 . ∼ = V2 Proof. Let L ∈ HomG (V1 , V2 ). Then ker L ⊂ V1 is an invariant subspace of V1 . Similarly, Range (L) ⊂ V2 is an invariant subspace of V2 . Thus, either ker L = 0 or ker L = V1 . The first situation forces Range L = 0 and, since V2 is irreducible, we conclude Range L = V2 . Hence, L has to be in isomorphism of G-modules. We deduce that, if V1 and V2 are not isomorphic as G-modules, then HomG (V1 , V2 ) = {0}. Assume now that V1 ∼ = V2 and S : V1 → V2 is an isomorphism of G-modules. According to the previous discussion, any other nontrivial G-morphism L : V1 → V2 has to be an isomorphism. Consider the automorphism T = S −1 L : V1 → V1 . Since V1 is a complex vector space T admits at least one (non-zero) eigenvalue λ. The map λV1 − T is an endomorphism of G-modules, and ker (λV1 − T ) = 0. Invoking again the above discussion we deduce T ≡ λV1 , i.e. L ≡ λS . This shows dim HomG (V1 , V2 ) = 1. Schur’s lemma is powerful enough to completely characterize S 1 − M od, the representations of S 1 . Example 3.4.39. (The irreducible (complex) representations of S 1 ). a complex irreducible S 1 -module S1 × V (eiθ , v ) −→ Tθ v ∈ V, Let V be where Tθ1 ·Tθ2 = Tθ1 +θ2 mod 2π , In particular, this implies that each Tθ is an S 1 -automorphism since it obviously commutes with the action of this group. Hence Tθ = λ(θ)V which shows that dim V = 1 since any 1-dimensional subspace of V is S 1 -invariant. We have thus obtained a smooth map λ : S 1 → C∗ , such that λ(eiθ · eiτ ) = λ(eiθ )λ(eiθ ). Hence λ : S 1 → C∗ is a group morphism. As in the discussion of the modular function we deduce that |λ| ≡ 1. Thus, λ looks like an exponential, i.e., there exists α ∈ R such that (verify!) λ(eiθ ) = exp(iαθ), ∀θ ∈ R. Moreover, exp(2π iα) = 1, so that α ∈ Z. Conversely, for any integer n ∈ Z we have a representation S 1 → Aut (C) ρn (eiθ , z ) → einθ z. 126 CHAPTER 3. CALCULUS ON MANIFOLDS The exponentials exp(inθ) are called the characters of the representations ρn . Exercise 3.4.40. Describe the irreducible representations of T n -the n-dimensional torus. Definition 3.4.41. (a) Let V be a complex G-module, g → T (g ) ∈ Aut (V ). The character of V is the smooth function χV : G → C, χV (g ) := tr T (g ). (b) A class function is a continuous function f : G → C such that f (hgh−1 ) = f (g ) ∀g, h ∈ G. (The character of a representation is an example of class function). Theorem 3.4.42. Let G be a compact Lie group, U1 , U2 complex G-modules and χUi their characters. Then the following hold. (a)χU1 ⊕U2 = χU1 + χU2 , χU1 ⊗U2 = χU1 · χU1 . (b) χUi (1) = dim Ui . (c) χUi∗ = χUi -the complex conjugate of χUi . (d) G χUi (g )dVG (g ) = dim UiG , where UiG denotes the space of G-invariant elements of Ui , UiG = {x ∈ Ui ; x = Ti (g )x ∀g ∈ G}. (e) G χU1 (g ) · χU2 (g )dVG (g ) = dim HomG (U2 , U1 ). Proof. The parts (a) and (b) are left to the reader. To prove (c), fix an invariant Hermitian metric on U = Ui . Thus, each T (g ) is an unitary operator on U . The action of G on U ∗ is given by T (g −1 )† . Since T (g ) is unitary, we have T (g −1 )† = T (g ). This proves (c). Proof of (d). Consider P : U → U, P u = G T (g )u dVG (g ). Note that P T (h) = T (h)P , ∀h ∈ G, i.e., P ∈ HomG (U, U ). We now compute T (h)P u = G T (hg )u dVG (g ) = G ∗ T (γ )u Rh −1 dVG (γ ), G T (γ )u dVG (γ ) = P u. 3.4. INTEGRATION ON MANIFOLDS Thus, each P u is G-invariant. Conversely, if x ∈ U is G-invariant, then Px = G 127 T (g )xdg = G x dVG (g ) = x, i.e., U G = Range P . Note also that P is a projector, i.e., P 2 = P . Indeed, P 2u = G T (g )P u dVG (g ) = G P u dVG (g ) = P u. Hence P is a projection onto U G , and in particular dimC U G = tr P = Proof of (e). χU1 · χU2 dVG (g ) = ∗ dVG (g ) = χU1 · χU2 ∗ dVG (g ) = χU1 ⊗U2 G tr T (g ) dVG (g ) = G χU (g ) dVG (g ). G G G G χHom(U2 ,U1 ) = dimC (Hom (U2 , U1 ))G = dimC HomG (U2 , U1 ), since HomG coincides with the space of G-invariant morphisms. Corollary 3.4.43. Let U , V be irreducible G-modules. Then (χU , χV ) = χU · χV dg = δU V = 1 , U 0 , U ∼ =V . ∼ =V G Proof. Follows from Theorem 3.4.42 using Schur’s lemma. Corollary 3.4.44. Let U , V be two G-modules. Then U ∼ = V if and only if χU = χV . Proof. Decompose U and V as direct sums of irreducible G-modules U = ⊕m 1 (mi Ui ) V = ⊕1 (nj Vj ). Hence χU = mi χVi and χV = nj χVj . The equivalence “representation” ⇐⇒ “characters” stated by this corollary now follows immediately from Schur’s lemma and the previous corollary. Thus, the problem of describing the representations of a compact Lie group boils down to describing the characters of its irreducible representations. This problem was completely solved by Hermann Weyl, but its solution requires a lot more work that goes beyond the scope of this book. We will spend the remaining part of this subsection analyzing the equality (d) in Theorem 3.4.42. Describing the invariants of a group action was a very fashionable problem in the second half of the nineteenth century. Formula (d) mentioned above is a truly remarkable result. It allows (in principle) to compute the maximum number of linearly independent invariant elements. Let V be a complex G-module. Explicitly. as the space of complex multi-linear skew-symmetric maps V × · · · × V → C. Hence χΛk (3. cV G To obtain a more concentrated formulation of the above equality we need to recall some elementary facts of linear algebra. cV These facts can be presented coherently by considering the Z-graded vector space I• c (V ) := k ∗ Λk inv V . k ∗ Denote by bc k (V ) the complex dimension of the space of G-invariant elements in Λc V . If g ∈ G acts on V by T (g ). (see Exercise 2.128 CHAPTER 3. then g acts on Λk V ∗ by Λk T (g −1 )† = Λk T (g ). Denote by Λ• r V the space of R-multi-linear.4. Its Poincar´ e polynomial is PI• (t) = c (V ) tk bc k (V ) = tk χΛk ∗ dVG (g ). skew-symmetric maps V × · · · × V → R. cV We conclude that PI• (t) = c (V ) tk σk T (g ) dVG (g ) = det V + t T (g ) dVG (g ). For each endomorphism A of V denote by σk (A) the trace of the endomorphism Λk A : Λk V → Λk V.8) G G Consider now the following situation. .25) the number σk (A) is the coefficient of tk in the characteristic polynomial σt (A) = det(V + tA). σk (A) is given by the sum σk (A) = 1≤i1 ···ik ≤n det aiα iβ (n = dim V ). Equivalently.7) ∗ = σk T (g ) . CALCULUS ON MANIFOLDS Let V be a complex G-module and denote by χV its character.4. (We implicitly assumed that each T (g ) is unitary with respect to some G-invariant metric on V ). The complex exterior ∗ algebra Λ• c V is a complex G-module. (3. One has the equality bc k (V ) = G χΛk ∗ dVG (g ).2. (3. p p . This requires a fibered version of orientability. br k (V ) is equal to the complex dimension of Λr V ⊗ C.5 we deduce • ∼ • ∗ Λ• r V ⊗ C = Λc V ⊗C Λc V = k ∗ ∗ j ⊕i+j =k Λi c V ⊗ Λc V ∗ . so that the Poincar´ e polynomial k is of I• ( V ) = ⊕ I k r r (t) = tk br PI• k (V ). so that Λr V ⊗ C is the space of R-multi-linear. The aim of this subsection is to discuss what happens when we deal with an entire family of objects parameterized by some smooth manifold.5. Sec. and (y j ) are local coordinates on the base B (the parameter space). Using (3. r (V ) k k ∗ On the other hand. The real dimension of the subspace Ik r (V ) of k ∗ r G-invariant elements in Λr V will be denoted by bk (V ).7) and (3. instead of a single manifold F . it is a complex G-module. the complex Grassmannians. We complexify it.g.4. y j ). On the total space E we will always work with split coordinates (xi . where (xi ) are local coordinates on the standard fiber F . y ) −→ y. INTEGRATION ON MANIFOLDS 129 ∗ • The vector space Λ• r V is a real G-module.4. We will first define a fibered version of integration.2.3. We will discuss only the fibered version of integration.10) = G det V + tT (g ) det V + t T (g ) dVG (g ) det V + tT (g ) 2 = G dVG (g ). Chapter 3. we have an entire (smooth) family of them (Fb )b∈B .. §3. Using the results of Subsection 2. The model situation is the bundle E = Rk × Rm → Rm = B.5 Fibered calculus In the previous section we have described the calculus associated to objects defined on a single manifold.9) Each of the above summands is a G-invariant subspace.4. and as such. We will have the chance to use this result in computing topological invariants of manifolds with a “high degree of symmetry” like.4. Assume now that. e. skew-symmetric maps V × · · · × V → C.9) we deduce PI• (t) = r (V ) k G i+j =k σi T (g ) σj T (g ) ti+j dVG (g ) (3. In more rigorous terms this means that we are given a smooth fiber bundle p : E → B with standard fiber F . (x. The interested reader can learn more about this operation from [40].4. The exterior derivative also has a fibered version but its true meaning can only be grasped by referring to Leray’s spectral sequence of a fibration and so we will not deal with it.4. and trivializations p−1 (Uα ) −→ F × Uα .. the diffeomorphism F preserves any orientation on F . Let p : E → B be a smooth bundle with standard fiber F . for each y ∈ Uαβ . α (b) There exists an open cover (Uα ).45. The bundle is said to be orientable if the following hold. CALCULUS ON MANIFOLDS Definition 3. . then so is the total space E (as an abstract smooth manifold).4.130 CHAPTER 3.4. Exercise 3.e. (a) The manifold F is orientable . such that the gluing maps Ψ 1 Ψβ ◦ Ψ− α : F × Uαβ → F × Uαβ (Uαβ = Uα ∩ Uβ ) are fiberwise orientation preserving.46. If the the base B of an orientable bundle p : E → B is orientable. i. In the sequel all orientable bundles will be given the fiber-first orientation. y ) ∈ F E/B uniquely defined by its action on forms supported on domains D of split coordinates p D∼ = Rr × Rm → Rm .Important convention. (x. There exists a linear operator p∗ = •−r : Ω• cpt (E ) → Ωcpt (B ).b) E is given by ΩF × ωB . where ωF ∈ det Tf F (respectively ωB ∈ det Tb B ) defines the orientation of Tf F (respectively Tb B ). Let p : E → B be an orientable fiber bundle with standard fiber F . The natural orientation can thus be called the fiber-first orientation. Proposition 3. |I | = r The operator E/B is called the integration-along-fibers operator. The natural orientation of the total space E is defined as follows. This convention can be briefly described as orientation total space = orientation fiber ∧ orientation base. then If ω = f dxI ∧ dy J . The general case reduces to this one since any bundle is locally a product. f ∈ Ccpt = E/B Rr f dxI dy J 0 .47. . Let p : E → B be an orientable bundle with oriented basis B . |I | = r . ∞ (Rr+m ). and the gluing maps are fiberwise orientation preserving. f → Ψαβ (f. If E = F × B then the orientation of the tangent space T(f.4. y ) → y. r = dim F. . ∂y j  Rr j  E/B dE ω = Rr i ∂f i dx ∧ dxI ∂xi + (−1)|I |  ∂f I  dx ∧ dy j ∧ dy J . INTEGRATION ON MANIFOLDS 131 The proof goes exactly as in the non-parametric case. Then for any ω ∈ Ω∗ cpt (E ) and η ∈ ωcpt (B ) such that deg ω + deg η = dim E we have E/B dE ω = (−1)r dB ω.e. Stokes’ formula shows that the first integral is always zero. . Proposition 3. y ) → y. Let p : E → B be an orientable bundle with an r-dimensional ∗ standard fiber F .4. (x.11) ∂f i dx ∧ dxI ∧ dy J + (−1)|I | ∂xi j ∂f I dx ∧ dy j ∧ dy J .3.. (Fubini) The last equality implies immediately the projection formula p∗ (ω ∧ p∗ η ) = p∗ ω ∧ η. when B is a point. E/B The details are left to the reader. i.4.48. E/B If B is oriented and ω η ar as above then ω ∧ p∗ (η ) = E B E/B ω ∧ η. and ω = f dxI ∧ dy J . E/B The second equality is left to the reader as an exercise. Proof. Then dE ω = i p (3. ∂y j The above integrals are defined to be zero if the corresponding forms do not have degree r. Hence  ∂ dE ω = (−1)|I | j  ∂y E/B  Rr dxI  ∧ dy j ∧ dy J = (−1)r dB j ω.4. It suffices to consider only the model case p : E = Rr × Rm → Rm = B. One shows using partitions of unity that these local definitions can be patched together to produce a well defined map •−r : Ω• cpt (E ) → Ωcpt (B ). Suppose that p : E → B is an oriented fibration. ∀b ∈ B. Definition 3. ∂E ) → B be a ∂ -bundle. Remark 3. (a) A smooth manifold E with boundary ∂E . a2 + b2 = 0. CALCULUS ON MANIFOLDS Exercise 3. (b) Prove that for every compactly supported smooth function f : E → R we have p∗ (f ωE ) b = Eb f ωE/B ωB (b). Standard Models • Interior models • Boundary models A ∂ -bundle is obtained by gluing two types of local models. there exists a unique volume form ωE/B on Eb = p−1 (b) with the property that. If p : Int E → B is orientable and the basis B is oriented as well.4. for every x ∈ Eb . p E f ωE = B Eb f ωE/B ωB . Rr × Rm → Rm m m Hr + × R → R . Compute the ∧dy Gelfand-Leray residue dxdt along the fiber p(x. (ii) A smooth map p : E → B such that the restrictions p : Int E → B and p : ∂E → B are smooth bundles. we have ∗ ωE (x) = ωE/B (x) ∧ (p∗ ωB )(x) ∈ Λdim E Tx E. One can think of a ∂ -bundle as a smooth family of manifolds with boundary. (ii) The induced orientation as the boundary of E . This form is called the Gelfand-Leray residue of ωE rel p.49. ωE is a volume form on E . · · · . A ∂ -bundle is a collection (E. ∂E. y ) = 0.4. Let p : (E. 1] × M → M (t. (c) Consider the fibration R2 → R.50. where Hr + := (x1 .51. (x.4. and ωB is a volume form on B .132 CHAPTER 3.52. Example 3. . The interior fiber is the open interval (0. then on ∂E one can define two orientations.4. B ) consisting of the following. x1 ≥ 0 . y ) → t = ax + by . p. The fiber of p : ∂ (I ×M ) → M is the disjoint union of two points. (Gelfand-Leray). (i) The fiber-first orientation as the total space of an oriented bundle ∂E → B . (a) Prove that. for every b ∈ B . m) → m. 1). xr ) ∈ Rr . The standard fiber of p : Int E → B is called the interior fiber. The projection p : [0. defines a ∂ -bundle. 3.53. ∂E ) → B be an orientable ∂ -bundle with an r-dimensional interior fiber. Prove the above theorem. Then for any ω ∈ Ω• cpt (E ) we have ω= ∂E/B E/B dE ω − (−1)r dB ω (Homotopy formula). Prove that the above orientations on ∂E coincide. Exercise 3. E/B This is “the mother of all homotopy formulæ”.4. 133 Theorem 3. Let p : (E.55. Exercise 3.4. INTEGRATION ON MANIFOLDS These two orientations coincide.4. .54. It will play a crucial part in Chapter 7 when we embark on the study of DeRham cohomology. E/B The last equality can be formulated as = ∂E/B E/B dE − (−1)r dB .4. we will study the problem formulated in Chapter 1: why a plane (flat) canvas disk cannot be wrapped in an one-to-one fashinon around the unit sphere in R3 . Classical vector analysis describes one method of measuring lengths of smooth paths in R3 . Loosely speaking.1 Metric properties §4. Answering this requires the notion of Riemann curvature which will be the central theme of this chapter. If γ : [0. this means that the only operations we can perform are those which do not change the lengths of curves on the surface. this problem can be solved if we assume that there is a procedure of measuring lengths of tangent vectors at any point on our manifold. one can think of “canvas surfaces” as those surfaces on which any “reasonable” curve has a well defined length. one can say that such surfaces are endowed with a clear procedure of measuring lengths of piecewise smooth curves. then its length is given by 1 length (γ ) = 0 |γ ˙ (t)|dt. 4. Adapting a more constructive point of view. Alternatively. Thus. the Riemannian geometry studies the properties of surfaces (manifolds) “made of canvas”. These are manifolds with an extra structure arising naturally in many instances.1. On such manifolds one can speak of the length of a curve. 134 . The simplest way to do achieve this is to assume that each tangent space is endowed with an inner product (which can vary from point to point in a smooth way).1 Definitions and examples To motivate our definition we will first try to formulate rigorously what do we mean by a “canvas surface”. 1] → R3 is such a paths.Chapter 4 Riemannian Geometry Now we can finally put to work the abstract notions discussed in the previous chapters. A “canvas surface” can be deformed in many ways but with some limitations: it cannot be stretched as a rubber surface because the fibers of the canvas are flexible but not elastic. We want to do the same thing on an abstract manifold. where |γ ˙ (t)| is the Euclidean length of the tangent vector γ ˙ (t). and the angle between two smooth curves. and we are clearly faced with one problem: how do we make sense of the length |γ ˙ (t)|? Obviously. In particular. 2) are said to be isometric if there exists a diffeomorphism φ : M1 → M2 such that φ∗ g2 = g1 .2.1. Let (xi α ) be a collection of local coordinates on Uα . g0 ) is the classical Euclidean geometry.1.1. i. Then RM is a non-empty convex cone in the linear space of symmetric (0. 2)–tensor field on M . . for any x ∈ M . g ) is a Riemann manifold then. then we get a local description of g as g = gij dxi dxj . gi ) (i = 1.1. 2)–tensors. The only thing that is not obvious is that RM is non-empty. METRIC PROPERTIES 135 Definition 4. ∂xi ∂xj Proposition 4. If (M. The space Rn has a natural Riemann metric g0 = (dx1 )2 + · · · + (dxn )2 . g ) consisting of a smooth manifold M and a metric g on the tangent bundle. then we define its length by b l (γ ) = a |γ ˙ (t)|γ (t) dt. We will use again partitions of unity. i. The geometry of (Rn . Let M be a smooth manifold. Proof. the restriction gx : Tx M × Tx M → R is an inner product on the tangent space Tx M .1. Example 4. Using these local coordinates we can construct by hand the metric gα on Uα by 2 n 2 gα = (dx1 α ) + · · · + (dxα ) . b] → M is a piecewise smooth path.e. (b) Two Riemann manifolds (Mi . We will frequently use the alternative notation (•. . and it is indeed a Riemann metric on M.. . (The Euclidean space). . xn ) on M . Then define g= β ∈B βgφ(β ) . . and denote by RM the set of Riemann metrics on M .4. |v |x := gx (v. pick a partition of unity B ⊂ C0 α α∈A . a smooth.e. If γ : [a. The tensor g is called a Riemann metric on M . •). •)x = gx (•. ). (a) A Riemann manifold is a pair (M. ∞ (M ) subordinated to the cover (U ) Now..3. v )1/2 . symmetric positive definite (0. The reader can check easily that g is well defined. there exits a map φ : B → A such that ∀β ∈ B supp β ⊂ Uφ(β ) . If we choose local coordinates (x1 . Cover M by coordinate neighborhoods (Uα )α∈A . gij = g ( ∂ ∂ . The length of a tangent vector v ∈ Tx M is defined as usual. For example. Figure 4. (Ax. RIEMANNIAN GEOMETRY Example 4.1). but they look “different ” (see Figure 4. g ) be a Riemann manifold and S ⊂ M a submanifold. . On any manifold there exist many Riemann metrics. 12 22 32 . then HIn is the unit sphere in Rn . where (•. HA has a natural metric induced by the Euclidean metric on Rn . For example. and the induced metric is called the round metric of S n−1 . when A = In . Remark 4. If ı : S → M denotes the natural inclusion then we obtain by pull back a metric on S gS = ı∗ g = g |S .1: The unit sphere and an ellipsoid look “different”. any invertible symmetric n × n matrix defines a quadratic hypersurface in Rn by HA = x ∈ Rn .2: A plane sheet and a half cylinder are “not so different”. Figure 4. Let (M. and there is no natural way of selecting one of them. the unit sphere x2 + y 2 + z 2 = 1 is diffeomorphic 2 2 2 to the ellipsoid x +y +z = 1. •) denotes the Euclidean inner product on Rn .136 CHAPTER 4. x) = 1.4. For example.1. However.1.5. (Induced metrics on submanifolds). One can visualize a Riemann structure as defining a “shape” of the manifold. X. Y g 1 (du2 + dv 2 ).1.7. endowed with the metric h= Show that the Cayley transform z = x + iy → w = −i establishes an isometry (D.1. Y ) = X. Example 4. g ) where D is the unit open disk in the plane R2 and the metric g is given by 1 g= (dx2 + dy 2 ).4.4. Y ∈ Tg G. (i) A line segment is the shortest path connecting two given points. where (Lg−1 )∗ : Tg G → T1 G is the differential at g ∈ G of the left translation map Lg−1 . · g on G defined by hg (X. ∀g ∈ G. Consider a Lie group G.1.8. L∗ g h = h ∀g ∈ G.6. . 4v 2 z+i = u + iv z−i = (Lg−1 )∗ X. The Poincar´ e model of the hyperbolic plane is the Riemann manifold (D. · on LG induces a Riemann metric h = ·.2 it is illustrated the deformation of a sheet of paper to a half cylinder. · g is a smooth tensor field. Example 4. v ) ∈ R2 . The conclusion to be drawn from these two examples is that we have to be very careful when we use the attribute “different”. but the metric structures are the same since we have not changed the lengths of curves on our sheet.1.2 to produce metrics which are both left and right invariant. (Left invariant metrics on Lie groups). (The hyperbolic plane).1. (Lg−1 )∗ Y . One checks easily that the correspondence G g → ·. The first question one may ask is whether there is a notion of “straight line” on a Riemann manifold. g ) ∼ = (H. we can use the averaging technique of Subsection 3. §4. In the Euclidean space R3 there are at least two ways to define a line segment. METRIC PROPERTIES 137 appearances may be deceiving. and it is left invariant. They look different. h). v > 0}. If G is also compact. and denote by LG its Lie algebra.e.2 The Levi-Civita connection To continue our study of Riemann manifolds we will try to follow a close parallel with classical Euclidean geometry. Let H denote the upper half-plane H = {(u. 1 − x2 − y 2 Exercise 4.. i. Then any inner product ·. In Figure 4. Z ) +g (X. Y ). Then there exists a unique symmetric connection ∇ on T M compatible with the metric g i. Z ∈ Vect (M ) we have Zg (X.1. Moreover. i ∂j = 0 ∀i.9. For any X. X ) + Xg (Y. ∇i ∂k ) = 0. . Using the symmetry of the connection we compute Zg (X. Condition (4. it has a natural trivial connection ∇0 defined by ∂ ∇0 . ∇Z Y ) − g (Z. Consider a Riemann manifold (M.. we would like this connection to be compatible with the metric. Y ]. j. Uniqueness. but even so.1. ∇Y X ) + g (Y. Of course. ∀X. Y ) = g (∇Z X. Y ) − Y g (Z. ∇g = 0.1. ∂k ) − g0 (∂j . ∂k ) = ∇i δjk − g0 (∇i ∂j . Let ∇ be such a connection. γ ˙ (t) γ (4.2) so that the problem of defining “lines” in a Riemann manifold reduces to choosing a “natural” connection on the tangent bundle. X ]. ∇X Z ) = g ([Z. the connection is compatible with the metric.. there are infinitely many connections to choose from.. if g0 denotes the Euclidean metric. all the Christoffel symbols vanish. we will use the second interpretation as our starting point.1.e. X ) + g (∇X Y.e.1). T (∇) = 0. Proof. Let us first reformulate (4. X ) + g ([X. then 0 0 0 ∇0 i g0 (∂j . Y ]. Y ) + 2g (∇X Z.1.1) can be rephrased as ∇0 ˙ (t) = 0. Proposition 4.138 CHAPTER 4. The connection ∇ is usually called the Levi-Civita connection associated to the metric g . Y. Y ∈ Vect (M ). Y ) + g (X.e. ∇g = 0 and ∇X Y − ∇Y X = [X. X ) = g (∇Z X. ∇i := ∇∂i .e. (4. ∇Z Y ) since ∇g = 0.1) Since we have not said anything about calculus of variations which deals precisely with problems of type (i). where ∂i := ∂xi i. i. We will soon see however that both points of view yield the same conclusion. the tangent bundle of R3 is equipped with a natural trivialization. As we know. i. 1] → R3 satisfying γ ¨ (t) = 0. Y ]. The following fundamental result will solve this dilemma. We will achieve this by producing an explicit description of a connection with the above two mproperties. RIEMANNIAN GEOMETRY (ii) A line segment is a smooth path γ : [0. and as such. Z ) + g ([Z. Y ) − g (∇Y Z. g ). . .. . If (g i ) denotes the inverse of (gi ) we deduce 1 Γij = g k (∂i gjk − ∂k gij + ∂j gik ) . Definition 4. The above equality establishes the uniqueness of ∇. We can now define the notion of “straight line” on a Riemann manifold. Z = ∂j = ∂x ).3) indeed defines a connection with the required properties. Z ) − Y g (Z. . Z ]. Using local coordinates (x1 .. It boils down to showing that (4. Y ) = {Xg (Y.3). Z ) + g ([Y. ∇dγ ˙ (t) = x ¨i ∂i + x ˙ i ∇ d ∂i = x ¨i ∂i + x ˙ ix ˙ j ∇j ∂i dt dt (4.5) =x ¨ ∂k + k Γk ˙ ix ˙ j ∂k ji x k (Γk ij = Γji ). we can use the classical Banach-Picard theorem on existence in initial value problems (see e. nonlinear system of ordinary differential equations. [4]).4.3) ∂ ∂xi . X ]. The routine details are left to the reader. with X = ∂i = ∂ ∂ Y = ∂k = ∂x .1.. ∂k ) = gk Γij = 1 (∂i gjk − ∂k gij + ∂j gik ) . ∇∂i ∂j = Γij ∂ .1. Set d =γ ˙ (t) = x ˙ i ∂i . b) → M. that j k g (∇i ∂j . . .1.1. (4. 139 (4. satisfying ∇γ ˙ (t) = 0.6) k Since the coefficients Γk ij = Γij (x) depend smoothly upon x. Y ) 2 −g ([X. so that the geodesic equation is equivalent to x ¨k + Γk ˙ ix ˙j = 0 ij x ∀k = 1. xn ) on M we deduce from (4. i.. ˙ (t) γ where ∇ is the Levi-Civita connection. .1. . the scalars Γij denote the Christoffel symbols of ∇ in these coordinates. .e. and the path γ is described by γ (t) = (x (t).. n. Y )}.1. x (t))..1. dt Then. 2 (4. we can rewrite the geodesic equation as a second order. g ) is a smooth path γ : (a. METRIC PROPERTIES We conclude that 1 g (∇X Z. X ) + Zg (X.4) Existence.1.. Y ]. xn ) with respect to which the Christoffel symbols are 1 n (Γk ij ). Using local coordinates (x1 .g.10. X ) − g ([Z. A geodesic on a Riemann manifold (M. 2 Above... We deduce the following local existence result. for any X ∈ LG .X : (−ε. [X. Exercise 4. ad(Y )].15.1. where the bracket in the right hand side is the usual commutator of two endomorphisms.1. i. |X | = r}. X ) = (γ (t). then the length of γ ˙ (t) is independent of time.X is defined at each moment of time t for any (x. γ ˙ (t)) γ = γx. Y ]. The geodesic flow has some remarkable properties. then the Riemann manifold is called geodesically complete.e. we deduce that Sr (M ) are invariant subsets of the geodesic flow. [X. The local flow defined above is called the geodesic flow of the Riemann manifold (M. the operator ad(X ) is skew-adjoint.140 CHAPTER 4.X .e. Z = − Y. For any compact subset K ⊂ T M there exists ε > 0 such that for any (x. RIEMANNIAN GEOMETRY Proposition 4. Y ]) = [ad(X ).1. γ dt dt Thus. · on LG such that. X ) ∈ K there exists a unique geodesic γ = γx. γ ˙ (0) = X . One can think of a geodesic as defining a path in the tangent bundle t → (γ (t).. i. Y ]. ε) → M such that γ (0) = x.1.14. g ).7) We can now extend this inner product to a left invariant metric h on G. We want to describe its geodesics. Any X ∈ LG defines an endomorphism ad(X ) of LG by ad(X )Y := [X.1. g ) be a Riemann manifold.. Proposition 4. Let (M. Assume that there exists an inner product ·. If the path γ (t) is a geodesic. .12. When the geodesic glow is a global flow. Describe the infinitesimal generator of the geodesic flow.11. if we consider the sphere bundles Sr (M ) = {X ∈ T M . The Jacobi identity implies that ad([X. X ) ∈ T M . ˙ (t) . γ ˙ (t)). The above proposition shows that the geodesics define a local flow Φ on T M by Φt (x. Example 4.1. (4. γ ˙ (t)) = 2g (∇γ ˙ (t)) = 0. any γx. and let LG be its Lie algebra. Definition 4. Let G be a connected Lie group. Proof.13 (Conservation of energy). Z ] . We have d d |γ ˙ (t)|2 = g (γ ˙ (t). on any compact Lie groups there exist metrics satisfying (4. Z ) = const. Y )}. Y }.1.1. METRIC PROPERTIES 141 First. Y ]. Prove that its restriction to LG satisfies (4. at 1 ∈ G. i. Xj ].. so that the first three terms in the above formula vanish. X − [Z.1. Let L be a finite dimensional real Lie algebra.. Y = so that.16.17. Z + [Y. 2 This formula correctly defines a connection since any X ∈ Vect (G) can be written as a linear combination X= αi Xi αi ∈ C ∞ (G) Xi ∈ LG . Y. Z ]. we have to determine the associated Levi-Civita connection. This means that γ is an integral curve of the left invariant vector field X so that the geodesics through the origin with initial direction X ∈ T1 G are γX (t) = exp(tX ). If γ (t) is a geodesic. Z ) + h([Y. The Killing pairing or form is the bilinear map κ : L × L → R. 2 Using the skew-symmetry of ad(X ) and ad(Z ) we deduce ∇X Z. κ(x. h(X.j Since [Xi .1. Using (4. Z ) − Y (Z. Exercise 4. these vector fields are left invariant. If we take X.3) we get 1 h(∇X Z.1. Z ] 2 ∀X. The Lie algebra L is said to be semisimple if the Killing pairing is a duality. i. Xi ]. Y . Z ]. A Lie group G is called semisimple if its Lie algebra is semisimple. . Y ) = const.e. Y ]. X ) + Zh(X. then h(Y. we can write γ ˙ (t) = 0 = ∇γ ˙ (t) = ˙ (t) γ i γi Xi . Y = {− [X. In particular.1. X ) − h([Z.8) 1 [X. X ].7). Xj ] = −[Xj .1.. (4. X ]. Z ∈ LG . Y ) 2 −h([X.e. we deduce γ ˙ i = 0. X ) = const. y ) := −tr (ad(x) ad(y )) x. Z ∈ LG . h(Z.7). Y ) = {Xh(Y. Z ]. y ∈ L. we have 1 ∇X Z = [X.4. so that γ ˙ i Xi + 1 2 γi γj [Xi . Let G be a Lie group and h a bi-invariant metric on G. i. γ ˙ (t) = γi (0)Xi = X.. We obtain the following equality (at 1 ∈ G) 1 ∇X Z. Definition 4. . The metric g on T n is bi-invariant. and we let the reader verify that these are in fact the great circles of S 3 . RIEMANNIAN GEOMETRY Exercise 4. On the contrary..αn is dense in T n !!! (see also Section 7. i. Prove that SO(n) and SU (n) and SL(n.. compact Lie group. we have seen examples in which one of the basic axioms of Euclidean geometry no longer holds: two distinct geodesic (read lines) may intersect in more than one point.7).1...1. This is known as Weyl’s theorem. (Geodesics on flat tori. when the α’s are linearly independent over Q then a classical result of Kronecker (see e. requires substantially more work. so that SU (2) is diffeomorphic with the unit sphere S 3 ⊂ R4 .. R) are semisimple Lie groups. The geodesics of this metric are the 1parameter subgroups of S 3 .αn (t) t → (eiα1 t . and on SU (2)). Thus.1. any semisimple Lie group with positive definite Killing form is compact.1... 0. 2 2 Example 4. . 0. then obviously γα1 .3 The exponential map and normal coordinates We have already seen that there are many differences between the classical Euclidean geometry and the the general Riemannian geometry in the large. which will be given later in the book. all the geodesics on S 3 are closed. 1 . Show that the parallel transport of X along exp(tY ) is (Lexp( t Y ) )∗ (Rexp( t Y ) )∗ X. If the numbers αk are linearly dependent over Q. and its restriction to (1. eiαn t ) αk ∈ R.1..e..1... a2 + b2 + c2 + d2 = 1 .142 CHAPTER 4. The global topology of the manifold is responsible for this “failure”.. Prove that the Killing form is positive semi-definite1 and satisfies (4.g. but U (n) is not. Let G be a compact Lie group. the circles of maximal diameter on S 3 .7) since the bracket is 0..e. Exercise 4.21.4 to come) The special unitary group SU (2) can also be identified with the group of unit quaternions a + bi + cj + dk . its restriction to the origin satisfies the skew-symmetry condition (4.. θn ) are the natural angular coordinates on T n . 0) satisfies (4. The converse of the above exercise is also true. i.18.1.20. Its proof. The geodesics through 1 will be the exponentials γα1 .. .1.. If (θ1 . §4.16 Exercise 4. Hint: Use Exercise 4. then the flat metric is defined by g = (dθ1 )2 + · · · (dθn )2 . In particular.19.1.. The round metric on S 3 is bi-invariant with respect to left and right (unit) quaternionic multiplication (verify this).. and obviously.7)..αn (t) is a closed curve. The n-dimensional torus Tn ∼ = S 1 × · · · × S 1 is an Abelian. [42]) states that the image of γα1 . 4. The tangent space Tq M is an Euclidean space. k. g ) and q ∈ M as above. We define the exponential map at q by expq : V ⊂ Tq M → M. Let q ∈ U be the point with coordinates (0. j. Via a linear change in coordinates we may as well assume that gij (q ) = δij . The positive real number ρM (q ) is called the injectivity radius of M at q . In this section we will define using the metric some special collections of local coordinates in which things are very close to being Euclidean. the geodesic γq.1.e.22..X (t). METRIC PROPERTIES 143 Locally however.22 relies on the following key fact. k ∂y ∂y k We now describe a geometric way of producing such coordinates using the geodesic flow.1. We have the following result. Local Riemannian geometry is similar in many respects with the Euclidean geometry.. for any X ∈ V .X (st). Then there exists r > 0 such that the exponential map expq : Dq (r) → M is a diffeomorphism onto. gij (q ) = δij ∂gij ∂δij (q ) = (q ) = 0. ∀s > 0 γq. We can formulate this by saying that (gij ) is Euclidean up to order zero. i. Definition 4. and we can define Dq (r) ⊂ Tq M . g ) be a Riemann manifold and U an open coordinate neighborhood with coordinates (x1 . The proof of Proposition 4. . all of the classical incidence axioms hold. ∀i. . things are not “as bad”.23.1.. To achieve this we try to find local coordinates (y j ) near q such that in these new coordinates the metric is Euclidean up to order one. the open “disk” of radius r centered at the origin. We would like to “spread” the above equality to an entire neighborhood of q . We will try to find a local change in coordinates (xi ) → (y j ) in which the expression of the metric is as close as possible to the Euclidean metric g0 = δij dy i dy j . locally. Proposition 4. The supremum of all radii r with this property is denoted by ρM (q ).1. X → γq. The infimum ρM = inf ρM (q ) q is called the injectivity radius of M . Let (M. Let (M. Hence..sX (t) = γq. there exists a small neighborhood V of 0 ∈ Tq M such that. 0). Denote as usual the geodesic from q with initial direction X ∈ Tq M by γq. For example.X (1).... .X (t) is defined for all |t| ≤ 1. Note the following simple fact. xn ). To prove the second equality we need the following auxiliary result. . By definition (D0 expq )X = γ ˙ q. en ) of Tq M ...X (0) = X. Now choose an orthonormal frame (e1 . Let (xi ) be normal coordinates at q ∈ M .26. form an orthonormal basis of Tq M and this Proof.. the open set expq (Dq (r)) is called a normal neighborhood. In particular. Then we have g ij (q ) = δij and ∂ g ij (q ) = 0 ∀i. The result in the above lemma can be formulated as g ∇ ∂ ∂ xj ∂ ∂ .144 CHAPTER 4.... .1.. i ∂ x ∂ xk = 0. and denote by (x1 . For 0 < r < ρM (q ). xn ) provides a coordinatization of the open set expq (Dq (r)) ⊂ M . . and will be denoted by Br (q ) for reasons that will become apparent a little later. The Fr´ echet differential at 0 ∈ Tq M of the exponential map D0 expq : Tq M → Texpq (0) M = Tq M is the identity Tq M → Tq M . Lemma 4. It defines a line t → tX in Tq M which is mapped via the exponential map to the geodesic γq. In normal coordinates (xi ) (at q ) the Christoffel symbols Γi jk vanish at q.. k.. and denote by g ij the expression of the metric tensor in these coordinates.25. By construction.22 follows immediately from the above lemma using the inverse function theorem.1. the normal coordinates provide a first order contact between g and the Euclidean metric. any point p ∈ expq (Dq (r)) can be uniquely written as p = expq (xi ei ). ∂ xk Thus. Proposition 4. xn ) the resulting cartesian coordinates in Tq M ..1.X (t).1.. j k j n Γi jk (0)m m = 0 ∀m ∈ R mi ∂∂ xi from which we deduce the lemma. . Consider X ∈ Tq M .. Proof.24. RIEMANNIAN GEOMETRY Lemma 4. j. Proposition 4. k . j. ∀i. Proof. mn ) ∈ Rn the curve xi = mi t is the geodesic t → expq so that j k Γi jk (x(t))m m = 0. so that the collection (x1 . the vectors ei = ∂∂ xi proves the first equality. For any (m1 . The coordinates thus obtained are called normal coordinates at q . 27. METRIC PROPERTIES so that.0) F ( ∂ ∂ ∂ ∂ ∂ . We begin with a technical result which is interesting in its own. and find local coordinates which produce a second order contact with the Euclidean metric. and we will devote an entire section to this subject. ∀i. the matrix defining D(q. The reader may ask whether we can go one step further. We get a smooth map F : V → M × M (m.0) F has the form  0 ∗  .4. there is a geometric way of measuring the “second order distance” between an arbitrary metric and the Euclidean metric. This thing is in general not possible and. D(q.1. We must warn the reader that the above result does not guarantee that the postulated connecting geodesic lies entirely in Br (q ). any two points of Br (q ) can be joined by a unique geodesic of length < ε. This is a different ball game. We can . This is where the curvature of the Levi-Civita connection comes in. As we have already mentioned.1. and ε > 0 such that. g ) be a Riemann manifold. We compute the differential of F at (q. j. at least locally. In particular. X j ) near (q. 0) ∈ T M such that expm X is well defined for all (m. 0) are D(q. 0). in fact. §4. ∇ Using ∇g = 0 we deduce ∂ g ij (q ) ∂ xk ∂ ∂ xj 145 ∂ = 0 at q.1. ∀m ∈ Br (q ). The partial derivatives of F at (q. At this step we are in for a big surprise. Lemma 4. it is nonsingular.9) ∂ = ( ∂x k g ij ) |q = 0. 0) ∈ T M diffeomorphically onto some neighborhood U of (q. Let (M. For each q ∈ M there exists 0 < r < ρM (q ).0) F ( )= . and in particular. this is not the unique way of extending the notion of Euclidean straight line to arbitrary Riemann manifolds. )= + i i i i ∂x ∂x ∂X ∂X ∂X i Thus. Proof.1. X ) ∈ V . we have ε < ρM (m) and Bε (m) ⊃ Br (q ). q ) ∈ M × M . 0) ∈ T M . It follows from the implicit function theorem that F maps some neighborhood V of (q. We will prove that the geodesics defined as in the previous subsection do just that.4 The length minimizing property of geodesics We defined geodesics via a second order equation imitating the second order equation defining lines in an Euclidean space. using normal coordinates (xi ) near q we get coordinates (xi . ∂ xi (4. First. X ) → (m. One may try to look for curves of minimal length joining two given points. Using the smooth dependence upon initial data in ordinary differential equations we deduce that there exists an open neighborhood V of (q. expm X ). and assume 0 < R < ρM (m). and consider the unique geodesic γ : [0.28. Let m ∈ M be an arbitrary point.e. In BR (m) ⊂ M .146 CHAPTER 4. X ) . m2 ∈ Br (q ) =⇒ (m1 . R] × [0. Clearly. the curve t → expm (tX ) is a geodesic of length < ε joining m to expm (X ). Lemma 4. We can now formulate the main result of this subsection.e. and Bε (m) = expm (Dε (m)) ⊃ Br (q ). Proof. 1] → M is a piecewise smooth path with the same endpoints as γ then 1 1 |γ ˙ (t)|dt ≤ 0 0 |ω ˙ (t)|dt. R) × (0.. It is the unique geodesic with this property since F : V → U is injective. and its differential is nowhere zero. If we use normal coordinates on BR (m) we can express f as the embedding (0. m ∈ Bδ (q )} for some sufficiently small ε and δ . |X |m < ε. Let q . t) = expm (sX (t)) (s. γ is the shortest path joining its endpoints. The proof relies on two lemmata.. 1]) = γ ([0. We have to prove that the curve t → expm (δX (t)) are orthogonal to the radial geodesics s → expm (sX (t)). 1 → Tm M. 0 ≤ s ≤ R. Thus. the geodesics through m are orthogonal to the hypersurfaces Σδ = expq (Sδ (0)) = expm (X ). Let t → X (t). i. t) → sX (t). Choose 0 < r < min(ε. 1] → M of length < ε joining two points of Br (q ). X (t) is a curve on the unit sphere S1 (0) ⊂ Tm M . the map expm : Dε (m) ⊂ Tm M → M is a diffeomorphism onto its image. 0 < δ < R. 1] → M. i. We assume that the map t → X (t) ∈ S1 (0) is an embedding. R) × (0. it is injective. 0 ≤ t ≤ 1 denote a smooth curve in Tm M such that |X (t)|m = 1. Consider the smooth map f : [0. t) ∈ (0. f (s. (s. for any m ∈ M .1. . RIEMANNIAN GEOMETRY choose V to have the form {(m. In particular.1. for any m ∈ Br (q ). ρM (q )) such that m1 .29 (Gauss). 1). m2 ) ∈ U. Theorem 4. If ω : [0. we deduce that. r and ε as in the previous lemma. with equality if and only if ω ([0. |X | = δ . 1]). Each ω (t) can be uniquely expressed in the form ω (t) = expm (ρ(t)X (t)) |X (t)| = 1 0 < |ρ(t)| < R. where ·. t). Using the metric compatibility of the Levi-Civita connection along the image of f we compute ∂s ∂s . · denotes the inner product defined by g .) are geodesics. ∇∂t ∂s = ∂t |∂s |2 = 0. we deduce (using the symmetry of the Levi-Civita connection) 1 ∂s . We have to show ∂s .30.4. piecewise smooth curve ω : [a. ∂t + ∂s . On the other hand. For s = 0 we have f (0. and as such we have the equality ∂s = X (t). ∂t ] = 0. The equality holds if and only if X (t) = const and ρ ˙ (t) ≥ 0. Lemma 4. so that ω (t) = f (ρ(t). we deduce ∇∂s ∂s = 0. t). We conclude that the quantity ∂s . as needed. t) = expm (0) so that ∂t |s=0 = 0. ∇∂s ∂t = ∂s . Using the normal coordinates on BR (m) we can think of ∂s as a vector field on a region in Tm M . Since the curves s → f (s. t). Now consider any continuous. since [∂s . Proof. t = const. ∂t is independent of s. and since ∂f = |X (t)| = 1 ∂ρ . the shortest path joining two concentrical shells Σδ is a radial geodesic. Then ω ˙ = Since the vectors ∂f ∂ρ ∂f ∂f ρ ˙+ . In other words. and therefore ∂s .1. b] → BR (m) \ {m}. Define ∂t similarly. ∂t = 0 ∀(s. METRIC PROPERTIES Set ∂s := f∗ 147 ∂ ∂s ∈ Tf (s. The length of the curve ω (t) is ≥ |ρ(b) − ρ(a)|. ∇∂s ∂t .t) M. ∂t = ∇∂s ∂s . ∂ρ ∂t and ∂f ∂t are orthogonal. ∂t = 0 ∀(s. 2 since |∂s | = |X (t)| = 1. Let f (ρ. and ∂t are sections of the restriction of T M to the image of f . t) := expm (ρX (t)).1. The objects ∂s . More precisely. q ) < r . X b |ω ˙ |dt ≥ a a |ρ ˙ |dt ≥ |ρ(b) − ρ(a)|. Let m0 . 1] → M of length < ε such that γ (i) = mi .e. any minimal path can be reparametrized such that it satisfies the geodesic equation. (4.1. q ) = inf l(ω ). Exercise 4. Theorem 4. 1] → M piecewise smooth path joining p to q A piecewise smooth path ω connecting two points p. i = 0.10) Corollary 4. and consider a geodesic γ : [0. 1. b] → M joining m0 to m1 . Prove the above corollary. Equality holds if and only if ρ(t) is monotone.1.33.148 we get CHAPTER 4. ˙ = 0.28 also implies that any two nearby points can be joined by a unique minimal geodesic. Letting δ → 0 we deduce l(ω ) ≥ R = l(γ ). RIEMANNIAN GEOMETRY |ω ˙ |2 = |ρ ˙ |2 + The equality holds if and only if b ∂f ∂t ∂f ∂t 2 ≥ |ρ ˙ |2 . 1]) then we obtain a strict inequality. This completes the proof of the lemma. Any Riemann manifold has a natural structure of metric space. and X (t) is constant. For any δ > 0 this path must contain a portion joining the shell Σδ (m0 ) to the shell ΣR (m0 ) and lying between them.10) .1. we set d(p.1. i. The image of any minimal path coincides with the image of a geodesic. Thus = 0.1. For any q ∈ M we have the equality ρM (q ) = sup r. r satisfies (4. By the previous lemma the length of this segment will be ≥ R − δ . ω : [0. . We can write γ (t) = expm0 (tX ) X ∈ Dε (m0 ). q such that l(ω ) = d(p. The proof of Theorem 4.1.28 we deduce immediately the following consequence. m1 ∈ Br (q ).1. Let q ∈ M . q ) is said to be minimal.28 is now immediate.32. In particular we have the following consequence. Set R = |X |. b]) does not coincide with γ ([0. In other words. From Theorem 4. Corollary 4. Consider any other piecewise smooth path ω : [a.1. Then for all r > 0 sufficiently small expq (Dr (0))( = Br (q ) ) = p ∈ M d(p.1. Corollary 4. If ω ([a.31.34. For any q ∈ M there exists 0 < R < ιM (q ) such that for any r < R the ball Br (q ) is convex. Proposition 4. and we have 2 r2 = (x1 )2 + · · · + (xn )2 . At this moment it is convenient to use normal coordinates (xi ) near q . m1 ) are sufficiently small. m1 in BR (q ) can be joined by a unique minimal geodesic [0.m1 (t) can be uniquely expressed as γm0 . We compute easily d2 2 ¨ n ) + |x ˙ |2 (r ) = 2r2 (¨ x1 + · · · + x dt2 ¨ i + Γi ˙ jx ˙k = 0 x jk (x)x Since Γi jk (0) = 0 (normal coordinates). The geodesic γm0 . Choose ax 0 < ε < 2 ρM (q ). we deduce that there exists a constant C > 0 (depending only on the magnitude of the second derivatives of the metric at q ) such that |Γi jk (x)| ≤ C |x|. m1 ∈ BR (q ) the map t → d(q.36.4. (4. 1. The same argument used in the proof of Theorem 4. We will prove that ∀m0 .m1 takes the form (xi (t)). and 0 < R < ε such that any two points m0 . γm0 .m1 (t) = expq (r(t)X (t)) X (t) ∈ Tq M with r(t) = d(q.11) ˙ (t) = where x ˙ i ei ∈ Tq M .1.m1 (t)). x We substitute the above inequality in (4. A subset U ⊂ M is said to be convex if any two points in U can be joined by a unique minimal geodesic which lies entirely inside U . Definition 4. (r ) ≥ 2|x dt2 If we chose from the very beginning R + ε ≤ (nC )−1/3 . METRIC PROPERTIES 149 Proof.1. m0 ) and d(q. 1] if d(q.1.1. Note that d(q.1. γm0 . γ (t)) < R + ε < ρM (q ). Using the geodesic equation we obtain ¨ i ≥ −C |x||x ˙ |2 . 1] t → γm0 . The geodesic γm0 .1. 1 Proof.11) to get d2 2 ˙ |2 1 − nC |x|3 .28 shows that for any 0 < r < ρM (q ) the radial geodesics expq (tX ) are minimal. d 2 It suffices to show dt 2 (r ) ≥ 0 for t ∈ [0.1. not necessarily contained in BR (q ). The path γ satisfies the equation x .m1 (t)) is convex and thus it achieves its maxima at the endpoints t = 0.35.m1 (t) of length < ε.12) (4. on any manifold there exist complete Riemann metrics. Think for example of a manifold where there exist closed geodesic.150 CHAPTER 4. §4.38.1. e. g ) be a Riemann manifold and let (Uα ) an open cover consisting of bounded geodesically convex open sets.12) is nonnegative. First. Let M = R3 with the Euclidean metric. (d) M is geodesically complete. We devote this subsection to describing this more general “vector analysis”. Kn ⊂ Kn+1 and n Kn = M such that if pn ∈ Kn then d(q. Theorem 4.. On a complete manifold there could exist points (sufficiently far apart) which can be joined by more than one minimal geodesic.g. and an orientation. g ) be an oriented Riemann manifold.1.41. Prove the Hopf-Rinow theorem.5 Calculus on Riemann manifolds The classical vector analysis extends nicely to Riemann manifolds.39. We now have two structures at our disposal: a Riemann metric. This establishes the convexity of t → r2 (t) and concludes the proof of the proposition. The following assertions are equivalent: (a) expq is defined on all of Tq M . RIEMANNIAN GEOMETRY then. (e) There exists a sequence of compact sets Kn ⊂ M . Moreover. the tori T n . Exercise 4. Let M be a Riemann manifold and q ∈ M . Let (M. using the metric we can construct by duality the lowering-the-indices isomorphism L : Vect (M ) → Ω1 (M ). (b) The closed and bounded (with respect to the metric structure) sets of M are compact.40. on a (geodesically) complete manifold any two points can be joined by a minimal geodesic. Hence. A vector field V on M has the form ∂ ∂ ∂ V =P +Q +R . In the last result of this subsection we return to the concept of geodesic completeness.1. the right-hand-side of (4. Using a partition of unity (ϕi ) subordinated to this cover we can form a new metric g ˜= i ϕi gα(i) (supp ϕi ⊂ Uα(i) ). (c) M is complete as a metric space. Set dα = (diameter (Uα ))2 . Example 4. because along the geodesic we have |x| ≤ R + ε. Exercise 4. Denote by gα the 1 metric on Uα defined by gα = d− α g so that the diameter of Uα in the new metric is 1. Remark 4. Prove that g ˜ is a complete Riemann metric.1.37 (Hopf-Rinow). pn ) → ∞. Let (M. ∂x ∂y ∂z . and we will use both of them to construct a plethora of operations on tensors.1.1. We will see that this can be described in terms of the metric space structure alone.1. 1.1. and set . Near p it can be written as ω= I ωI dxI . for any ordered multi-index I : (1 ≤ i1 < · · · < ik ≤ n). Let ω ∈ Ωk (M ). Then the exterior derivative d is related to the Levi-Civita on Λk T ∗ M connection via the equality d = ε ◦ ∇. .42. where as usual. at p. Denote temporarily by D the operator ε ◦ ∇.42. Note that ∂i := ∂∂ xi D= i dxi ∧ ∇i . β ∈ Ωk (M ). Thus. then W is the infinitesimal work of V . ∀α ∈ Ω1 (M ). METRIC PROPERTIES Then W = LV = P dx + Qdy + Rdz. Proof. The equality d = D is a local statement.1. 151 If we think of V as a field of forces in the space. Proposition 4. I = I dxi ∧ (∂j ωI dxI + ωI ∇i (dxI )) = Since the point p was chosen arbitrarily this completes the proof of Proposition 4. Our discussion about normal coordinates will payoff. Dω = I dxi ∧ ∇i (ωI dxI ) dxi ∧ ∂j ωI = dω. we set dxI := dxi1 ∧ · · · ∧ dxik . In normal coordinates at p we have (∇i ∂i ) |p = 0 from which we get the equalities (∇i dxj ) |p (∂k ) = −(dxj (∇i ∂k )) |p = 0.4. We will use a strategy useful in many other situations. ∇i = ∇∂i . On a Riemann manifold there is an equivalent way of describing the exterior derivative. Choose (xi ) normal coordinates at an arbitrary point p ∈ M . and it suffices to prove it in any coordinate neighborhood. Let ε : C ∞ (T ∗ M ⊗ Λk T ∗ M ) → C ∞ (Λk+1 T ∗ M ) denote the exterior multiplication operator ε(α ⊗ β ) = α ∧ β. 1. (∇ denotes the Levi-Civita connection. and ei = ∂i |p .1. ∂n ) = (det(g ij ))1/2 . Such a factor is zero since we are working in . .. (4.. Choose normal coordinates (xi ) near p. RIEMANNIAN GEOMETRY Exercise 4. g ) the exterior derivative dω can be expressed by k dω (X0 . given two tensor fields T1 . xn ) are local coordinates such that dx1 ∧ · · · ∧ dxn is positively oriented. . .1. .) r (M ) which we continue The Riemann metric defines a metric in any tensor bundle Ts to denote by g . . .. ... We have the following not so surprising result.. we may as well assume X = ∂k .1. (∇X ∂i ) |p . ∇X dVg = 0. Note first that dVg (∂1 ..14) We consider each term separately.. . If (x1 .. Thus... . en ) = 0.2. Since the expression in (4. ..p = (T.. T2 of the same type (r. In particular. Show that for any k -form ω on the Riemann manifold (M. . . any such tensor has a pointwise norm M /2 p → |T |g. ∂n ).. X for all X0 . Using the orientation we can construct (using the results in subsection 2. where |g | := det(gij ). .4) a natural volume form on M which we denote by dVg . T )1 p .. n. This is the positively oriented volume form of pointwise norm ≡ 1. xi g ij = g (∂i . In particular. Xk ) = i=0 ˆ i . and we call it the metric volume.g ) def M ∞ f dvg ∀f ∈ C0 (M ). Proof. .13) where e1 . en ) = X (dVg (∂1 . .. . Proposition 4.1. so that X (det(g ij ))1/2 |p = ∂k (det(g ij ))1/2 |p is a linear combination of products in which each product has a factor of the form ∂k g ij |p . we can integrate (compactly supported) functions on M by f = (M. . s) we can form their pointwise scalar product M p → g (T. .. ..44.. . Xk ). . . We compute (∇X dVg )(e1 .. Xk ∈ Vect (M ). T2 (p)).43... for some k = 1. (−1)i (∇Xi ω )(X0 ... . then dVg = |g |dx1 ∧ · · · ∧ dxn . ∀X ∈ Vect (M )..13) is linear in X . ep is a basis of Tp M . . ∂n )) |p − i dvg (e1 . S )p = gp (T1 (p). Set ∂i = ∂∂ . ∂k )... We have to show that for any p ∈ M (∇X dVg )(e1 .152 CHAPTER 4. X1 . . (4. 45.1. or div g X .1. The infinitesimal energy flux of F is the 2-form ΦF = ∗WF = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy.1. X ) = df (X ) = X · f ∀X ∈ Vect (M ). ∗1 = dVg . Thus.44 is proved. the smooth function defined by the equality LX dVg = (div X )dVg .14) is zero. or gradg f . The divergence of F is the scalar defined as div F = ∗d ∗ WF = ∗dΦF = ∗ {(∂x P + ∂y Q + ∂z R)dx ∧ dy ∧ dz } = ∂x P + ∂y Q + ∂z R. METRIC PROPERTIES 153 normal coordinates. Proposition 4. The exterior derivative of WF is the infinitesimal flux of the vector field curl F dWF = (∂y R − ∂z Q)dy ∧ dz + (∂z P − ∂x R)dz ∧ dx + (∂x Q − ∂y P )dx ∧ dy = Φcurl F = ∗Wcurl F . In other words g (grad f.1. the vector field g -dual to the 1-form df . The other terms are zero as well since in normal coordinates at p we have the equality ∇X ∂i = ∇∂k ∂i = 0. g ) we denote by grad f . ∀α β ∈ Ωk (M ). uniquely determined by α ∧ ∗β = (α. . (a) For any smooth function f on the Riemann manifold (M. and X ∈ Vect (M ). (4.46. then we compute easily 2 2 2 ∗d ∗ df = ∂x f + ∂y f + ∂x f = ∆f.15) Definition 4. the first term in(4. g ) is an oriented Riemann manifold. we denote by div X . we also have the Hodge ∗-operator ∗ : Ωk (M ) → Ωn−k (M ).1. Once we have an orientation. In particular. Example 4.4. (b) If (M. If f is a function on R3 . β )dVg . To any vector field F = P ∂x + Q∂y + R∂z on R3 we associated its infinitesimal work WF = L(F ) = P dx + Qdy + Rdz.1. The first integral in the right-hand-side vanishes by the Stokes formula since α ∧ ∗β has compact support. . Then M 1 |g | ∂i ( |g |X i ). where we view ∇X as an element of C ∞ (End (T M )) via the identifications ∇X ∈ Ω1 (T M ) ∼ = C ∞ (T ∗ M ⊗ T M ) ∼ = C ∞ (End (T M )).48.k = nk + n + 1. (b) div X = ∗d ∗ α. (a) Express g in the spherical coordinates (r.154 CHAPTER 4.k M (α. instead of the more precise g (•. . then div X = where X = X i ∂i . (dα.1. and denote by g the Riemann metric on S 2 induced by the Euclidean metric g0 = dx2 + dy 2 + dz 2 on R3 . β )dVg = dα ∧ ∗β = M M d(α ∧ ∗β ) + (−1)k M α ∧ d ∗ β. ϕ) defined as in Example 3. Then (a) div X = tr (∇X ). β )dVg = (−1)k+(n−k+1)(n−k) α ∧ ∗δβ. z ) = z . z ) ∈ R3 . δβ )dVg . y. where νn. Consider the unit sphere S2 = (x. Let X be a vector field on the oriented Riemann manifold (M.. For simplicity.1.4. The proof will rely on the following technical result which is interesting in its own. •). Let α be a (k − 1)-form and β a k -form such that at least one of them is compactly supported.k (mod 2). Proof. Express gradg f in (b) Denote by h the restriction to S 2 of the function h spherical coordinates. x2 + y 2 + z 2 = 1 . •). xn ) are local coordinates such that dx1 ∧ · · · ∧ dxn is positively oriented. β )dVg = (−1)νn. RIEMANNIAN GEOMETRY Exercise 4.1. Proposition 4. We have M (dα. Denote by δ the operator δ = ∗d∗ : Ωk (M ) → Ωk−1 (M ).14. Lemma 4. θ. (c) If (x1 ... Since d ∗ β ∈ Ωn−k+1 (M ) and ∗2 = (−1)(n−k+1)(k−1) on Ωn−k+1 (M ) we deduce M (dα.47. y.49. M This establishes the assertion in the lemma since (n − k + 1)(k − 1) + k ≡ νn. and denote by α the 1-form dual to X . g ). ˆ (x. we will denote the inner products by (•. X )dVg = M (df. ..1. . Xn ) = X (Ω(X1 .. (4. Xn ) = i Ω(X1 .50..1.. Hence {(Xf ) + f (div X )} dVg = d(iX f Ω)..1.. On the other hand... Xi ]. Xn )... Xn ). Define d∗ : Ωk (M ) → Ωk−1 (M ) by d∗ := (−1)νn. . . Then (LX Ω)(X1 . Xn ).. Set Ω := dVg . ... LX (f Ω) = (iX d + d iX )(f Ω) = d iX (f Ω).49 we deduce − M f (div X )dVg = − M ∞ f δαdVg ∀f ∈ C0 (M ). Using the above equality in (4.16) we deduce from ∇X Y − [X. .. ∇X Xi .17).4. ∞ (M ) we have Proof of (b) For any f ∈ C0 LX (f ω ) = (Xf )Ω + f (div X )Ω. METRIC PROPERTIES Definition 4.. we can find smooth functions fij . ∇Xi X...17) Over the neighborhood where the local moving frame is defined. .1... Xn ) be a local moving frame of T M in a neighborhood of some point. Xn )) = i Ω(X1 . Part (a) of the proposition follows after we substitute the above equality in (4.... . and let (X1 .. ..16) Since ∇Ω = 0 we get X · (Ω(X1 . [X.. . 155 Proof of the proposition. . Xn )) − i Ω(X1 .. Y ] = ∇Y X that (LX Ω)(X1 ..1. Since the form f Ω is compactly supported we deduce from Stokes formula d(iX f Ω) = 0.1..k ∗ d ∗ .1. M We have thus proved that for any compactly supported function f we have the equality − M f (div X )dVg = = M M (Xf )dVg = M df (X )dVg (grad f... (4.. Using Lemma 4. such that ∇Xi X = fij Xj ⇒ tr (∇X ) = fii .k δ = (−1)νn. α)dVg . ... Along ∂M we have a closed subset of a boundary-less manifold M a vector bundle decomposition ˜ ) |∂M = T (∂M ) ⊕ n (T M where n = (T ∂M )⊥ is the orthogonal complement of T ∂M in (T M ) |∂M .e.1. Z ) + g (Y. One uses the fact that LX is an even s-derivation and the equalities LX dxi = ∂i X i dxi (no summation). g ). Let (M. . β )dVg = α ∧ ∗β. g ) be an oriented Riemann manifold (possibly with boundary). ∇Z X ) = 0 for all Y. β M = M (α. Show that δη = 0. Using partitions of unity we can extend ν to a vector field defined on M . Exercise 4. The desired formula follows derivating in the left-hand-side. the product σx ∧ ωx is a positively oriented element of det Tx M .156 This completes the proof of (b).3. β define α. for each x ∈ ∂M . Definition 4. n admits nowhere vanishing sections.1. g ) be a Riemann manifold and X ∈ Vect (M ).. For any k -forms α. over the boundary det T M = det(T ∂M ) ⊗ n so that n∼ = det T M ⊗ det(T ∂M )∗ . div (X ) = 0. g ) be an oriented Riemann manifold with boundary ∂M .1. M whenever the integrals in the right-hand-side are finite. RIEMANNIAN GEOMETRY LX ( |g |dx1 ∧ · · · ∧ dxn ) = div (X )( |g |dx1 ∧ · · · ∧ dxn ). M is ˜ of the same dimension. This will be called the unit outer normal. Consider a Killing vector field X on the oriented Riemann manifold (M. Let (M.53. and denote by η the 1-form dual to X .1. β = α. Proof of (c) We use the equality CHAPTER 4. and each such section defines an orientation in the fibers of n. Show that the following conditions on X are equivalent. (A vector field X satisfying the above equivalent conditions is called a Killing vector field). and any positively oriented ωx ∈ det Tx ∂M . i. (a) LX g = 0. and will be denoted by ν . Let (M.52. Since both M and ∂M are oriented manifolds it follows that ν is a trivial line bundle. An outer normal is a nowhere vanishing section σ of n such that. Since n carries a fiber metric. (b) g (∇Y X. In particular. Z ∈ Vect (M ). Exercise 4. By definition. proved in Subsection 3.51. we can select a unique outer normal of pointwise length ≡ 1. Indeed. . so that J = − J . Let I be an ordered (k − 1)-index. METRIC PROPERTIES 157 Proposition 4.49. x ≥ 0}. Note that..19) We have to compare the two signs J and J . d∗ β )dVg where ˆ ∗ denotes the Hodge ∗-operator on ∂M with the induced metric g ˆ and orientation. then 1 ∈ J c so that (α ∧ ∗β ) |∂M = 0 = α |∂M ∧ˆ ∗(iν β ) |∂M ... β )dVg = dα ∧ ∗β = d(α ∧ ∗β ) + (−1)k α ∧ d ∗ β. Let (M. Similarly. n}.1. 1∈J . since the boundary ∂M has the orientation −dx2 ∧ · · · ∧ dxn .. As in the proof of Lemma 4.1. first J and then J c . .19). d∗ β M. in (4. α ∈ Ωk−1 (M ) and β ∈ Ωk (M ). Obviously sign (J ∪J c ) = sign (J ∪J c ). and therefor. .4. The sign J is the signature of the permutation J ∪J c of {1. On the boundary we have the equality ˆ ∗(iν dxJ ) = −ˆ ∗(dxJ ) = − J dxJ c ( J = ±1).18) and (4.54 (Integration by parts).18) and (4. Denote by J c the ordered (n − k )-index complementary to J so that (as sets) J ∪ J c = {1.1. it suffices to consider only the cases α = dxI . x ) ∈ R . the only nontrivial situation left to be discussed is 1 ∈ J ..1. n} obtained by writing the two increasing multi-indices one after the other.1. β M = α. To prove the second part we have to check that (α ∧ ∗β ) |∂M = α |∂M ∧ˆ ∗(iν β ) |∂M . and J be an ordered k -index.19). 1 ∈ J where J = J \ {1}..1.1.. We have ∗dxJ = and iν dxJ = Jc J dx ( J = ±1) (4. iν β ∂M + α. if 1 ∈ J .. we deduce that the sign J is (−1)×(the signature of the permutation J ∪J c of {2. d∗ β )dVg α |∂M ∧ˆ ∗(iν β ) |∂M + M (α.1. g ) be a compact.. By linearity. −dxJ .53 we can rephrase the above equality as dα.. (4. n}). oriented Riemann manifold with boundary.. β )dVg = = ∂M ∂M (α ∧ ∗β ) |∂M + M (α. The first part of the proposition follows from Stokes formula arguing precisely as in Lemma 4. This is a local (even a pointwise) assertion so we may as well assume 1 n n 1 M = Hn + = {(x . The proposition now follows from (4. . . · notation of Definition 4.1. Note that ν = −∂1 . Using the ·. Then M (dα. and that the metric is the Euclidean metric.18) 0 . β = dxJ . Proof.49 we have (dα. .1.1. . ∗d ∗ α)dVg = − (X. Proof..1. g ∂M − f.55 (Gauss). RIEMANNIAN GEOMETRY Corollary 4. β so that the previous results hold for noncompact manifolds as well provided all the integrals are finite. the geometer’s Laplacian takes the form ∆M = − 1 |g | ∂i |g |g ij ∂j . dg M − f. Let (M.54). Denote by α the 1-form dual to X . Corollary 4. where (g ij ) denotes as usual the matrix inverse to (gij ). ∆M g M = df. (1. Using Proposition 4. . and X a vector field on M .57. ν )dvg∂ . Then div (X )dVg = (X. A smooth function f on M satisfying the equation ∆M f = 0 is called harmonic.. g ) be an oriented Riemann manifold. ν )dvg∂ . with α = f .1.1. M ∂M where g∂ = g |∂M .1.56. Let (M. Note that when g is the Euclidean metric.54. then the geometers’ Laplacian is 2 2 ∆0 = −(∂i + · · · + ∂n ).1. ∂ν and ∂f ∂g f. We have div (X )dVg = 1 ∧ ∗d ∗ αdVg = α(ν )dvg∂ = (1. xn ). . The second identity is now obvious.48 we deduce that in local coordinates (x1 . Then ∂g f.158 CHAPTER 4. Definition 4. ∂M . ∆M g M − ∆M f. oriented Riemann manifold with boundary. and β = dg . d∗ α)dVg M M M M = ∂M ∂M Remark 4. which differs from the physicists’ Laplacian by a sign. Let (M. g M = . g ) be a compact.1. ∂ν ∂ν Proof. g ) as in Proposition 4.58 (Green). The geometric Laplacian is the linear operator ∆M : C ∞ (M ) → C ∞ (M ) defined by ∆M = d∗ df = − ∗ d ∗ df = − div (grad f ). The first equality follows immediately from the integration by parts formula (Proposition 4. The compactness assumption on M can be replaced with an integrability condition on the forms α. g ∈ C ∞ (M ).1. ∂M . and f. (b) Show that the n-dimensional “area” of S n is σn = where Γ is Euler’s Gamma function ∞ Sn dvg0 = 2π (n+1)/2 . Y. Y ]. In particular. . or [101]. Consider the Killing form on su(2) (the Lie algebra of SU (2)) defined by X. and the classical Beta integrals (see [39]. METRIC PROPERTIES 159 Exercise 4. Z .61. Chapter XII) ∞ 0 rn−1 (Γ(n/2))2 . Compute SU (2) B . where ei ∈ su(2) are the Pauli matrices e1 = i 0 0 −i e2 = 0 1 −1 0 e3 = 0 i i 0 (b) Show that the trilinear form on su(2) defined by B (X. f ∈ C ∞ (M ) are such that ∆M u = f show that M f = 0. Γ( n+1 2 ) Γ(s) = 0 ts−1 e−t dt. The group SU (2) is given the orientation defined by e1 ∧ e2 ∧ e3 ∈ Λ3 su(2). Y = −tr X · Y. and the computations in the previous exercise.1. dr = (1 + r2 )n 2Γ(n) Exercise 4.59. and then compute the volume of the group with respect to this metric. Exercise 4. is skew-symmetric. . (a) Show that the round metric g0 on S n is given in these coordinates by g0 = 4 (du1 )2 + · · · + (dun )2 . (a) Show that the Killing form defines a bi-invariant metric on SU (2). (b) If u.1. .1. un ) the coordinates on the round sphere S n → Rn+1 obtained via the stereographic projection from the south pole.1. (a) Prove that the only harmonic functions on a compact oriented Riemann manifold M are the constant ones. (c) B has a unique extension as a left-invariant 3-form on SU (2) known as the Cartan form on SU (2)) which we continue to denote by B .4. Hint: Use the natural diffeomorphism SU (2) ∼ = S 3 . Hint: Use the “doubling formula” π 1/2 Γ(2s) = 22s−1 Γ(s)Γ(s + 1/2). .60. B ∈ Λ3 su(2)∗ . . Denote by (u1 . Z ) = [X. 1 + r2 where r2 = (u1 )2 + · · · + (un )2 . The Riemann curvature tensor achieves just that.2 The Riemann curvature Riemannian geometry Roughly speaking. Thus. we do not need to go beyond this order of approximation to recover all the information. there is a lot more information encoded in the Riemann manifold than just its size. However. To recover it. and go beyond the first order approximations we have used so far. In this section we introduce the reader to the Riemann curvature tensor and its associates. We will describe some special examples. It is an object which is very rich in information about the “shape” of a manifold. a very short (in diameter) manifold is different from a very long one. provides a second order approximation to the geometry of the manifold. a Riemann metric on a manifold has the effect of “giving a shape” to the manifold.160 4. and a large (in volume) manifold is different from a small one. and we will conclude with the GaussBonnet theorem which shows that the local object which is the Riemann curvature has global effects. . and loosely speaking. As Riemann himself observed. we need to look deeper in the structure. ∂ )∂j . defined as R(g ) = F (∇).2 (The symmetries of the curvature tensor). Y )U. ∂ )∂j . In terms of the Christoffel symbols we have m Rijk = ∂j Γik − ∂k Γij + Γmj Γm ik − Γmk Γij . Theorem 4. Y )V. ∂k )∂i .2. . . V ) = −g (R((Y. V ) = −g (R(X.1 Definitions and properties by ∇ the Levi-Civita connection. ∂i = ∂i .2. R(∂k . U. . .2.1. Y )Z = [∇X . Z. §4. X ). U ). xn ) we have the description Rijk ∂ = R(∂j . ∇Y ]Z − ∇[X. The Riemann curvature tensor R satisfies the following identities (X. . Let (M. (a) g (R(X. In local coordinates (x1 .Y ] Z. Y. (b) g (R(X. where F (∇) is the curvature of the Levi-Civita connection. U. The Riemann curvature is the tensor R = R(g ). Lowering the indices we get a new tensor m Rijk := gim Rjk = R(∂k . and denote Definition 4. Y )U. g ) be a Riemann manifold. we will use Einstein’s summation convention. V ∈ Vect (M )). The Riemann curvature is a tensor R ∈ Ω2 (End (T M )) explicitly defined by R(X. V ). Unless otherwise indicated. Z ) + (∇Y R)(Z. Y )U. since (locally) X . (a) It follows immediately from the definition of R as an End(T M )-valued skewsymmetric bilinear map (X. Y . Z )X = 0. V )X. X (Y g (U. V ) + g (R(X. Y ] = 0. U )) = 2g (∇X ∇Y U. U ) + 2g (∇X U. (d) We will use the following algebraic lemma ([60]. ∇Y U ). U ) + 2g (∇X U. Y ) → R(X. Y ) = 0 In local coordinates the above identities have the form Rijk = −Rjik = −Rij k . j 161 i i Rjk + Ri jk + Rk = 0. Y can be written as linear combinations (over C ∞ (M )) of commuting vector fields. Y ). (c) As before. Y )V. U )) = 2Y g (∇X U. The 1st Bianchi identity is then equivalent to ∇X ∇Y Z − ∇Y ∇X Z + ∇Z ∇X Y − ∇X ∇Z Y + ∇Y ∇Z X − ∇Z ∇Y X = 0. V ) = g (R(X. j j (∇i R)j mk + (∇ R)mik + (∇k R)m i = 0. (d) g (R(X. Y )Z + R(Z.g. X )Y + R(Y. X ) + (∇Z R)(X. U ) = 2g (∇Y ∇X U. Y )U. The identity now follows from the symmetry of the connection: ∇X Y = ∇Y X etc.The Riemann curvature (c) (The 1st Bianchi identity) R(X. Z pairwise commute. V ) = g (R(U. . (b) We have to show that the symmetric bilinear form Q(U. We may as well assume that [X. U ). Chapter 5). (E. and similarly. Then Q(U. we can assume the vector fields X . Proof. U ) = g ((∇X ∇Y − ∇Y ∇X )U. Y ). it suffices to check Q(U. X = X i ∂i ). (e) (The 2nd Bianchi identity) (∇X R)(Y. Subtracting the two equalities we deduce (b). U ) = 0. We compute Y (Xg (U. ∇Y U ). Rijk = Rk ij . Thus. U ) is trivial. 2)-tensor (X. Y ) → Ric (X. X2 . X1 .e. X1 ) 2 − S (X3 . This concludes the proof of (d). g ) be a Riemann manifold with curvature tensor R. X1 .162 Riemannian geometry Lemma 4.4. We begin by presenting the simplest ones. X3 .2. the 1st Bianchi identity shows that the associated form S is identically zero. X2 ) . The Ricci curvature is the trace of this endomorphism. X4 . Y on M define an endomorphism of T M by U → R(U. X3 . X3 . 4 +1 n − 2 4 = 1 n 2 2 n +1 2 − (Hint: Consult [13]. X4 ) + R(X2 . X4 . X2 . X1 . Define S (X1 . X )Y. X4 ). Y ) = tr (U → R(U. X4 ) = −R(X1 .e. Let (M. X2 . The Riemann curvature R = g (R(X1 . X4 ) = −R(X2 . X2 . X2 . X4 . Chapter 1.2. ∀i. X4 ) = R(X1 . j. X3 . Rijk + Ri n 2 jk + Rik j = 0. X )Y ).3. X4 . X3 . X3 . X3 ).. Denote by Cn of n-dimensional curvature tensors. We view it as a (0. X2 . Ric (X. X1 . X4 ) + R(X3 . tensors (Rijkl ) ∈ (Rn )⊗4 satisfying the conditions.. n . X2 )X3 . Any two vector fields X . X3 .5. Exercise 4. X1 . X4 ) − S (X2 . X2 ) 1 = S (X1 . k. X4 ) satisfies the symmetries required in the lemma and moreover. X3 . X2 . . X3 . then R(X1 . . Let R : E × E × E × E → R be a quadrilinear map on a real vector space E . Y ) ∈ C ∞ (M ). X2 . X4 ) R(X1 . X1 . X2 ) + S (X4 . i. X4 ) − R(X3 .3.) The Riemann curvature tensor is the source of many important invariants associated to a Riemann manifold. If R satisfies the symmetry conditions R(X1 . i. Section G.23). Rijk = Rk Prove that dim Cn = ij = −Rjik . X3 . The proof of the lemma is a straightforward (but tedious) computation which is left to the reader. Definition 4.2. (e) This is the Bianchi identity we established for any linear connection (see Exercise 3. this implies Ricgλ = Ricg .Y)= span(Z. then the sectional curvatures are measured in meter−2 . Z. we obtained a new. However. the less curved is the corresponding sphere. Y )Y.) Remark 4. X ) (Y. W ). Y ). Let X. In general. Y ) = λ−2 Kg (X.6. Y ) = Kp (Z.7. For any linearly independent X. the larger the constant λ.2. Y ∈ Vect (M ) we have Kgλ (X. Kp (X. X ) . for any two linearly independent vectors X. (Y. . . The scalar curvature s of a Riemann manifold is the trace of the Ricci tensor.2. In local coordinates. the scalar curvature changes by the same factor upon rescaling the metric.W) is a 2-dimensional subspace of Tp M prove that Kp (X. Y ) . . Intuitively. Y ) = |X ∧ Y | where |X ∧ Y | denotes the Gramm determinant |X ∧ Y | = (X. In particular.8. the scalar curvature is described by s = g ij Ricij = g ij Ri j . and a constant λ > 0. if g is the canonical metric on the 2-sphere of radius 1 in R3 .2. the sectional curvatures are sensitive to metric rescaling.2)-tensor (as the metric). Y. rescaled metric gλ = λ2 g . . Definition 4.The Riemann curvature 163 If (x1 . W ∈ Tp M such that span(X. X ) (X. Given a metric g on a smooth manifold M .1) which is non-zero since X and Y are linearly independent. Y ∈ Tp M set (R(X. In particular. where (g ij ) is the inverse matrix of (gij ). xn ) are local coordinates on M and the curvature R has the local expression R = (Rkij ) then the Ricci curvature has the local description Ric = (Ricij ) = Rj i . Y ) (4. A simple computation shows that the Christoffel symbols and Riemann tensor of gλ are equal with the Christoffel symbols and the Riemann tensor of the metric g . If we thing of the metric as a quantity measured in meter2 . g ) be a Riemann manifold and p ∈ M . then gλ is the induced metric on the 2-sphere of radius λ in R3 . . Exercise 4.2. The symmetries of the Riemann curvature imply that Ric is a symmetric (0. For example. (|X ∧ Y |1/2 measures the area of the parallelogram in Tp M spanned by X and Y . Let (M. . Z ] = − [[X. Z ].2 Examples Example 4. • is the restriction of a bi-invariant metric m on G. n = dim M . Y = Y j Ej ∈ LG we have 1 Ric (X. W 4 4 1 1 [X.2. Z ]+ [Y. To compute the Ricci curvature we pick an orthonormal basis E1 . Y ]. W = − =− 1 1 [[X. Z ]] − [Y. 2 We can now easily compute the curvature R(X. Z ] 4 2 1 1 1 1 1 (Jacobi identity) = [[X. 4 Denote the Killing form by κ(X. Gr2 (M ) (p. For any X = X i Ei .1. Y ) = tr (Z → [[X. [X. §4.15. Y ]. Y ]. In other words. Thus Kp is in fact a function on Gr2 (p) the Grassmannian of 2-dimensional subspaces of Tp M . We have shown that the Levi-Civita connection of this metric is 1 ∇X Y = [X. [Z. Y )Z = 1 1 {[X. The function Kp : Gr2 (p) → R defined above is called the sectional curvature of M at p. Z ]. Z = − Y. Y ) = −tr ad(X ) ad(Y ) . viewed as left invariant vector fields on G.164 Riemannian geometry According to the above exercise the quantity Kp (X. Y ]. Y ]. if M is a Riemann manifold.11. 4 4 4 2 4 We deduce R(X. If (X. •. ad(X )Z . Y ∈ LG . [X. Definition 4. such that. 4 4 Now let π ∈ Gr2 (Tg G) be a 2-dimensional subspace of Tg G. Y ]) 4 . Y ].2. En of LG . Prove that Gr2 (M ) = disjoint union of Gr2 (p) p ∈ M can be organized as a smooth fiber bundle over M with standard fiber Gr2 (Rn ).10.2. Z ]] − [[X. . . Thus. Y ]. • is a metric on the Lie algebra LG satisfying ad(X )Y. Y ) depends only upon the 2-plane in Tp M generated by X and Y . Y ] ≥ 0. Y ]. . for some g ∈ G. Z ]. Consider again the situation discussed in Example 4. and •. G is a Lie group. W = ad(Z )[X.9. Z ]]} − [[X. [Y. Y ]. Z ]] − [Y. π ) → Kp (π ) is a smooth map. Y )Z. [X. [X. ∀X.2. ad(Z )W = − [X. Y ) is an orthonormal basis of π . then the sectional curvature along π is 1 Kg (π ) = [X. Exercise 4. W ] . Y ]. s= 4 4 i In particular. where |g | = det(gij ). we deduce that the only nontrivial component of the Riemann tensor is R = R1212 . Let M be a 2-dimensional Riemann manifold (surface).12. Rijkl = −Rijlk = Rklij . 4 Remark 4. Ei ]. and the form dx1 ∧ dx2 defines the orientation. [Y. we can construct a 2-form ε(g ) = 1 1 1 Kdvg = s(g )dVg = R1212 dx1 ∧ dx2 . Using the same notations as above we get 1 1 Ric (Ei . ad(Y )Ei i i 1 1 ad(Y ) ad(X )Ei . Ei ]. The sectional curvature is simply a function on M K= 1 1 R1212 = s(g ).13. Many problems in topology lead to a slightly more general situation than the one discussed in the above example namely to metrics on Lie groups which are only left invariant. if M is oriented. 4 4 In particular. We can now easily compute the scalar curvature. |g | 2 In this case. if G is a compact semisimple group and the metric is given by the Killing form then the scalar curvature is s(κ) = 1 dim G. Example 4. Ei = − tr ad(Y ) ad(X ) = κ(X. 2)-tensor.2. Ei ) = − i 165 1 4 ad(Y )[X. . Ei ). the scalar K is known as the total curvature or the Gauss curvature of the surface. In particular. Y ). Although the results are not as “crisp” as in the bi-invariant case many nice things do happen.2. x2 ). it is a scalar multiple of the Killing metric. on a compact semisimple Lie group the Ricci curvature is a symmetric positive definite (0. Due to the symmetries of R. and consider local coordinates on M . Y ]. We want to emphasize that this form is defined only when M is oriented.The Riemann curvature = = =− 1 4 1 4 1 4 [[X. Ei i [X. (x1 . For details we refer to [73]. and more precisely. Ei ] = i 1 4 ad(X )Ei . Ei ]. Ei ) = κ(Ei . 2π 4π 2π |g | The 2-form ε(g ) is called the Euler form associated to the metric g. Cartan’s insight was that the local properties of a manifold equipped with a geometric structure can be very well understood if one knows how the frames of the tangent bundle (compatible with the geometric structure) vary from one point of the manifold to another. 2π 2π 1 1 g R(∂1 . x2 ) plane. x ) are the usual Cartesian coordinates. .4 can be rephrased as sin θ(t) = −tg (R(∂1 . ´ Car§4. This is an orthogonal transformation of Tq M . ∂1 )dx1 ∧ dx2 = R1212 dx1 ∧ dx2 . α = 1. A = R 0 2112 |g | Assume for simplicity that (x1 .4.e.2. . ∂2 ) of Tq M . i ∂ . t] × [0.14. . Throughout this subsection we will use Einstein’s convention. .3. Example 4. i. x ) are normal coordinates at q .166 Riemannian geometry We can rewrite this using the pfaffian construction of Subsection 2. dVg = dx1 ∧ dx2 . n. ∂2 ∂2 . so that ˙(0). it has a matrix description as Tt = cos θ(t) − sin θ(t) sin θ(t) cos θ(t) .2. and with respect to the orthogonal basis (∂1 . Assume √ √ as before 1 2 that (x . The curvature R is a 2-form with coefficients in the bundle of skew-symmetric endomorphisms of T M so we can write 1 0 R1212 R = A ⊗ dVg . .. ∂2 )∂2 . t] in the (x1 .. Xα = Xα i ∂ 1 n where (x . and ε(g ) = Hence we can write ε(g ) = 1 1 P f g (−A)dvg =: P f g (−R). 2π 2π The Euler form has a very nice interpretation in terms of holonomy. We can think of the Euler form as a “density” of holonomy since it measures the holonomy per elementary parallelogram.2. R1212 = θ Hence R1212 is simply the infinitesimal angle measuring the infinitesimal rotation suffered by ∂1 along St . and consider the square St = [0. Denote by (θα ) the dual coframe. and ∂i := ∂xi . ∂2 is an orthonormal basis of Tq M .. Consider an orthonormal moving frame on Rn ..3 Cartan’s moving frame method This method was introduced by Elie tan at the beginning of the 20th century. . ∂1 ) + O(t2 ). the moving frame of T ∗ Rn defined by α θα (Xβ ) = δβ . x2 ) are normal coordinates at a point q ∈ M . at q . We will begin our discussion with the model case of Rn . |g | = 1 since ∂1 . The result in Subsection 3. Denote the (counterclockwise) holonomy along ∂St by Tt . Hence. Thus at q . 2. and θ as a column matrix. ∀α.2. The significance of these structural equations will become evident in a little while. α ) which measures the infinitesimal rotation We can form the matrix valued 1-form ω = (ωβ suffered by the moving frame (Xα ) following the infinitesimal displacement x → x + dx. We will find them define dXα to produce the forms ωβ using a different (dual) search strategy.2) we deduce γ α α dωβ = −ωγ ∧ ωβ .2.2. We choose a local orthonormal moving frame (Xα )1≤α≤n .4) are called the structural equations of the Euclidean space. (a) ωβ α α .The Riemann curvature 167 The 1-forms θα measure the infinitesimal displacement of a point P with respect to the frame (Xα ). (4.2) where we set dXα := i ∂Xα ∂xj dxj ⊗ ∂i . and we construct similarly its dual coframe (θα )1≤α≤n . dω = −ω ∧ ω.2. There exists a collection of 1-forms (ωβ 1≤α. or equivalently. (4. and we can rewrite this as α dθα = θβ ∧ ωβ . (b) dθα = θβ ∧ ωβ . Xβ = ω · Xα . We now try to perform the same computations on an arbitrary Riemann manifold (M. Since θ = d we deduce α 0 = d2  = dθ = dθα ⊗ Xα − θβ ⊗ dXβ = (dθα − θβ ∧ ωβ ) ⊗ Xα . g ).2. Note that the T M -valued 1-form θ = θα Xα is the differential of the identity map  : Rn → Rn expressed using the given moving frame. Cartan). or a n × 1 matrix whose entries are 1-forms. (4.3)–(4. Using the equality d2 Xβ = 0 in (4. α ) is a skew-symmetric matrix since In particular. there is no natural way to α entering the structural equations. α defined by Introduce the 1-forms ωβ α dXβ = ωβ Xα . Unfortunately. Xβ + Xα .4) The equations (4.3) Above. dim M = n.2. in the last equality we interpret ω as a n × n matrix whose entries are 1-forms. ω = (ωβ 0 = d Xα . or dθ = −ω ∧ θ. β . ∀α.15 (E. ω · Xβ .β ≤n uniquely defined by the requirements α = −ω β . α) Proposition 4. consider the Levi-Civita connection ∇. and we have dθα (Xβ .5) α = f α θ γ . the matrix valued 1-form (ωβ connection in this local moving frame.5). Define ωβ βγ α that the forms ωβ satisfy both (a) and (b). α ) is the 1-form associated to the Levi-Civita In other words.2. uniquely determined by the conditions there exist functions gβγ 1 α β α α θ ∧ θγ . The differential of θα can be computed in terms of the Levi-Civita connection (see Subsection 4.5). Since the collection of two forms (θα ∧ θβ )1≤α<β ≤n defines a local frame of Λ2 T ∗ Rn . γ 1 α γ β α + gγα − gαβ ) fβγ = (gβγ 2 (4. We let the reader check Existence.2.2.3. (a1) fβγ αγ while (b) gives α − f α = gα (b1) fβγ γβ βγ The above two relations uniquely determine the f ’s in terms of the g ’s via a cyclic permutation of the indices α. β. and after all. Suppose that there exist forms ωβ α there exist functions fβγ such that α α γ ωβ = fβγ θ .15 so that we must have α α ω ˆβ = ωβ . Thus the ω ˆ ’s satisfy both conditions (a) and (b) of Proposition 4. Since ∇ is compatible with the Riemann metric. what is their significance? α by To answer these questions. Then the condition (a) is equivalent to α = −f β . gβγ = −gγβ . Xγ ). What have we gained by constructing the forms ω . α .12 we deduce that the 2-form Ω = (dω + ω ∧ ω ) . Hence α ∇Xγ Xβ = ω ˆβ (Xγ )Xα . Then Uniqueness. dθα = gβγ 2 α satisfy the conditions (a)&(b) above. using the computation in Example 3. The reader may now ask why go through all this trouble. Xγ ) = Xβ θα (Xγ ) − Xγ θα (Xβ ) − θα (∇Xβ Xγ ) + θα (∇Xγ Xβ ) α δ δ (use θα (Xβ ) = δβ = const) = −θα (ˆ ωγ (Xβ )Xδ ) + θα (ˆ ωβ (Xγ )Xδ ) α α α =ω ˆβ (Xγ ) − ω ˆγ (Xβ ) = (θβ ∧ ω ˆβ )(Xβ .1. In particular. we deduce in standard manner that β α = −ω ω ˆβ ˆα . Define ω ˆβ α ∇Xβ = ω ˆβ Xα . where the f ’s are given by (4.168 Riemannian geometry Proof. The Cartan structural equations of a Riemann manifold take the form dθ = −ω ∧ θ. The technique of orthonormal frames is extremely versatile in concrete computations. (4.2.2. Using the moving frames method. and S a k -dimensional submanifold in M . 1 0 − y12 0 The Gauss curvature is K= 1 1 g (Ω(∂x .2. describe the scalar curvature of gu in terms of u and the scalar curvature of g . |g | y Exercise 4.. y∂y ) is an orthonormal moving frame. We want to analyze the relationship between the Riemann geometry of M (assumed to be known) and the geometry of S with the induced metric. We will use the moving frame method to compute the curvature of the hyperbolic plane.2. The restriction of g to S induces a Riemann metric gS on S . i. the upper half space H+ = {(x. Example 4. Note that ω ∧ ω = 0. y y y 1 1 dθy = d( dy ) = 0 = (− dx) ∧ θx . §4. We refer to [92] for more details on this aspect of the Riemann tensor. g ) be a Riemann manifold of dimension m. dω + ω ∧ ω = Ω. y y Thus the connection 1-form in this local moving frame is ω= 1 0 −y 1 0 y dx.2. We compute easily 1 1 1 dθx = d( dx) = 2 dx ∧ dy = ( dx) ∧ θy . g ) is a Riemann manifold.6) we deduce that the Riemann curvature is 1 0 0 −1 y2 Ω = dω = dy ∧ dx = θx ∧ θy .The Riemann curvature 169 is the Riemannian curvature of g .6) Comparing these with the Euclidean structural equations we deduce another interpretation of the Riemann curvature: it measures “the distance” between the given Riemann metric and the Euclidean one”. and u ∈ C ∞ (M ).4 The geometry of submanifolds We now want to apply Cartan’s method of moving frames to discuss the the local geometry of submanifolds of a Riemann manifold.16. ∂x ) = y 4 (− 4 ) = −1. Using the structural equations (4.e. θy = y dy ) is its dual coframe. . y > 0} endowed with the metric g = y −2 (dx2 + dy 2 ). ∂y )∂y . Define a new metric gu := e2f g . Let (M. 1 1 The pair (y∂x . y ) .17. Suppose (M. and (θx = y dx. This can be equivalently rephrased as M τ ∇S ∀X. Xk . M by this frame...8) B. splits into two components: a tangential component X τ . . and its is the orthogonal complement of T S in (T M ) |S . .. A. We distinguish two situations.2. β µα β = −µα 1 ≤ α. .2. The NS is called the normal bundle of S → M .. . . . We will analyze the structural equations of M restricted to S → M . We deduce k 0= β =1 θβ ∧ µα β. .170 Riemannian geometry Denote by ∇M (respectively ∇S ) the Levi-Civita connection of (M.. ωp ∈ Λ1 V . Xk+1 . X Y = (∇X Y ) (4. Xm ) such that the first k vectors (X1 . Xk ) are tangent to S . g ) (respectively of (S.7) dθα = β =1 θ β ∧ µα β. β ≤ m) the connection 1-forms associated to ∇ α S and let σβ . Exercise 4. (1 ≤ α.2. .. The metric g produces an orthogonal splitting of vector bundles (T M ) |S ∼ = T S ⊕ NS S. gS ). k < α ≤ m. i=1 then there exist scalars Aij . dθα = θβ ∧ µα β 1 ≤ α... Thus. Denote the dual coframe by (θα )1≤α≤m .2. β ≤ k ) be the connection 1-forms of ∇ corresponding to the frame (X1 .. . . If θ1 .2.. . that is a vector field X of M along S . .15 implies that along S α σβ = µα β 1 ≤ α. β ≤ k. X ν . Now choose a local orthonormal moving frame (X1 . The uniqueness part of Proposition 4. a section of (T M ) |S . Let V be a d-dimensional real vector space and consider p linearly independent elements ω1 . 1 ≤ α ≤ k . 1 ≤ i. θp ∈ Λ1 V are such that p θi ∧ ωi = 0. Denote by (µα β ). β ≤ k. (1 ≤ α. and a normal component. Note that θα |S = 0 for α > k. p ≤ d. j ≤ p such that Aij = Aji and p θi = j =1 Aij ωj ..7) yields k (4.18 (Cartan Lemma). Since θβ |S = 0 for β > k the equality (4. Y ∈ Vect (S ). At this point we want to use the following elementary result. Xk ). β ≤ m... and V = V γ Xγ = θγ (V )Xγ 1 ≤ γ ≤ k. Y )Z. β (V )U The last term is precisely the normal component of ∇M V U . T heorema Egregium (Gauss). Z ) .N = ∇M U V. N = N(V. V ). (S . Choose U. γ ≤ k satisfying Using Cartan lemma we can find smooth functions fβγ λ λ λ γ fβγ = fγβ . T ∈ Vect (S ) we have RM (X. N = ∇M U V ν . Y )Z. We have thus proved the following equality. g ) (resp. T ) . V ∈ Vect (S ). Now form λ β N = fβγ θ ⊗ θ γ ⊗ Xλ . If we write g (•. then N(U. T 2 = RS (X. V ) = λ>k      λ>k β β γ   λ γ fβγ θ (V ) θβ (U ) X λ   = β µλ Xλ . There is an alternative way of looking at N. τ We have thus established − V.9) The map N is called the 2nd fundamental form 2 of S → M . (4.2. Z ) . so that we have established ∇M V U ν = N(U. N ∈ C ∞ (NS ). V ). ∇U N = − V. V ∈ Vect (S ) U = U β Xβ = θβ (U )Xβ 1 ≤ β ≤ k. g |S ). We can view N as a symmetric bilinear map Vect (S ) × Vect (S ) → C ∞ (NS ).The Riemann curvature 171 λ . • . λ > k . N = − U.2. ∇M V N . N − V. More precisely we have the following celebrated result. N(Y. (4. . If U. T ) − N(X.10) The 2nd fundamental form can be used to determine a relationship between the curvature of M and that of S . then N(U. RS ) denote the Riemann curvature of (M. and µλ β = fβγ θ .11) The first fundamental form is the induced metric. U ). V ) = N(V. (4. U ) = ∇M U V ν .2. Let RM (resp. 1 ≤ β. ∇M U N . Z. ∇M U N τ = N(U. Then for any X. N τ M = ∇M U V. N(Y. •) = •. T + N(X. Y. 2.2. Take the inner product with T of both sides above. ∇Y ]Z − ∇[X. T ) − ∇S [X. N(Y. . T ) + N(X. and assume M = Rm .e. Let us further specialize. all hypersurfaces are transversally oriented since NS is locally trivial by definition. T − N(Y. Y )Z. Y )Z. Z ). and the second fundamental form is completely described by Nn (X.e. ∇Y ]Z.172 Proof. Gauss formula becomes RS (X. This is a truly remarkable result. Xm−1 . Then RS (X. X ) · Nn (Y. Y )Y. Nn (Y. Z ). T S = ∇M X ∇Y Z. T ) Nn (X. T = RM (X.Y ] Z. T ) Nn (Y.. Z ) Nn (Y. Z ) (4. N(X. Y ). Y )X = [∇M X . and choose an orthonormal moving frame (X1 . Y In this case. T + ∇M X N(Y. T ) = − ∇M Y n.10) we have Nn (X..10) RM (X. Z ) .Y ] Z.Y ] Z S M S S = ∇M X ∇Y Z + N(Y.. T . Nn (Y. Pick an orthonormal frame n of NS . Z ). n . S is a a codimension 1 submanifold such that the normal bundle NS is trivial3 . Y ) − |Nn (X. while in the left-hand-side we have a purely intrinsic term (which is defined entirely in terms of the internal geometry of S ). . Z ) Nn (X. •) is NS -valued. X .Y ] Z − N([X.8)-(4. Then (X1 . i.11). Xm−1 ) of T S .2. X = Nn (X. Historically.. . T − ∇[X. Y )|2 . The above result is especially interesting when S is a transversally oriented hypersurface. On the right-hand-side we have an extrinsic term (it depends on the “space surrounding S ”). 3 Locally. T − Nn (X. a length 1 section of NS . Y ) := N(X. Y is RS (X.. T ) . Y ]. Y )Z. T S − ∇M Y N(X. Z ) − ∇Y ∇X Z + N(X. Y )Z.. T S − ∇M Y ∇X Z. Y ). Z ). the sectional curvature along the plane spanned by X.. Note that S ∇M X Y = ∇X Y + N(X. Riemannian geometry We have M M RM (X. Z ) − ∇[X. Nn is a bona-fide symmetric bilinear form. Y ) = − ∇M X n.2. T = Nn (X. Z ).12) In particular. we deduce using (4. and moreover. T S = [∇S X . i.2. n) is an orthonormal moving frame of (T M ) |S . This is precisely the equality (4. according to (4. Since N(•.. and set ∂i := ∂x . Thus. Let u0 ∈ QA .13) In particular.19. i Extend (e1 . Au0 . ∂j ) |u0 = Ku0 = 1 |Au0 |2 Ae1 . Set n = |Au | Au. e2 ) of R such that e0 = n(u0 ). Au + 3 / 2 |Au| |Au| 1 A∂i u. We can now explain rigorously why we cannot wrap a plane canvas around the sphere. e1 Ae2 .The Riemann curvature 173 the extrinsic term was discovered first (by Gauss). e2 . and very much to Gauss surprise (?!?) one does not need to look outside S to compute it. e2 Ae2 . Au 1 ∂i Au. ej |u0 |Au0 | 1 1 = ∂i u. The second fundamental form of QA at u0 is Nn (∂i . e1 . invertible linear operator with at least one positive eigenvalue. x1 . any intrinsic quantity is invariant under “bending”.2. e2 ) to a local moving frame of T QA near u0 . This implies the quadric QA = {u ∈ R3 .2. Nn (∂i . u = 1}. In particular. Au. while the intrinsic geometry is not changed since the lengths of the“fibers” stays the same. (Quadrics in R3 ). |Au0 | |Au0 | We can now compute the Gaussian curvature at u0 . x = 0} = (Au0 )⊥ . Denote the ∂ Cartesian coordinates in R3 with respect to this frame by (x0 . Au −1/2 )Au + 1 A∂i u |Au| ∂i Au. the only thing that changes is the extrinsic geometry. r2 . is nonempty and smooth (use implicit function theorem to check this). when A = r−2 I so that QA is the round sphere of radius r we deduce Ku = 1 ∀|u| = r. x2 ). We compute ∂i n = ∂i Au |Au| =− Hence = ∂i ( Au. Aej . e1 Ae1 . It changed dramatically the way people looked at manifolds and thus it fully deserves the name of The Golden (egregium) Theorem of Geometry. Then Tu0 QA = {x ∈ R3 . QA is a transversally oriented hypersurface in R3 since the map QA u → Au defines a 1 nowhere vanishing section of the normal bundle. 3 Consider an orthonormal frame (e0 . when we deform a plane canvas. Aej |u0 = ei . Example 4. no matter how we deform the plane canvas we will always get a surface with Gauss curvature 0 which cannot be wrapped on a surface of constant positive curvature! Gauss himself called the total curvature a“bending invariant”. (4. This marked the beginning of a new era in geometry. ∂j |u0 . Let A : R3 → R3 be a selfadjoint. ∂j ) = ∂i n. Notice that. 2.174 Thus. taking into account our orientation conventions. Riemannian geometry Example 4. Note that the Whitney sum NΣ ⊕ T Σ is the trivial bundle R3 Σ . compact surface in R3 .2. ∂1 ) Nn (∂2 . We can think of n. ∂2 ) At q . 0 −Nn (∂2 . ∂2 ) = R1212 = Now notice that ∂i n = −Nn (∂i . ∂i n. Nn (∂2 . ∂1 ) + Nn (∂i . ∂1 ⊥ ∂2 so that. We obtain in this way a map G : Σ → S 2 = {u ∈ R3 . ∂2 )∂2 .14) . the round sphere has constant positive curvature. ∂2 ) . ∂2 ) .20. Then. (Gauss).2. The map G is called the Gauss map of Σ → S 2 . and let (x1 . Let n be the unit section of NΣ defining the above orientation. ∂1 )∂1 − Nn (∂i . (4. We orient NΣ such that orientation NΣ ∧ orientation T Σ = orientation R3 . e. ∂1 ) Nn (∂1 . ∂2 )∂2 .g. ∂j n = Nn (∂i . ∂1 )∂1 − Nn (∂i . Consider the Euler form εΣ on Σ with the metric induced by the Euclidean metric in R3 . ∂2 ). It really depends on how Σ is embedded in R3 so it is an extrinsic object. ∂1 )Nn (∂j . we have 2πεΣ (∂1 ..15) Nn (∂1 . Let Σ be a transversally oriented. x2 ) be normal coordinates at q ∈ Σ such that orientation Tq Σ = ∂1 ∧ ∂2 . because the orientability issue is decided by the sign of the determinant 1 0 0 0 −Nn (∂1 . The last equality can be rephrased by saying that the derivative of the Gauss map G∗ : Tq Σ → Tn(q) S 2 acts according to ∂i → −Nn (∂i . ∂1 |q and ∂2 |q as defining a (positively oriented) frame of R3 . ∂1 ) −Nn (∂2 . we deduce G∗ preserves (reverses) orientations ⇐⇒ R1212 > 0 (< 0). ∂2 )Nn (∂j . ∂2 ) (4. |u| = 1}. Σ x → n(x) ∈ S 2 . Denote by Nn the second fundamental form of Σ → R3 . ∂1 ) −Nn (∂1 . In particular. a connected sum of a finite number of tori. (Gauss-Bonnet Theorem. 1) are the corresponding Euler forms then εg0 (S ) = εg1 (S ). (4. S S . If g0 and g1 are two Riemann metrics on S and εgi (S ) (i = 0. ∂2 n ∂1 n.5 The Gauss-Bonnet theorem for oriented surfaces We conclude this chapter with one of the most beautiful results in geometry.16) If we denote by dv0 the metric volume form on S 2 induced by the restriction of the Euclidean metric on R3 . ∂2 n × Nn (∂1 .2.2. ∂2 n ∂2 n. ∂1 ) Nn (∂2 . In Chapter 9 we will investigate in greater detail the Gauss map of arbitrary submanifolds of an Euclidean space. This may sound like forcing our way in through an open door since everybody can “see” they are different. G∗ (∂2 ))|.15) we get §4. ∂1 n = Hence Nn (∂1 . we see that (4. ∂2 n . 1 ∗ 1 ∗ GΣ dv0 = G ε 2.2. From this point of view the curvature is a “distortion factor”. Its meaning reaches deep inside the structure of a manifold and can be viewed as the origin of many fertile ideas. (4. ∂2 ) . ∂2 ) Nn (∂2 . But it will do more than that. Preliminary version. ∂1 ) Nn (∂1 . ∂2 n)| = |dv0 (G∗ (∂1 ).2. ∂2 ) t Nn (∂1 . ∂1 ) Nn (∂1 . ∂1 n ∂2 n. Note that the last equality offers yet another interpretation of the Gauss curvature. ∂1 n ∂1 n.2. Recall one of the questions we formulated at the beginning of our study: explain unambiguously why a sphere is “different” from a torus. εΣ = Using (4.2.) Let S be a compact oriented surface without boundary.16) put together yield 2π |εΣ (∂1 . We will have more to say about it in the next subsection.14) and (4. ∂2 ) Nn (∂1 . Unfortunately this is not a conclusive explanation since we can see only 3-dimensional things and possibly there are many ways to deform a surface outside our tight 3D Universe.21. ∂1 ) Nn (∂2 . ∂2 )| = |dv0 (∂1 n. = ∂1 n. ∂1 ) Nn (∂1 . ∂2 ) 2 175 ∂1 n. ∂2 ) Nn (∂2 . Theorem 4.2. The Gauss curvature describes by what factor the area of this parallelogram was changed.The Riemann curvature We can rephrase this coherently as an equality of matrices ∂1 n. The Gauss map “stretches” an infinitesimal parallelogram to some infinitesimal region on the unit sphere. The elements of Riemann geometry we discussed so far will allow us to produce an invariant powerful enough to distinguish a sphere from a torus. ∂2 n ∂2 n.17) 2π 2π Σ S This is one form of the celebrated Gauss-Bonnet theorem . ∂2 ) Nn (∂2 . Let M be a compact oriented manifold. 1].2)-tensors while the dot denotes the t-derivative.2.2. dt S t It is convenient to consider a more general problem. g ˙t 2 t dVgt . then by taking the traces of both sides of the above equality we deduce λ(x) = s(g )(x) . Then d E(g t ) = − dt M 1 Ricgt − s(g t )g t .22. . We have the following remarkable result. g ) of dimension n is said to be Einstein if the metric g satisfies Einstein’s equation Ricg = where s(x) denotes the scalar curvature.2. The functional g → E(g ) is called the Hilbert-Einstein functional. Observe that if the Riemann metric g satisfies the condition Ricg (x) = λ(x)g (x) (4.176 Riemannian geometry Hence the quantity S εg (S ) is independent of the Riemann metric g so that it really depends only on the topology of S !!! The idea behind the proof is very natural. Let M be a compact oriented manifold without boundary and g t = (gij be a 1-parameter family of Riemann metrics on M depending smoothly upon t ∈ R. We will show d εg = 0 ∀t ∈ [0. Example 4. In the above formula ·. n := dim M.2. Definition 4.2. Denote by gt the metric gt = g0 + t(g1 − g0 ). t ) Lemma 4.25.23. Using the computations in Example 4.11 we deduce that a certain constant multiple of the Killing metric on a compact semisimple Lie group is an Einstein metric. For any Riemann metric g on E define EM (M. n for some smooth function λ ∈ C ∞ (M ). A Riemann manifold (M. g ) = M s(g )dVg . where s(g ) denotes the scalar curvature of (M. the Riemann manifold is Einstein if and only if it satisfies (4. We refer to [13] for an in depth study of the Einstein manifolds.18). n Thus. · t denotes the inner product induced by g t on the space of symmetric (0.18) s(x) g.2. g ). Definition 4.24. ∀t.2. (Schouten-Struik.23. and we list a few of them which will be used later.27. For typographical reasons we will be forced to introduce new notations. ˆ i ∂j = 0.2. Many nice things happen at q. Hint: Use the 2nd Bianchi identity. . . .2. this shows that on a Riemann 3-manifold the full Riemann tensor is completely determined by the Einstein tensor. Γ ˆ i = 0.19) at t = 0. Exercise 4.2. Proof of the lemma We will produce a very explicit description of the integrand d s(g t )dVgt dt = d d s(g t ) dVgt + s(g t ) dV t dt dt g (4. 2)-tensor 177 1 Eij := Rij (x) − s(x)gij (x) 2 is called the Einstein tensor of (M. xn ) a collection of g ˆ-normal i coordinates at q . g ˆ will denote (g t ) for t = 0. (4. 2 In particular.2. We will adopt a “roll up your sleeves.26. Exercise 4. at q. ∂k g ˆij = 0. Notice that when (S. g ).2. Consider a 3-dimensional Riemann manifold (M. g ). g ) is a compact oriented Riemann surface two very nice things happen. δg ˆα = i (4.21 is thus an immediate consequence of Lemma 4. Theorem 4. and we will keep track of the zillion indices we will encounter. the computations are not as hopeless as they may seem to be. while g t will be denoted simply by g .22) . Show that s Rijk = Eik gj − Ei gjk + Ej gik − Ejk gi + (gi gjk − gik gj ). (i) (S. Thus. [87]). Denote by ∇ the Levi-Civita connection of g and let Γjk denote its Christoffel symbols in the coordinates (xi ). and just do it” strategy reminiscent to the good old days of the tensor calculus frenzy. (ii) E(g ) = 2 S εg . . ∇ jk If α = αi dxi is a 1-form then. Let q be an arbitrary point on S . while a dot means differentiation with respect to t at t = 0. g ) is Einstein. The general case is entirely analogous. By this we mean that we will work in a nicely chosen collection of local coordinates. Prove that the scalar curvature of an Einstein manifold of dimension ≥ 3 is constant.2. (Recall that only R1212 is nontrivial).2. A hat over a quantity means we think of that quantity at t = 0.19) d of dt E(g t ). where δ = ∗d ∗ .The Riemann curvature The (0.2. and denote by (x1 .20) (4. g ˆij = g ˆij = δij . As we will see.21) ∂i αi . We will study the integrand (4.2. 2. ˙ we use the known formulæ To evaluate the derivatives Γ’s 1 km Γm (∂i gjk − ∂k gij + ∂j gik ) . ij = g 2 which. Differentiating s at t = 0. The tensor h is a symmetric (0. We get The term g ˙ ij can be computed by derivating the equality g ik gjk = δk g ˙ ik g ˆjk + g ˆik hjk = 0.178 In particular. (4.2. upon differentiation at t = 0.2. Its g ˆ-trace is the scalar trg ˆij hij = tr L−1 (h). for any smooth function u we have (∆M. (4. at q trg ˆh = i hii .27) . the scalar curvature is k k m k m s = trg Rij = g ij Rij = g ij ∂k Γk ij − ∂j Γik + Γmk Γij − Γmj Γik . and then evaluating at q we obtain ij ˆk ˆk ˙k ˙k s ˙=g ˙ ij ∂k Γ ∂k Γ ij − ∂j Γik + δ ij − ∂j Γik ˆ ij + =g ˙ ij R i ˙k ˙k ∂k Γ ii − ∂i Γik .2.g ˆ u)(q ) = − i Riemannian geometry ∂i 2 u. Finally.23) Set h = (hij ) := (g ˙ ) = (g ˙ ij ).2. In particular. The Ricci tensor is k k k m k m Rij = Rikj = ∂k Γk ij − ∂j Γik + Γmk Γij − Γmj Γik . (4. ˆh = g where L is the lowering the indices operator defined by g ˆ.24) The curvature of g is given by m Rikj = −Rijk = ∂k Γij − ∂j Γik + Γmk Γm ij − Γmj Γik . 2 (4.26) (4. yield 1 1 km ˙m Γ (∂i g ˆjk − ∂k g ˆij + ∂j g ˆik ) + g ˆ (∂i hjk − ∂k hij + ∂j hik ) ij = 2 2 1 = (∂i hjm − ∂m hij + ∂j him ) . so that g ˙ ij = −hij .25) i at t = 0. 2)-tensor. 22) we deduce that the last term in (4. g 2 g ˆ dVg ˆ+ M ˆ ˙ ) dVg ∆M.g ˙ + δ trg ˆ trg ˆg ˆ (∇g (4.26) -(4.2.2.2. ∂m ) = ∂k him .2.2. Green’s formula shows the last two terms vanish and the proof of the Lemma is concluded.28) = − Ric.e.19) is a lot easier to compute. δ trg ˆ (∇h) = δ trg ˆ (∇g We have thus established that s ˙ = − Ric. In the local coordinates (xi ) g ˆ is the raising the indices operator defined by g we have ˆ ˆ i h)jk dxk . 2 dt At q the metric is Euclidian..2. 1)-tensor.k To get a coordinate free description of the last term note that. .k =− ∂i hkk + i.k 1 2 (∂i 2 hkk − ∂i ∂k hik ) i. and we get. (4. trg ˆij (∇ ˆ (∇h) = g Using (4.25).g ˙+ ˆ trg ˆg i. g ˙ ˆ (g i ˆ|. ˆ k h)(∂i .20). + ∆M. and (4.28) can be rewritten (at q ) as ˆ ˆ ˙ ). (∇ ˆ h is a (0. and d |g | = dt Hence ˙ (g ) = E M g ˙ ii = |g ˆ| · trg ˙) = g ˆ. g ˙ g ˆ ˆ ˙ ).2.g ˙ + δ trg ˆ trg ˆg ˆ (∇g ˆ. g ˆ |g 1 s(ˆ g )ˆ g − Ricg ˙ ˆ .j i. Using the g The total covariant derivative ∇ ˆ-trace we can construct a (0. 3)-tensor. at q ˆ ij + 1 hij R 2 ˆ ij − hij R i. g ˆij = δij .29) The second term of the integrand (4. a 1-form −1 ˆ ˆ trg ˆ (∇h) = tr(Lg ˆ ∇h). i. g ˙ + ∆M.27) in (4.The Riemann curvature We substitute (4.2.2. dVg = ± |g |dx1 ∧ · · · ∧ dxn . 1 where L− ˆ.j (∂k ∂i hik − ∂k 2 hii + ∂k ∂i hik ) − i.k g ˆ 179 s ˙=− i. so that d ˙ g = ± 1 |g dV ˆ|−1/2 |g |dx1 ∧ · · · ∧ dxn .k 2 ∂i ∂k hik ∂i ∂k hik . at q. . The number 1 g (S ) = (2 − χ(S )) 2 is called the genus of the surface. To compute χ(S 2 ) we use the round metric g0 for which K = 1 so that χ(S 2 ) = 1 2π dvg0 = 1 areag0 (S 2 ) = 2. The reason for this terminology will become apparent when we discuss DeRham cohomology.2 that its curvature is zero. the Euler characteristic is independent of the metric used to define it.2.31. . . So far we have no idea whether χ(S ) is even an integer. the last equality can be rephrased as k g (S1 # · · · #Sk ) = i=1 g (Si ). In terms of genera. S where g is an arbitrary Riemann metric on S . . Thus upon iteration we get k χ(S1 # · · · #Sk ) = i=1 χ(Si ) − 2(k − 1). Remark 4. Sk . the Euler characteristic is a topological invariant of the surface. Proof. According to the theorem we have just proved.2. Since the Lie bracket is trivial we deduce from the computations in Subsection 4. We define its Euler characteristic as the number χ(S ) = 1 2π ε(g ). χ(S 2 ) = 2 and χ(T 2 ) = 0. Hence. This concludes the proof of the proposition. for any compact oriented surfaces S1 . Let S be a compact. 2π S2 To compute the Euler characteristic of the torus we think of it as an Abelian Lie group with a bi-invariant metric.2. Proposition 4. Proposition 4.2) are two compact oriented surfaces without boundary then χ(S1 #S2 ) = χ(S1 ) + χ(S2 ) − 2.28. .30.180 Riemannian geometry Definition 4. If Si (i=1.2.2. a Z-graded vector space naturally associated to a surface whose Euler characteristic coincides with the number defined above. oriented surface without boundary.29. Because of properties (i)-(iv). −3.The Riemann curvature 181 f S f S f S + f Figure 4. Set ± ±1 Sf := Sf ∩ {±x > 0}.3: Generating a hot-dog-shaped surface In the proof of this proposition we will use special metrics on connected sums of surfaces which require a preliminary analytical discussion.2. Sf := Sf ∩ {±x > 1} Since f is even we deduce Kg dvg = 1 2 Kg dvg = 2π. We denote by g the metric on Sf induced by the Euclidean metric in R3 .5. One such function is graphed in Figure 4.30) ± Sf 4 Sf One such diffeomorphism can be explicitly constructed projecting along radii starting at the origin. . Consider f : (−4.5]. 4) → (0. even function such that (i) f (x) = 1 for |x| ≤ 2. Sf is a smooth surface diffeomorphic4 to S 2 .0]. (4. (iv) f is non-decreasing on [-4. (iii) f (x) = 1 − (x − 3)2 for x ∈ [3. Since Sf is diffeomorphic to a sphere Sf Kg dVg = 2πχ(S 2 ) = 4π.3 Denote by Sf the surface inside R3 obtained by rotating the graph of f about the xaxis. ∞) a smooth. 4]. (ii) f (x) = 1 − (x + 3)2 for x ∈ [−4. We can now compute χ(S1 #S2 ) = 1 2π 1 = 2π (4.31 Let Di ⊂ Si (i = 1. so that over this region Kg = 0.2.182 Riemannian geometry S 1 D1 S2 D 2 O Figure 4. .2.30) + 1 2π Kg2 dVg2 − 1 2π Kg dVg = χ(S1 ) + χ(S2 ) − 2. g1 ) is + − isometric with Sf and (D2 . Pick a metric gi on Si such that (D1 .4). on the neck C = {|x| ≤ 2} the metric g is locally Euclidean g = dx2 + dθ2 .2.31) C Proof of the Proposition 4.32 (Gauss-Bonnet). The connected sum S1 #S2 is obtained −1 1 by chopping off the regions Sf from D1 and Sf from D2 and (isometrically) identifying ± the remaining cylinders Sf ∩ {|x| ≤ 1} = C and call O the overlap created by gluing (see Figure 4.4: Special metric on a connected sum On the other hand.31) S1 #S2 Kg ˆ dvg ˆ Kg1 dVg1 + Kg1 dVg1 1 2π 1 − 2π D2 S1 \D1 S2 \D2 Kg2 dVg2 + 1 2π O Kg dVg = 1 2π S2 S1 D1 Kg dVg (4. Then χ(Σg ) = 2 − 2g and g (Σg ) = g. This completes the proof of the proposition. 2) be a local coordinate neighborhood diffeomorphic with a disk in the plane. Denote the metric thus obtained on S1 #S2 by g ˆ.2. (4. Hence Kg dvg = 0. g2 ) is isometric to Sf . Let Σg denote the connected sum of g -tori. (By definition Σ0 = S 2 . Corollary 4.2. 33.2.The Riemann curvature In particular. 183 Remark 4. a sphere is not diffeomorphic to a torus. . It is a classical result that the only compact oriented surfaces are the connected sums of g -tori (see [68]). so that the genus of a compact oriented surface is a complete topological invariant. Note that.1 The 1-dimensional Euler-Lagrange equations From a very “dry” point of view. The action of a path (trajectory) γ : [0. which in our opinion is the most eloquent argument for the raison d’ˆ etre of the calculus of variations. and we will barely touch its physical significance. We recommend to anyone looking for an intellectual feast the Chapter 16 in vol. Given a smooth function U : R3 → R called the potential.1. 0 In the calculus of variations one is interested in those paths as above with minimal action. 1] → M connecting these points is the real number S (γ ) = SL (γ ) defined by 1 S (γ ) = SL (γ ) := L(t. and let L : R × T M → R by a smooth function called the lagrangian. The limited space we will devote to this subject will hardly do it justice. 1] → R3 is a quantity called the Newtonian action. given by 1 ˙|2 − U (q ). L = Q − U = kinetic energy − potential energy = m|q 2 The scalar m is called the mass. γ (t))dt. Example 5. γ ˙ (t). p1 ∈ M . we can form the lagrangian L(q. ˙ q ) : R3 × R3 ∼ = T R3 → R.Chapter 5 Elements of the Calculus of Variations This is a very exciting subject lieing at the frontier between mathematics and physics. Consider a smooth manifold M . the Newtonian action is measured in the same units as the energy. Fix two points p0 . The action of a piecewise smooth path γ : [0.1 The least action principle §5. 184 .1. as a physical quantity.2 of “The Feynmann Lectures on Physics” [35]. 5.1. the fundamental problem of the calculus of variations can be easily formulated as follows. k = 1. 1] → M satisfying the Euler-Lagrange equations with respect to some lagrangian L is said to be an extremal of L. q i ) = k (t. q ) = gq (v. 1. 2 We see that the action defined by L1 coincides with the length of a path. q i ). v ). .1. L2 : T M → R defined by L1 (v. q ˙n . dt ∂ γ ˙ ∂γ More precisely. γ. k dt ∂ q ˙ ∂q Definition 5. Tangent bundles are very peculiar manifolds. . . v )1/2 (v ∈ Tq M ). where ∂i := i . if (q ˙j . . q ) = gq (v. for any smooth path γ ˜ : [0. To any Riemann manifold (M. q n ) of local coordinates on a smooth manifold M automatically induces local coordinates on T M . v has a decomposition ∂ v = v i ∂i . This will be the only type of local coordinates we will ever use. . γ. The collection (q ˙1 . p1 ∈ M two fixed points. Theorem 5. q ). . where q ∈ M . .5. (i) γ (i) = pi . Furthermore. Suppose γ : [0. Let L : R × T M → R be a lagrangian. . then γ is a solution of the system of nonlinear ordinary differential equations d ∂L ∂L (t. (ii) SL (γ ) ≤ SL (˜ γ ). Before we present the main result of this subsection we need to introduce a bit of notation. . v ∈ Tq M . . ˙ γ ). A path γ : [0. . and p0 . . q n ) defines local coordinates on T M . i = 0.3 (The least action principle). THE LEAST ACTION PRINCIPLE 185 Example 5. q ˙j . Then the path γ satisfies the Euler-Lagrange equations d ∂ ∂ L(t.1.1. . . 1] → M is a smooth path such that the following hold. q i ) are holonomic local coordinates on T M such that γ (t) = (q i (t)). and . n = dim M.2. The action defined by L2 is called the energy of a path. Any point in T M can be described by a pair (v. . 1] → M joining p0 to p1 . 1 L2 (v. ˙ γ) = L(t. g ) one can naturally associate two lagrangians L1 . ∂q We set q ˙i := v i so that v=q ˙i ∂i . .1. Any collection (q 1 . q 1 . j and γ ˙ = (q ˙ (t)). These are said to be holonomic local coordinates on T M . q ˙j .4. ε) × [0. ∂s ∂s ∂s dt ∂s ∂ q ˙j dαs = (q ˙i (s. In fact. 1 then there exists at least one variation rel endpoints α such that δα = X . .1. 1] → M be a smooth path. ˙ L = −∇U (q ). . 1] → M. In the proof of the least action principle we will use the notion of variation of a path.1). αs (i) = pi ∀s. Exercise 5. Then. ds Assume for simplicity that the image of γ is entirely contained in some open coordinate neighborhood U with coordinates (q 1 . Proof of Theorem 5. t)) and Following the tradition. 2 Then ∂ ∂ L = mq.186 CHAPTER 5.1 1 L = m|q ˙|2 − U (q ).1) These are precisely Newton’s equation of the motion of a particle of mass m in the force field −∇U generated by the potential U .3. Then SL (α0 ) ≤ SL (αs ) ∀s. Let αs be a variation of γ rel endpoints. d |s=0 SL (αs ) = 0.6. t)). ∂q ˙ ∂q and the Euler-Lagrange equations become mq ¨ = −∇U (q ). Let γ : [0. Definition 5. we can write αs (t) = (q i (s. Prove that if t → X (t) ∈ Tγ (t) M is a smooth vector field along γ .1. If moreover. 1.1. q n ). (5. A variation of γ is a smooth map α = αs (t) : (−ε. dt so that The quantity δα is a vector field along γ called infinitesimal variation (see Figure 5. i = 0. the pair (δα. . δ α ˙ ) ∈ T (T M ) is a vector field along t → (γ (t). such that α0 (t) = γ (t). we set δα := ∂α ∂q i ∂ dαs ∂q ˙j ∂ |s=0 = ∂i δ α ˙ := |s=0 = . for very small |s|.1. .5. ELEMENTS OF THE CALCULUS OF VARIATIONS To get a better feeling of these equations consider the special case discussed in Example 5. Hint: Use the exponential map of some Riemann metric on M . then we say that α is a variation rel endpoints. .1. Note that d δα ˙ = dt δα. and at endpoints δα = 0. γ ˙ (t)) ∈ T M . such that X (t) = 0 for t = 0. (5. . are objects independent of any choice of local coordinates.6 deduce it holds for any vector field δαi ∂i along γ . and for the above arguments to work it suffices to choose only a special type of variations. Using this result in (5. At this point we use the following classical result of analysis.1. αs ) = 0 ∂L i δα dt + ∂q i 1 0 ∂L j δα ˙ dt. THE LEAST ACTION PRINCIPLE 187 γ p q Figure 5.1.1.1. then f is identically zero. α ˙s . From Exercise 5.5. This is true locally. (a) In the proof of the least action principle we used a simplifying assumption. 1). namely that the image of γ lies in a coordinate neighborhood. if any.7.2) The last equality holds for any variation α. localized on small intervals of [0. ∂q ˙j s Integrating by parts in the second term in the right-hand-side we deduce 1 0 ∂L d − ∂q i dt ∂L ∂q ˙i δαi dt. We check this directly. (b) The Euler-Lagrange equations were described using holonomic local coordinates.1: Deforming a path rel endpoints We compute (at s = 0) 0= d d SL (αs ) = ds ds 1 0 1 L(t. The minimizers of the action.1] such that 1 0 ∞ f (t)g (t)dt = 0 ∀g ∈ C0 (0.1. Remark 5. 1]. We leave the reader fill in the details. If f (t) is a continuous function on [0. so that the Euler-Lagrange equations have to be independent of such choices. In terms of infinitesimal variations this means we need to look only at vector fields along γ supported in local coordinate neighborhoods.2) we deduce the desired conclusion. . namely on symplectic manifolds. xi ) are the coordinates induced on T M .1. and they are currently the focus of very intense research.. The Euler-Lagrange equations for an extremal γ (t) can then be rewritten as a “geodesics equation” ∇L ˙ γ ˙ γ. For more details and examples we refer to the monographs [5. in which the extremals will turn out to be precisely the geodesics on a Riemann manifold. Via the Legendre transform the Euler-Lagrange equations become a system of first order equations on the cotangent bundle T ∗ M known as Hamilton equations. ∂q k ∂ ∂xj ∂ ∂ 2 xj k ∂ = + q ˙ ∂q i ∂q i ∂xj ∂q k ∂q i ∂ x ˙j ∂ ∂x ˙i ∂ ∂xj ∂ = = ∂q ˙j ∂q ˙j ∂ x ˙j ∂q i ∂ x ˙j Then ∂L ∂xj ∂L ∂ 2 xj k ∂L = + q ˙ . The æsthetically conscious reader may object to the way we chose to present the Euler-Lagrange equations. One method is to consider a natural nonlinear connection ∇L on T M as in [77] . These are manifolds carrying a closed 2-form whose restriction to each tangent space defines a symplectic duality (see Subsection 2. ∂q i ∂q i ∂xj ∂q k ∂q i ∂ x ˙j d dt ∂L ∂q ˙i = d dt ∂xj ∂L ∂q i ∂ x ˙j = ∂ 2 xj k ∂L ∂xj d q ˙ + ∂q k ∂q i ∂ x ˙j ∂q i dt ∂L ∂x ˙j . these equations are independent of coordinates.2. These equations have the advantage that can be formulated on manifolds more general than the cotangent bundles. then we have the transition rules xi = xi (q 1 . Is there any way of writing these equation so that the intrinsic nature is visible “on the nose”? If the lagrangian L satisfies certain nondegeneracy conditions there are two ways of achieving this goal. These are intrinsic equations we formulated in a coordinate dependent fashion.2.e. ELEMENTS OF THE CALCULUS OF VARIATIONS If (xi ) is another collection of local coordinates on M and (x ˙ j . . Another far reaching method of globalizing the formulation of the Euler-Lagrange equation is through the Legendre transform. . 25].188 CHAPTER 5.4. The example below will illustrate this approach on a very special case when L is the lagrangian L2 defined in Example 5. q n ).) Much like the geodesics equations on a Riemann manifold. . . We now see that the Euler-Lagrange equations in the q -variables imply the Euler-Lagrange in the x-variable. the Hamilton equations carry a lot of information about the structure of symplectic manifolds. x ˙j = so that ∂xj k q ˙ . which again requires a nondegeneracy condition on the lagrangian. i. e.5. we get Since g km gjm = δj so that ∂L2 1 ∂gij i j = q ˙q ˙ . ˙ q ) = (gij q ˙i q ˙j )1/2 .1. 2 ∂L2 = gjk q ˙j ∂q ˙k The Euler-Lagrange equations are q ¨j gjk + m . |γ ˙ 0 | = 1.4).1.5) The coefficient of q ˙i q ˙j in (5. L2 in Example 5. (5.1. is the geodesic equation in disguise.3) q ¨m + g km ∂gjk 1 ∂gij − i ∂q 2 ∂q k q ˙i q ˙j = 0.4) When we derivate with respect to t the equality gik q ˙i = gjk q ˙j we deduce g km ∂gjk i j 1 q ˙q ˙ = g km ∂q i 2 ∂gjk ∂gik + ∂q j ∂q i q ˙i q ˙j . we may as well assume it is parameterized by arclength.3) which. g ) be a Riemann manifold.5) is none other than the Christoffel symbol Γm ij so this equation is precisely the geodesic equation. 2 ∂q k ∂q k ∂gjk i j 1 ∂gij i j q ˙q ˙ = q ˙q ˙ . i. Example 5.1.1. (5. i ∂q 2 ∂q k (5. dt 2 ∂q k We recognize here the equation (5. when we express the Euler-Lagrange equations for a minimizer γ0 of this action.2.. and we get 1 q ¨m + g km 2 ∂gjk ∂gij ∂gik + − k j i ∂q ∂q ∂q q ˙i q ˙j = 0.1.9. Note that the action p1 SL q (t) = L(q. 1 L2 (q.1.1. ˙ q ) = gij (q )q ˙i q ˙j . Thus. We substitute this equality in (5. We will compute the Euler-Lagrange equations for the lagrangians L1 . The Euler-Lagrange equations for L1 are d dt gkj q ˙j gij q ˙i q ˙j = ∂gij i j q ˙q ˙ ∂q k . This fact almost explains why the geodesics are the shortest paths between two nearby points. THE LEAST ACTION PRINCIPLE 189 Example 5. Let (M.1. . ˙ q )dt. p0 is independent of the parametrization t → q (t) since it computes the length of the path t → q (t). 2 gij q ˙i q ˙j Along the extremal we have gij q ˙i q ˙j = 1 (arclength parametrization) so that the previous equations can be rewritten as d 1 ∂gij i j gkj q ˙j = q ˙q ˙ .8. as we have seen.1. Consider now the lagrangian L1 (q. 2 Noether’s conservation principle This subsection is intended to offer the reader a glimpse at a fascinating subject touching both physics and geometry. ELEMENTS OF THE CALCULUS OF VARIATIONS §5. conservation became almost synonymous with symmetry. L = m|q 2 The momenta associated to this lagrangian are the usual kinetic momenta of the Newtonian mechanics pi = mq ˙i . A point in T M is said to be a state.190 CHAPTER 5. 1 ˙|2 − U (q ). γ ˙ ) = (pi q ˙i − L) = dt dt dt = d dt ∂L ∂q ˙i − ∂L ∂q i ∂L ∂q ˙i q ˙i + ∂L i ∂L i ∂L i q ¨ − iq ˙ − iq ¨ ∂q ˙i ∂q ∂q ˙ q ¨i = 0 (by Euler − Lagrange). Consider a smooth manifold M . and in fact. F = −∇U .. Emmy Noether discovered that many of the conservation laws of the classical mechanics had a geometric origin: they were. A lagrangian L : R × T M → R associates to each state several meaningful quantities. Then the energy is conserved along γ . Proposition 5.1. and one can say it is a very important building block of quantum .e. • The generalized momenta : pi = • The energy : H = pi q ˙i − L. ˙ q ) as describing the motion of a particle under the influence of the generalized force. while H is simply the total energy 1 ˙|2 + U (q ). most of them. Let γ (t) be an extremal of a time independent lagrangian L = L(q. ˙ q ). ∂q i ∂L .10 (Conservation of energy). γ ˙ ) = 0. We first need to introduce a bit of traditional terminology commonly used by physicists. d H (γ. dt Proof. q. • The generalized force : F = ∂L . ∂q ˙i This terminology can be justified by looking at the lagrangian of a classical particle in a potential force field. At the beginning of the 20th century (1918). It eased the leap from classical to quantum mechanics. By direct computation we get d d d H (γ. The tangent bundle T M is usually referred to as the space of states or the lagrangian phase space. reflecting a symmetry of the lagrangian!!! This became a driving principle in the search for conservation laws. i.1. H = m|q 2 It will be convenient to think of an extremal for an arbitrary lagrangian L(t. 5. q i ). and by LX the Lie derivative on T M along X. We see that L is X -invariant if and only if LX L = 0.1. What is then this Noether principle? To answer this question we need to make some simple observations.1. Set q ˙j (s). Let L be a lagrangian on T M . One can think of Ψs as defining an action of the additive group R on T M . THE LEAST ACTION PRINCIPLE 191 physics in general.13. The lagrangian L is X -invariant if and only if Xi j ∂L k ∂X ∂L + q ˙ = 0. q i (s) := Ψs (q ˙j . Denote by X ∈ Vect (T M ) the infinitesimal generator of Ψs .1. Hence X = Xi ∂ ∂X j ∂ +q ˙k k .1. Let M be the unit round sphere S 2 ⊂ R3 . ds ∂ q ˙j (s) ∂q j we use the definition of the Lie derivative on M To compute − d j ∂ ∂ q ˙ = LX (q ˙i i ) = j ds ∂q ∂q Xk j ∂q ˙j k ∂X − q ˙ ∂q k ∂q k j ∂ ∂ i ∂X = − q ˙ . The lagrangian L is said to be X .12. Then d ds |s=0 d i |s=0 q i (s) = X k δk . ∂q j ∂q i ∂q j since ∂ q ˙j /∂q i = 0 on T M . Definition 5. This flow induces a flow Ψs on the tangent bundle T M defined by s Ψs (v. q i ).1. In the few instances of conservation laws where the symmetry was not apparent the conservation was always “blamed” on a “hidden symmetry”.6) .11. and X a vector field on M . ∂q i ∂q k ∂ q ˙j (5. x) = Φs ∗ (v ). physicists say that X is an infinitesimal symmetry of the given mechanical system described by the lagrangian L. Alternatively. where θ is the z -axis define a 1-parameter group of isometries of S 2 generated by ∂θ 2 longitude on S . Example 5. Φ (x) .invariant if L ◦ Ψs = L. ∀s. The rotations about the ∂ . i ∂q ∂q ∂ q ˙j Corollary 5. We describe this derivative using the local coordinates (q ˙j . Let X be a vector field on a smooth manifold M defining a global flow Φs . Example 5. Take X = ∂ ∂q i dpi dt ∂L ∂q i = 0. q. ˙ q. Consider an extremal γ = γ (q i (t)) of L.e. i.192 CHAPTER 5. Consider a surface of revolution S in R3 obtained by rotating about the z -axis the curve y = f (z ) situated in the yz plane.e. ELEMENTS OF THE CALCULUS OF VARIATIONS z r y O x Figure 5.1.2: A surface of revolution Theorem 5. . z ) we can describe S as r = f (z ). Consider a lagrangian L = L(t.1.1.16. The classical conservation-of-momentum law is a special consequence of Noether’s theorem.14 (E. If component of the force is zero. We compute d d PX (γ. The conservation of momentum has an interesting application in the study of geodesics. the i-th component in Noether’s conservation law. Noether). γ ˙) = dt dt = X i (γ (t)) ∂L ∂q ˙i = ∂X i k ∂L d q ˙ + Xi k i ∂q ∂q ˙ dt (5. θ. Corollary 5. the i-th = 0 along any extremal. ) on Rn . then the quantity PX = X i is conserved along the extremals of L. In these coordinates.1.15. the Euclidean metric in R3 has the form ds2 = dr2 + dz 2 + r2 dθ2 . i. Proof. (Geodesics on surfaces of revolution). If the lagrangian L is X -invariant. Proof. then of the momentum is conserved).6) ∂L = X i pi ∂q ˙i ∂L ∂q ˙i Euler-Lagrange ∂X i k ∂L ∂L q ˙ + Xi i i k ∂q ˙ ∂q ∂q = 0... If we use cylindrical coordinates (r. Consider a geodesic γ (t) = (z (t). The conservation of energy implies that |γ ˙ |2 = 2L = H is constant along the geodesics.2 The variational theory of geodesics We have seen that the paths of minimal length between two points on a Riemann manifold are necessarily geodesics. . θ) as local coordinates on S . q1 it may happen that it is not a minimal path. This fact can be given a nice geometric interpretation. π ) is the oriented angle the geodesic makes with the parallels z = const.2. 5. Exercise 5. then the induced metric has the form gS = {1 + (f (z ))2 }dz 2 + f 2 (z )dθ2 = A(z )dz 2 + r2 dθ2 . On a a surface of revolution r = f (z ) the quantity r cos φ is constant along any geodesic. θ(t)).1. This should be compared with the situation in calculus. The lagrangian defining the geodesics on S is L= We see that L is independent of θ: ∂L ∂θ 193 1 ˙2 . intuitive approach of [71].17 (Clairaut). Describe the geodesics on the round sphere S 2 . This situation is a bit more complicated since the action functional 1 S= |γ ˙ |2 dt 2 is not defined on a finite dimensional manifold. so that the generalized momentum ∂L ˙ = r2 θ ˙ ∂θ is conserved along the geodesics. We get ˙ r2 θ γ. when a critical point of a function f may not be a minimum or a maximum. r = f (z ). Az ˙ 2 + r2 θ 2 = 0.1. With some extra effort this space can be organized as an infinite dimensional manifold. This is precisely what we intend to do in the case of geodesics.. but rather follow the ad-hoc. We will not attempt to formalize these prescriptions. We deduce the following classical result. where φ ∈ (−π. However. THE VARIATIONAL THEORY OF GEODESICS We can choose (z. ˙ ∂/∂θ = . r cos φ = r2 θ ˙ |−1 .e. Theorem 5.5. It is a function defined on the “space of all paths” joining the two given points. given a geodesic joining two points q0 . |γ ˙ | · |∂/∂θ| r|γ ˙| ˙|γ i.18. and on the cylinder {x2 + y 2 = 1} ⊂ R3 . and compute the angle φ between γ ˙ and cos φ = ∂ ∂θ . To decide this issue one has to look at the second derivative. ε) × (ti−1 . 1] → M connecting p to q . ∆t γ ˙ − t ˙ dt. The velocity γ ˙ (t) has a finite number of discontinuities. and consider p. 1] → M such that (i) ∀s ∈ (−ε.1 Variational formulæ Let M be a connected Riemann manifold. Theorem 5.q .2. ELEMENTS OF THE CALCULUS OF VARIATIONS §5. Denote by Ωp. Let γ ∈ Ωp. E∗ (δα) := d |s=0 E (αs ) = − ds 1 (δα)(t). ti ) is a smooth map.2.1] such that the restriction of α to each (−ε.q . An infinitesimal variation of a path γ ∈ Ωp. ∇ d γ 0 dt (5. ε) × [0. q ∈ M . ∂s Exercise 5.3 (The first variation formula). Given V ∈ Tγ construct a variation α such that δα = V .) . piecewise smooth paths γ : [0.2. The space of infinitesimal variations of γ is an infinite dimensional linear space denoted by Tγ = Tγ Ωp. Consider now the energy functional E : Ωp. and let α be a variation of γ .1.2. piecewise smooth vector field V along γ such that V (0) = 0 and V (1) = 0 and h lim V (t ± h) 0 exists for every t ∈ [0. 1]. A variation of γ is a continuous map α = αs (t) : (−ε. (Note that the right-hand-side depends on α only through δα so it is really a linear function on Tγ .q → R.q is a continuous. E (γ ) = 1 2 1 0 |γ ˙ (t)|2 dt. (ii) There exists a partition 0 = t0 < t1 · · · < tk−1 < tk = 1 of [0. so that the quantity ∆t γ ˙ = lim (γ ˙ (t + h) − γ ˙ (t − h)) h→0+ is nonzero only for finitely many t’s.194 CHAPTER 5. and V (t)+ = V ( t)− for all but finitely many t-s. αs ∈ Ωp. Definition 5.1) where ∇ denotes the Levi-Civita connection.q .q (M ) the space of all continuous.2.q .q = Ωp. ε).2. they are vectors V (t)± ∈ Tγ (t)M . Every variation α of γ defines an infinitesimal variation δα := ∂αs |s=0 . Fix γ ∈ Ωp. δα. ∀V ∈ Tγ .. γ ˙ = ∇ ∂ δα. We will imitate the finite dimensional case which we now briefly analyze. Corollary 5.e.2. ∇ ∂ γ ˙ ∂t ∂t ∂t to integrate by parts. Definition 5. Since αs = γ for s = 0 we conclude k ti E∗ (δα) = i=1 ˙ . then we can define the Hessian at x0 f∗∗ : Tx0 X × Tx0 X → R .7. THE VARIATIONAL THEORY OF GEODESICS Proof. 195 We differentiate under the integral sign using the equality ∂ |α ˙ s |2 = 2 ∇ ∂ α ˙ s. γ ˙ + δα.q .q is called critical if E∗ (V ) = 0. and we obtain k k We use the equality E∗ (δα) = i=1 δα. α ˙ s |s=0 dt. γ ˙ ti ti−1 ti − i=1 ti−1 δα. ∂s ∂s and we get d |s=0 E (αs ) = ds 0 1 ∇∂ α ˙ s. δα → E∗ (δα) is called the first derivative of E at γ ∈ Ωp. The above corollary shows that this is not the case: the criticality also implies smoothness. ∇ ∂ δα.1] as in Definition 5. df (x0 ) = 0. γ ti−1 ∂t ∂ δα. a critical path may have a discontinuous first derivative. Exercise 5.2. a priori.1.2. Remark 5. We want to define a second derivative of E in order to address the issue raised at the beginning of this section. We will have more to say about it in Chapter 11.5. ∇ ∂ γ ˙ dt.5.2.2. i. This is a manifestation of a more general analytical phenomenon called elliptic regularity. Let f : X → R be a smooth function on the finite dimensional smooth manifold X . The map E∗ : Tγ → R.1). Set α ˙s = ∂αs ∂t .6. A path γ ∈ Ωp. α ˙s . Note that.4.2. A path γ ∈ Ωp. ∂s ∂ ∂ ∂α and ∂t commute we have ∇ ∂ ∂α Since the vector fields ∂s ∂ ∂t = ∇ ∂t ∂s . Prove the above corollary.2. ∂t This is precisely equality (5. If x0 is a critical point of f . ∂s Let 0 = t0 < t2 < · · · < tk = 1 be a partition of [0.q is critical if and only if it is a geodesic. Proof. ε) ⊂ R2 and γ := α0. V2 ∈ Tx0 X . V2 ∈ Tγ construct a 2-parameter variation α such that Vi = δi α. Consider a 2-parameter variation of γ α := αs1 . or t = const. ˙ δ1 α)γ ˙ dt. 2. s2 ) ) |(0. (s1 . Given V1 .s2 (t). consider a smooth map (s1 .2. ∂t ∂t Since γ is a geodesic.196 CHAPTER 5.2. Given V1 . 0) = Vi . ∇2 ∂ δ1 α − R(γ. (5. 1 E∗∗ (δ1 α.q be a critical path. si = const.0) . i = 1. We now return to our energy functional E : Ωp.2.3) where R denotes the Riemann curvature.s2 ) |(0. ∆t t ∂α − ∂t 1 0 δ2 α. V2 ) = ∂ 2 f ( α(s1 . We can now define the Hessian of E at γ by E∗∗ (δ1 α. E∗∗ is a bilinear functional on Tγ . Let γ ∈ Ωp.s2 : (−ε. i = 1. Note that δi α ∈ Tγ . ELEMENTS OF THE CALCULUS OF VARIATIONS as follows. ∆t δ1 α − 0 δ2 α. 1] → M.9 (The second variation formula).2. s2 ) ∈ X such that ∂α α(0. ε) × [0.2) ∂si Now set f∗∗ (V1 . ∂t ∂α ∂t (5. ε) × (−ε. ∇ δ2 α. s2 . t) → αs1 .0 . In particular. ∂s1 ∂s2 Theorem 5. and has second order derivatives everywhere except maybe on finitely many “coordinate” planes ∂α | . 0) = x0 and (0. ∇ ∂ ∂t ∂α dt.2.0) Exercise 5. Set U := (−ε. The map α is continuous. Set δi α := ∂s i (0. ∇ ∂ γ ˙ dt − ∂t 0 ∂ ∂s1 ∇∂ ∂α dt. ∆1 γ ˙ − t 1 δ2 α.8. δ2 α) := ∂ 2 E (αs1 .q → R. δ2 α) = − t δ2 α. According to the first variation formula we have ∂E =− ∂s2 Hence ∂2E =− ∂s1 ∂s2 1 δ2 α. the Hessian f∗∗ (V1 . ∂t . ∇ ∂ ∂s1 ∆t − 0 ∇ ∂ ∂s1 δ2 α. 2.2. ε) × (−ε. ∂s1 ∂s2 Note that since x0 is a critical point of f . we have ∆t γ ˙ = 0 and ∇ ∂ γ ˙ = 0.2). s2 ) → α(s1 .4) ∇ t ∂ ∂s1 δ2 α.0) . ∂t (5. V2 ) is independent of the function α satisfying (5. §5. ˙ δα) ∂α ∂α = R(γ. ∂t ∂t Definition 5. V1 ). (b) Show that a vector field J along γ which vanishes at endpoints.15. γ ˙ ). ˙ t J = R(γ. ∀V ∈ Tγ . Definition 5. the definition of the curvature implies ∇ ∂ ∂s1 ∇∂ =∇∂ ∇ ∂t ∂t ∂ ∂s1 + R(δ1 α. Putting all the above together we deduce immediately the equality (5. Let γ ∈ Ωp.3).11. ∂t Proof.q be a geodesic. t δα = R(γ. Corollary 5. α0 = γ . THE VARIATIONAL THEORY OF GEODESICS Using the commutativity of ∇ ∂ ∂s1 197 ∂ ∂t with ∂ ∂s1 we deduce ∂ ∂s1 ∆t ∂α ∂t = ∆t ∇ ∂α ∂t = ∆t ∇ ∂ δ1 α .2. A smooth vector field J along a geodesic γ is called a Jacobi field if it satisfies the Jacobi equation ∇2 ˙ J )γ.10. Show that if J is a Jacobi field along a geodesic γ . Then the infinitesimal variation δα satisfies the Jacobi equation ∇2 ˙ δα)γ ˙ (∇t = ∇ ∂ ). Exercise 5. ∀V1 .2 Jacobi fields In this subsection we will put to work the elements of calculus of variations presented so far. Let γ ∈ Ωp. ε) × [0.2. ∂t Finally. is a Jacobi field if any only if E∗∗ (J.2.2. 1] → M such that.2.12.2. W ) = 0. Exercise 5. A geodesic variation of γ is a smooth map αs (t) : (−ε.13. V2 ) = E∗∗ (V2 .q .14. V2 ∈ Tγ . We set as usual δα = ∂α ∂s |s=0 . .2.2.2. g ) be a Riemann manifold and p.5. for all vector fields W along γ . Let γ ∈ Ωp. Proposition 5.q be a geodesic and (αs ) a geodesic variation of γ . E∗∗ (V1 . (a) Prove that J is a Jacobi field if and only if E∗∗ (J. ˙ δα) . Let (M. then there exists a geodesic variation αs of γ such that J = δα. and t → αs (t) is a geodesic for all s. ∇2 t δα = ∇t ∇t = ∇s ∇t ∂α ∂t ∂α ∂s = ∇t ∇s ∂α ∂t + R(γ. V ) = 0. and J a vector field along γ . q ∈ M . Let γ ∈ Ωp.16.19. Theorem 5. δγ is a Jacobi field vanishing at the poles. Consider γ : [0. We conclude that the poles are conjugate along any meridian (see Figure 5.2. 2. Definition 5. 2π ] → S 2 a meridian on the round sphere connection the poles. In particular. The following result (partially) explains the geometric significance of conjugate points. ELEMENTS OF THE CALCULUS OF VARIATIONS N γ S Figure 5. Then p is conjugate with no point on γ other than itself.17. i = 1.q is said to be nondegenerate if q is not conjugated to p along γ .q be a geodesic. Exercise 5. minimal geodesic. One can verify easily (using Clairaut’s theorem) that γ is a geodesic. Show that dim Jp = dim M .198 CHAPTER 5.2.18. Thus (γθ ) is a geodesic variation of γ with fixed endpoints. Define Jp to be the space of Jacobi fields V along γ such that V (p) = 0.2. Definition 5.q be a nondegenerate. Two points γ (t1 ) and γ (t2 ) on γ are said to be conjugate along γ if there exists a nontrivial Jacobi field J along γ such that J (ti ) = 0.20. Let γ ∈ Ωp.2. hence a new geodesic γθ .2.3: The poles are conjugate along meridians. and moreover. A geodesic γ ∈ Ωp. The counterclockwise rotation by an angle θ about the z -axis will produce a new meridian. Example 5.3). the evaluation map evq : Jp → Tp M V → ∇t V (p) is a linear isomorphism. Let γ (t) be a geodesic. a geodesic segment containing conjugate points cannot be minimal ! . t ≥ t1 . Let p1 = γ (t1 ) be a point on γ conjugate with p.5) and Step 2 we deduce 0 = E∗∗ (V.21.2.5) Indeed. (5. ∀U ∈ Tγ . . Define V ∈ Tγ by Vt = Jt .2. THE VARIATIONAL THEORY OF GEODESICS 199 Proof.2. ds2 This proves (5. t ∈ [0.2. From (5. E∗∗ (U. U ).5). Denote by Jt a Jacobi field along γ |[0. The final conclusion (that V is a Jacobi field) follows from Exercise 5. 2 Hence d2 |s=0 E (αs2 ) ≥ 0. V ) ≤ E∗∗ (V + τ U. ds2 Since γ is minimal for any small s we have length (αs2 ) ≥ length (α0 ) so that E (αs2 ) ≥ 1 2 1 0 2 |α ˙ s2 |dt 1 1 = length (αs2 )2 ≥ length (α0 )2 2 2 1 = length (γ )2 = E (α0 ). Exercise 5. V ) = 0.5.15. We argue by contradiction. E∗∗ (V. One computes easily that d2 E (αs2 ) = 2E∗∗ (U. U ) ≥ 0 ∀U ∈ Tγ . Step 3 follows from the above equality using the bilinearity and the symmetry of E∗∗ . This follows immediately from the second variation formula and the fact that the nontrivial portion of V is a Jacobi field. t1 ] 0. Step 3. γ (t) is conjugate to γ (0) is discrete. Let γ : R → M be a geodesic.t1 ] such that J0 = 0 and Jt1 = 0. Thus. Step 2.2. τ = 0 is a global minimum of fU (τ ) so that fU (0) = 0. V + τ U ) = fU (τ ) ∀τ.2. E∗∗ (U. Step 1. V ) = 0. We will prove that Vt is a Jacobi field along γ which contradicts the nondegeneracy of γ . Prove that the set t ∈ R . let αs denote a variation of γ such that δα = U . ˙ J dt = − 0 0 R(J. 1.2.21 is finite. and we deduce ∇t J. We define its index.6) 1 0 |∇t J |2 dt ≤ 0. (5.2. Often. ˙ J dt. Thus ∇2 ˙ J )γ.q be a geodesic.. Y )Y.23.e. Proof.6) Then for any p. Riemann manifold and q0 ∈ M . Let (M. 0 Since J (τ ) = 0 for τ = 0.2.20 can be reformulated as follows: the index of a nondegenerate minimal geodesic is zero. 1] → M the point γ (1) is not conjugated to γ (0). This implies ∇t J = 0 which coupled with the condition J (0) = 0 implies J ≡ 0. this dependence is very powerful.24. X ≤ 0 ∀X. ˙ J )γ. Theorem 5. Theorem 5.22.q . denoted by ind (γ ). ind (γ ) = 0. complete. A point q ∈ M is conjugated to q0 along some geodesic if and only if it is a critical value of the exponential map expq0 : Tq0 M → M. . is conjugate to γ (0) which by Exercise 5. The notion of conjugacy is intimately related to the behavior of the exponential map. Let M be a Riemann manifold with non-positive sectional curvature. g ) be a connected. q ∈ M and any geodesic γ ∈ Ωp. γ ˙ )γ.2. we deduce using (5. The proof is complete.2. ˙ J dt. R(X.200 CHAPTER 5. ˙ t J = R(γ. γ ˙ )γ. Let Jt be a Jacobi field along γ vanishing at the endpoints.2. ELEMENTS OF THE CALCULUS OF VARIATIONS Definition 5. J dt = R(γ. Y ∈ Tx M ∀x ∈ M.2. J |1 0− 1 0 |∇t J |2 dt = − 1 R(J. The index of a geodesic obviously depends on the curvature of the manifold. Theorem 5. so that 0 1 1 1 ∇2 t J. Let γ ∈ Ωp. We integrate by parts the left-hand-side of the above equality. It suffices to show that for any geodesic γ : [0. i. 1) . as the cardinality of the set Cγ = t ∈ (0. A possible shape one can obtain may look like in Figure 5. Stretch the round two-dimensional sphere of radius 1 until it becomes “very long”.2. Assume first that q is a critical value for expq0 . For any X ∈ Tv (Tq0 M ) denote by JX the Jacobi field along γv such that JX (q0 ) = 0. where X ∈ Tv (Tq0 M ) satisfies Dv expq0 (X ) = 0. Since v is not a critical point. and v is a critical point. Obviously W (0) = 0. Let q = expq0 v . denote by Jq0 the space of Jacobi fields J along γv such that J (q0 ) = 0. Let v (s) be a path in Tq0 M such that v (0) = v and v ˙ (0) = X . v ∈ Tq0 M . Thus. Consider now the following experiment. ∂t On the other hand this is a nontrivial field since ∇t W = ∇s |s=0 This proves q0 and q are conjugated along γv . Let M be an n-dimensional complete Riemann manifold. the vector field W = ∂ |s=0 expq0 (tv (s)) ∂s is a Jacobi field along γv . If for all X ∈ Vect (M ) (n − 1) Ric (X. t) → expq0 (tv (s)) is a geodesic variation of the radial geodesic γv : t → expq0 (tv ). The lesson to learn from this intuitive experiment is that the price we have to pay for lengthening the sphere is decreasing the curvature. Equivalently.2. assume v is not a critical point for expq0 .5. a Jacobi field along γv vanishing at both q0 and q must have the form JX .25.16. Theorem 5. The map Tv (Tq0 M ) → Jq0 X → JX is a linear isomorphism. in other words is very small.4.2. Our next result offers a more quantitative description of this phenomenon.2. a highly curved surface cannot have a large diameter. Conversely. ∂s ∂ expq0 (tv (s)) = ∇s v (s) |s=0 = 0. We will see in the next chapter that this corollary has a lot to say about the topology of M . X ) ≥ |X |2 . On a complete Riemann manifold M with non-positive sectional curvature the exponential map expq has no critical values for any q ∈ M . (5. Hence. THE VARIATIONAL THEORY OF GEODESICS 201 Proof. Corollary 5.7) The existence of such a Jacobi field follows from Exercise 5.2.26 (Myers). r2 . and moreover W (1) = ∂ |s=0 expq0 (v (s)) = Dv expq0 (X ) = 0. As in that exercise. Then Dv expq0 (X ) = 0. The map (s. The long tube is very similar to a piece of cylinder so that the total (= scalar) curvature is very close to zero. for some X ∈ Tv (Tq0 M ). this means X = 0 so that JX ≡ 0. n−1 E∗∗ (Wi .4: Lengthening a sphere. . . Set Wi = sin(πt/ )ei . n − 1. γ ˙ )γ ˙ dt 2 0 = 0 sin2 (πt/ ) π 2 / − R(ei . V ) ≥ 0 ∀V ∈ Tγ . then (n − 1)π 2 / 2 − Ric (γ. then every geodesic of length ≥ πr has conjugate points and thus is not minimal. Set q0 = γ (0). ˙ γ ˙ ) < 0. ∇t Wi + R(Wi . Hopf-Rinow theorem implies that M must be compact. and let ei (t) be an orthonormal basis of vector fields along γ such that en (t) = γ ˙ (t). . Wi ) < 0. Hence diam (M ) = sup{dist (p. Since γ is minimal we deduce E∗∗ (V. ] → M of length . ˙ ei dt. If > πr. and q1 = γ ( ). γ ˙ )γ. i=1 . Fix a minimal geodesic γ : [0. We sum over i = 1. and in particular. p. Then E∗∗ (Wi . Wi ) = i=1 sin2 πt/ 0 (n − 1)π 2 / 2 − Ric (γ. Wi ) = − Wi .202 CHAPTER 5. ELEMENTS OF THE CALCULUS OF VARIATIONS Figure 5. Proof. . so that. q ∈ M } ≤ πr. ˙ γ ˙ ) dt ≥ 0. and we obtain n−1 E∗∗ (Wi . q ) . Y q t4 + O(t5 ).2.2. We refer to [27] or [71] and the extensive references therein for a presentation of some of the most attractive results in this direction. 3 1 det gij (x) = 1 − Rij xi xj + O(3). Y ) = κ(X.2. (c) Denote by xi a collection of normal coordinates at q . by Hopf-Rinow theorem G has to be complete. Prove that 1 gk (x) = δk − Rkij xi xj + O(3).19. we have computed the Ricci curvature of the Killing metric. Remark 5. Prove the following. Proof. and we found 1 Ric (X. A semisimple Lie group G with positive definite Killing pairing is compact. For the unitary vectors X. Y ∈ LG . see Exercise 4. we have E∗∗ (Wi . Prove that if the Ricci curvature is negative definite at q then vol0 (Dr (q )) ≤ volg (expq (Dr (q )) for all r sufficiently small. Denote by Wt = δγs the associated Jacobi field along γ0 (t).5.29. The next result shows that the converse is also true. (a) Wt = DtX expq (Y ) = Frechet derivative of v → expq (v ) 1 (b) f (t) = t2 − 3 R(Y. at least for some Wi . This contradicts the minimality of γ . 4 The corollary now follows from Myers’ theorem.27. 3 (e) Let Dr (q ) = {x ∈ Tq M . semisimple Lie group is positive definite. Wi ) < 0. Let M be a Riemann manifold and q ∈ M . On the other hand. .28.1.2. Form f (t) = |Wt |2 . The Killing form defines in this case a bi-invariant Riemann metric on G. |x| ≤ r}. THE VARIATIONAL THEORY OF GEODESICS 203 Hence. vol0 denotes the Euclidean volume in Tq M while volg denotes the volume on the Riemann manifold M . Its geodesics through the origin 1 ∈ G are the 1-parameter subgroups exp(tX ) which are defined for all t ∈ R. Hence. Corollary 5. The proof is complete. Y ∈ Tq M consider the family of geodesics γs (t) = expq t(X + sY ). Y ) ∀X. Exercise 5. X )X. The interdependence “curvature-topology” on a Riemann manifold has deep reaching ramifications which stimulate the curiosity of many researchers. We already know that the Killing form of a compact. 1). We noticed an interesting phenomenon: the global “shape” (topology) of a manifold restricts the types of structures that can exist on a manifold. This chapter will set the above observations on a rigorous foundation. 204 . This study is interesting only if some additional structure is present since otherwise all manifolds are locally alike. Figure 6. this means the sphere is not diffeomorphic to a torus. For example. We used the Gauss-Bonnet theorem in the opposite direction. The Gauss-Bonnet theorem involves a heavy analytical machinery which may obscure the intuition. the GaussBonnet theorem implies that on a connected sum of two tori there cannot exist metrics with curvature everywhere positive because the integral of the curvature is a negative universal constant. In particular.1: Looking for unshrinkable loops. Notice that S 2 has a remarkable property which distinguishes it from T 2 : on the sphere any closed curve can be shrunk to a point while on the torus there exist at least two “independent” unshrinkable curves (see Figure 6.Chapter 6 The Fundamental group and Covering Spaces In the previous chapters we almost exclusively studied local properties of manifolds. and we deduced the intuitively obvious fact that a sphere is not diffeomorphic to a torus because they have distinct genera. x) → Ft (x). 1. The space of loops in X based at x0 is denoted by Ω(X. k = 1. 2 are different though they have the same image. 1] → X.6. x0 ) ∀t ∈ [0. (b) Two loops γ0 . such that γ (0) = γ (1) = x0 . The annulus {1 ≤ |z | ≤ 2} is homotopy equivalent to the unit circle.1. Y are said to be homotopy equivalent if there exist maps f : X → Y and g : Y → X such that f ◦ g Y and g ◦ f X .1 Basic notions nected spaces. Example 6. We write this X Y . The unit disk in the plane is contractible. and (s → Γt (s)) ∈ Ω(X. 1. A loop based at x0 is a continuous map γ : [0.1.1. 0 ≤ s1/2 γ1 ∗ γ2 (s) = . The fundamental group 205 In the sequel all topological spaces will be locally path con- Definition 6. Two continuous maps f0 . such that.1. The two loops γ1 . We write this as f0 f1 . . (a) Let X and Y be two topological spaces.1 §6. f1 : X → Y are said to be homotopic if there exists a continuous map F : [0.1. Fi ≡ fi for i = 0. such that Γi (s) = γi (s) i = 0. (c) A topological space is said to be contractible if it is homotopy equivalent to a point. s) → Γt (s). THE FUNDAMENTAL GROUP 6.4. 1] × X → Y (t. γ2 be two loops based at x0 ∈ X .5. The product of γ1 and γ2 is the loop γ1 (2s) . it is a closed curve + a description of a motion of a point around the closed curve. γk (t) = exp(2kπt). (a) Let γ1 . (a) Let X be a topological space and x0 ∈ X . (c) The identity loop is the constant loop ex0 (s) ≡ x0 . x0 ). γ2 (2s − 1) . We write this as γ0 x0 γ1 .2.1.1. 1/2 ≤ s ≤ 1 The inverse of a based loop γ is the based loop γ − defined by γ − (s) = γ (1 − s). Definition 6. 1].3. (t. (b) Two topological spaces X . γ1 : I → X based at x0 are said to be homotopic rel x0 if there exists a continuous map Γ : [0. 1] × I → X. Note that a loop is more than a closed curve.1. γ2 : I → C. Definition 6. Example 6. ·) is called the fundamental group (or the Poincar´ e group) of the topological space X . and one can check immediately that γ Hence the correspondence Ω(X. (c) α0 ∗ ex0 x0 α0 . Then (a) α0 ∗ β0 x0 α1 ∗ β1 . (t. y0 ). The following result is left to the reader as an exercise. the product operation descends to an operation “·” on Ω(X. Let α0 x0 α1 . the set of homotopy classes of based loops. x0 ) are the “unshrinkable loops” discussed at the beginning of this chapter. x0 )/ x0 . Then f (γ ) ∈ Ω(Y. x0 ) → π1 (Y.x0 ) . β0 loops. descends to a map f : π1 (X. x0 ). x0 ) → π1 (Y. − (b) α0 ∗ α0 x0 ex0 . The group (Ω(X. The image of a based loop γ in π1 (X.. Lemma 6.e. ·) is a group. y0 ).1. x0 ).1. Let γ ∈ Ω(X. f1 : (X. (c) Let f0 .. x0 ) → (Y. x0 )/ x0 . Fix two points. We prove (c).206 The Fundamental group and covering spaces Intuitively. 1 and Ft (x0 ) ≡ y0 . The fundamental group π1 (X. then (g ◦ f )∗ = g∗ ◦ f∗ . and it is denoted by π1 (X. x0 )/ x0 .1. . Let X and Y be two topological spaces. (d) (α0 ∗ β0 ) ∗ γ0 x0 α0 ∗ (β0 ∗ γ0 ). there exists a continuous map F : I × X → Y . i. and then γ2 (twice faster). Hence (Ω(X. (b) If (X. (a) (X )∗ = π1 (X. y0 ) be two base-point-preserving continuous maps. satisfying the following functoriality properties. Fi (x) ≡ fi (x). for i = 0. x0 ) “sees” only the connected component of X which contains x0 . x0 ) is denoted by [γ ]. y0 ) x0 γ f g ⇒ f (γ ) y0 f (γ ).8. This is clearly a group morphism. x0 ) → (Y. it has a unit and each element has an inverse. x0 ) γ → f (γ ) ∈ Ω(Y. y0 ) → (Z. Then (f0 )∗ = (f1 )∗ . x0 β1 and γ0 x0 γ1 be three pairs of homotopic based Hence. Definition 6.6. Assume f0 is homotopic to f1 rel x0 . x0 ∈ X and y0 ∈ Y . y0 ). Then any continuous map f : X → Y such that f (x0 ) = y0 induces a morphism of groups f∗ : π1 (X. Proposition 6. The elements of π1 (X.7. z0 ) are continuous maps. The statements (a) and (b) are now obvious. x)}x∈X . To get more information about X one should study all the groups {π1 (X. the product of two loops γ1 and γ2 is the loop obtained by first following γ1 (twice faster). Moreover the induced operation is associative. such that f (x0 ) = y0 and g (y0 ) = z0 . Proof. x) → Ft (x) such that. 2: Connecting base points. pt) = {1} is said to be simply connected. all the fundamental groups π1 (X. y0 ) be two continuous maps. x0 ) we have β0 = f0 (γ ) y0 f1 (γ ) = β1 .1. . Definition 6. Example 6. Exercise 6. x).1. Two homotopically equivalent connected spaces have isomorphic fundamental groups. (a) π1 (Rn . and Ft a homotopy rel x0 connecting them. For any γ ∈ Ω(X. α∗ ([γ ]) := [α− ∗ γ ∗ α]. Corollary 6. 1] → X joining x0 to x1 induces an isomorphism α∗ : π1 (X.6.11.1. pt) = {1}. The above homotopy is realized by Bt = Ft (γ ). Proposition 6. Any continuous path α : [0.9.10.1. x0 ) → (Y. However. Prove that the spheres of dimension ≥ 2 are simply connected. Prove Proposition 6.2. the fundamental group of a topological space X may change as the base point varies and it almost certainly does if X has several connected components. A priori. (b) π1 (annulus) ∼ π1 (S 1 ). A connected space X such that π1 (X. Let X be a connected topological space.1. and thus path connected since it is locally so.14.9. pt) to underscore this weak dependence on the base point.1. defined by. We will write π1 (X. x ∈ X are isomorphic. Exercise 6. see Figure 6. the fundamental group of a connected space X is independent of the base point modulo some isomorphism.1. Let f0 . pt) ∼ π1 (pt. f1 : (X. x0 ) → π1 (X. x1 ). if X is connected.1.13. THE FUNDAMENTAL GROUP 207 x γ 0 − α α x 1 Figure 6. Thus.12. . and identify the fiber Ex0 with Cr . .e. . Notice first that the natural embedding Ck+1 → Cn+1 induces an embedding CPk → n CP .1. . . . zn−1 (1 − tzn ). . . . 0. 0] ∈ CPn . . (a) Prove that α β 1 α ∗ β . For any continuous. and ∀ζ ∈ H. and let ∇ be a flat connection on E . so that Tγ ∈ GL (Cr ).208 The Fundamental group and covering spaces Exercise 6. This proves that CPn is simply connected. 1) by (α β )(s) = α(s) · β (s). πt flows the point ζ along the line P ζ until it reaches the hyperplane H.17. 1] since we can homotop it out of any neighborhood of P . Our induction hypothesis shows that γ1 can be shrunk to pt inside H . We project γ from P to the hyperplane H = CPn−1 → CPn . (b) Tβ ∗α = Tα ◦ Tβ . and let γ ∈ Ω(CPn . Geometrically.. . This morphism (representation) is called the monodromy of the connection. . piecewise smooth γ ∈ Ω(X. 0. in terms of homogeneous coordinates this embedding is given by [z0 . Exercise 6. 0] ∈ CPn .1. For t ∈ [0. . More precisely. Clearly. zn−1 . . .1. we want to show it is simply connected. . F (∇) = 0. . πt is a homotopy rel pt connecting γ = π0 (γ ) to a loop γ1 in H ∼ = CPn−1 based at pt. where · denotes the group multiplication. (b) Prove that π1 (G. (a) Prove that α x0 β ⇒ Tα = Tβ .16. We may assume γ avoids the point P = [0. 1] define πt (ζ ) = [z0 (1 − tzn ). we denote by π (ζ ) the intersection of the line P ζ with the hyperplane H. 0. 1) is Abelian. Let E → X be a rank r complex vector bundle over the smooth manifold X . and by Exercise 6.1.15. . Define a new operation “ ” on Ω(G. zn−1 (1 − zn ). . if ζ = [z0 . In homogeneous coordinates π (ζ ) = [z0 (1 − zn ). . zk . We will establish this by induction. zn ] ∈ CPn . Let G be a connected Lie group. Example 6. . More precisely. 0] (= [z0 .14. We want to compute the fundamental group of the complex projective space CPn . More precisely. . . any flat connection induces a group morphism −1 T : π1 (X. 0] when zn = 1). CP1 ∼ = S 2 . . . the sphere S 2 is simply connected. We now use a classical construction of projective geometry. pt). . . . . x0 ) denote by Tγ = Tγ (∇) the parallel transport along γ . Clearly π is continuous. : zk ] → [z0 . . . . . Pick x0 ∈ X . . . For n = 1. Thus. x0 ) → GL(Cr ) γ → Tγ . ∀t. We k next assume CP is simply connected for k < n and prove the same is true for n. Note that πt (ζ ) = ζ . i. (1 − t)zn ]. . Choose as base point pt = [1. while Ab denotes the category of Abelian groups. The morphisms (X. Y ). and ∗ is a distinguished point of X. f ◦ X = f g ◦ X = g ∀f ∈ Hom(X. W ) h ◦ (g ◦ f ) = (h ◦ g ) ◦ f. Z ). X ) such that. Y ). (ii) Hom(C) is a family of sets Hom (X. F(f )). The morphisms are the obvious ones.6. Y ) × Hom(Y. which satisfies the following conditions. g ∈ Hom(Y.1. (i) Ob(C) is a set whose elements are called the objects of the category. ) are the continuous maps f : X → Y such that f (∗) = . Example 6. ∗) is the category of marked topological spaces. Z ) → Hom(X. This brief subsection is a minimal introduction to this language. Y ) are called the morphisms (or arrows) from X to Y . X ). The elements of Hom (X. THE FUNDAMENTAL GROUP 209 §6. there exists a unique element X ∈ Hom(X. Here we have to be careful about one foundational issue. The morphisms are the F-linear maps. Definition 6. (ii) F(g ) ◦ F(f ) = F(g ◦ f ) (respectively F(f ) ◦ F(g ) = F(g ◦ f )). f ) → (F(X ).1. 67]. where R is some ring. Let C1 and C2 be two categories.18. To avoid this problem we need to restrict to topological spaces whose subjacent sets belong to a certain Universe. ◦) satisfying the following conditions.2 Of categories and functors The considerations in the previous subsection can be very elegantly presented using the language of categories and functors. one for each pair of objects X and Y . (C2) ∀f ∈ Hom(X. • F Vect is the category of vector spaces over the field F. such that. A category is a triplet C = (Ob(C). then F(X ) −→ F(Y ) (respectively F(X ) ←− F(Y )). The objects are pairs (X. Hom(C). ∗) → (Y. h ∈ Hom(Z. • (Top. f F(f ) F(f ) f .1. A covariant (respectively contravariant ) functor is a map F : Ob(C1 ) × Hom(C1 ) → Ob(C2 ) × Hom(C2 ). The objects are topological spaces and the morphisms are the continuous maps. and (i) F(X ) = F(X ) . (iii) ◦ is a collection of maps ◦ : Hom(X. For more about this foundational issue we refer to [54]. Y ).1. ∗).19. • R Mod denotes the category of left R-modules. • Top is the category of topological spaces. (C1) For any object X . The interested reader can learn more about it from the monograph [54. Namely. Z ). • Gr is the category of groups. (X. the collection of all topological spaces is not a set. (f. ∀g ∈ Hom(Z. g ) → g ◦ f. where X is a topological space. if X → Y . Definition 6.1.1 Definitions and examples As in the previous section we will assume that all topological spaces are locally path connected. This type of covering space is said to be trivial.2.e. g ) and (M ˜) be two Riemann manifolds of the same dimension ˜. Let D be a discrete set. Lt The fundamental group construction of the previous is a covariant functor π1 : (Top. the product X × D is a covering of X with covering projection π (x. Exercise 6. 6.20. Example 6. However.g Exercise 6. ∞) is no longer a a covering map. U U ⊗ V. (c) If π : X → Y is a covering map. (Prove this!). On the other hand. Prove that φ is a covering map. Let V be a real vector space. t → exp(2π it) is a covering map. Such a neighborhood U is said to be an evenly covered neighborhood.2.2.g ˜ → M a surjective local isometry i. (U1 → U2 ) L (U1 ⊗ V L⊗1V → U2 ⊗ V ). the operation of taking the dual defines a contravariant functor. for any y ∈ Y . ∗ : R Vect → R Vect.2. The exponential map exp : R → S 1 . Let φ : M and ˜ |v |g = |Dφ(v )|g ˜ ∀v ∈ T M . (a) A continuous map π : X → Y is said to be a covering map if. then for any y ∈ Y the set π −1 (y ) is called the fiber over y . Then the operation of right tensoring with V is a covariant functor ⊗V : R Vect → R Vect. The sets Vi are called the sheets of π over U . there exists an open neighborhood U of y in Y . d) = x. ∗) → Gr. (U → V ) L (V ∗ → U ∗ ). V V ∗ . for any topological space X . In Chapter 7 we will introduce other functors very important in geometry.2. A covering space of Y is a topological space X . together with a covering map π : X → Y . Example 6.210 The Fundamental group and covering spaces Example 6.4. Φ is smooth such that (M ˜) is complete.5. (b) Let Y be a topological space.1.2 Covering Spaces §6. . Show that a fibration with standard fiber a discrete space is a covering. Let (M. Then.2. For more information about categories and functors we refer to [54.3.2. such that π −1 (U ) is a disjoint union of open sets Vi ⊂ X each of which is mapped homeomorphically onto U by π . its restriction to (0. 67]. ˜. 7. ∀q ∈ S 3 . 211 Theorem 6. π π X1 π1 F M X2 π2 Y .6. By Hopf-Rinow theorem we deduce that (Tq M..2. The sphere bundle of det T M (with respect to this metric) is a 2 : 1 covering space of M called the orientation cover of M . A Riemann metric on M induces a metric on the determinant line bundle det T M . Moreover the map q → Tq ∈ SO(3) is a group morphism. (a) Prove that qxq −1 ∈ Im H. (b) Prove that for any q ∈ S 3 the linear map Tq : Im H → ImH S3 x → qxq −1 is an isometry so that Tq ∈ SO(3).e. Identify S 3 ⊂ R4 with the group of unit quaternions S 3 = {q ∈ H . Exercise 6.2. the diagram below is commutative. The pull-back h = exp∗ q (g ) is a symmetric non-negative definite (0. g ) be a complete Riemann manifold with non-positive sectional curvature.10. Then for every point q ∈ M . (c) Prove the above group morphism is a covering map.2. surjective group morphism.6 (Cartan-Hadamard). the exponential map expq : Tq M → M is a covering map. 2)-tensor field on Tq M .5. Let G smooth. The linear space R3 can be identified with the space of purely imaginary quaternions R3 = Im H = {xi + y j + z k}.2. i. ˜ and G two Lie groups of the same dimension and φ : G ˜ →Ga Exercise 6. The theorem now follows from Exercise 6. Proof.9. Let (M.2. |q | = 1}. COVERING SPACES The above exercise has a particularly nice consequence. In particular. Example 6.2. 1 2 Definition 6. Let X1 → Y and X2 → Y be two covering spaces of Y . It is in fact positive definite since the map expq has no critical points due to the non-positivity of the sectional curvature. this explains why exp : R → S 1 is a covering map.2. The lines t → tv through the origin of Tq M are geodesics of h and they are defined for all t ∈ R.8. h) is complete. Prove that φ is a covering map. Let M be a smooth manifold. A morphism of covering spaces is a continuous map F : X1 → X2 such that π2 ◦ F = π1 . i.2 Unique lifting property π Definition 6. Let X → Y be a covering space and F : Z → Y a continuous map. Γ1 (t) = Γ2 (t). Show that Deck (R → S 1 ) ∼ = Z.14 (Unique Path Lifting). if X → Y is a covering space then its automorphisms are called deck transformations. Obviously S is closed so it suffices to prove that it is also open. Theorem 6. i. Then there exists at most one lift F : Z → X of f such that F (z0 ) = x0 . where y0 = f (z0 ).2. Γ1 .212 The Fundamental group and covering spaces If F is also a homeomorphism we say F is an isomorphism of covering spaces. The set S is nonempty since 0 ∈ S ..r0 ] . The proposition is proved. Using a metric on this n R line bundle form the associated sphere bundle S (U1 ) → RP . the paths Γ1 = Fi (αz ). For each z ∈ Z let αz be a continuous path connecting z0 to z . Assume there exist two such lifts. X F Z f K MY π Proposition 6. π ). exp Exercise 6. From Proposition 6. We argue by contradiction. Since π ◦ Γ1 = π ◦ Γ2 . §6. Pick a small open neighborhood U of x0 such that π restricts to a homeomorphism onto π (U ). Then there exists at most one lift of γ .12. n (b) Denote by UR 1 the tautological (real) line bundle over RP . For simplicity. 1].e.15.2.14 we deduce that Γ1 ≡ Γ2 . i = 1.2. for any z ∈ Z .2. the diagram below is commutative. 1] → Y such that Γ(0) = x0 . The general situation is entirely similar. and f : Z → Y be a continuous map.2. we deduce Γ1 |[0. 2. Proof. Let X → Y be a covering space.r0 ] = Γ2 |[0. There exists r0 > 0 such that γi [0. x0 ∈ π −1 (y0 ). F2 are two lifts of f such that F1 (z0 ) = F2 (z0 ) = x0 then.2. Exercise 6. γ : [0.e. We will prove that there exists r0 > 0 such that [0. Γ : [0. Prove that this fiber bundle defines a covering space isomorphic with the one described at part (a). 1] ⊂ S . F1 (z ) = F2 (z ). . s+ε]∩[0. The deck transformations form a group denoted by Deck (X. Proof. A lift of f is a continuous map F : Z → X such that π ◦ F = f . and x0 ∈ π −1 (y0 ). Fix z0 ∈ Z . 1] → X . π π . there exists ε > 0 such that [s−ε. where Z is a connected space.13. π Finally. Set S := t ∈ [0.11. r0 ] ⊂ U . If F1 . and Γ2 = F2 (αz ) are two lifts of γ = f (αz ) starting at the same point. 1] → Y a path in Y and x0 a point in the fiber over y0 = γ (0). (a) Prove that the natural projection S n → RPn is a covering map. We will prove that for every s ∈ S .2. for any z ∈ Z . Let X → Y be a covering map. r0 ] ⊂ S . we consider the case s = 0. Γ2 : [0. and F : Z → X be a lift of f . 1] → Y is a continuous path starting at y0 . Corollary 6. Proof.18. π π π (t. and y0 ∈ Y . y0 ) are homotopic rel y0 . Let X → Y be a covering space. this path must be constant. Proof. y0 ∈ Y . and a partition 0 = t0 < t1 < . Corollary 6. Let X → Y be a covering space. ti ] × Uz into an evenly covered neighborhood of hti−1 (z ). Let X → Y be a covering space. Clearly x · ([γ ] · [ω ]) = (x · [γ ]) · [ω ].18. Hence. and any γ ∈ Ω( Y. z ) → ht (z ) is a homotopy of f (h0 (z ) ≡ f (z )). By unique lifting property this lift connects Γ0 to Γ1 . < tn = 1.e. y0 ) is homotopic to γ rel y0 . z2 ∈ Z . x · γ depends only upon the equivalence class [γ ] ∈ π1 (Y. we can now successively lift h |I ×Uz to a continuous map H = H z : I × Uz → X such that H0 (ζ ) = F (ζ ). f (pt) = γ (0).3 Homotopy lifting property π 213 Theorem 6.16 (Homotopy lifting property). and ht (pt) = γ (t). . then any lifts Γ0 . y0 ). 1] × Z → Y (t. if ω ∈ Ω(Y.2.2. .2. Γ1 which start at the same point also end at the same point.2. Use the previous theorem with f : {pt} → Y . z ) → Ht (z ). We thus get a continuous path Γt (1) inside the fiber π −1 (y0 ) which connects Γ0 (1) to Γ1 (1). then x · γ = x · ω.2.17 (Path lifting property). Proof. If h : [0. Suppose that X → Y is a covering map. Following this partition. γ1 ∈ Ω(Y.6. and γ : [0. for any z1 . and hence we can glue all these local lifts together to obtain the desired lift H on I × Z . and y0 ∈ Y . i. for every x ∈ π −1 (y0 ). COVERING SPACES §6. For each z ∈ Z we can find an open neighborhood Uz of z ∈ Z . Lift the homotopy connecting γ0 to γ1 to a homotopy in X . Then. Set x · γ := Γx (1). Γ0 (1) = Γ1 (1). y0 ). By Corollary 6. ∀ζ ∈ Uz . denote by Γx the unique lift of γ starting at x. 1] → X of γ starting at x0 . f : Z → Y be a continuous map. 1] × Z → X such that H0 (z ) ≡ F (z ). for every x0 ∈ π −1 (y0 ) there exists a unique lift Γ : [0. Since the fibers are discrete. then there exists a unique lift of h H : [0. such that h maps [ti−1 .2. . Then. the liftings on I × Uz1 and I × Uz2 agree on I × (Uz1 ∩ Uz2 ). depending on z . By unique lifting property. If γ0 . let γ ∈ Ω(X. This action is called the monodromy of the covering. The map x → x · γ is called the monodromy along γ . f : Z → Y a continuous map and z0 ∈ Z such that f (z0 ) = y0 . We claim that F is a well defined map. so that the correspondence π −1 (y0 ) × π1 (Y.1) Proof. we deduce f∗ = π∗ ◦ F∗ .2. π∗ is injective. x0 ) → π1 (Y. Construct the loop based at z0 − βz = ωz ∗ γz . Let X → Y be a covering space. From (6. and denote by Λz its unique lift in X starting at x0 . For any ζ ∈ V pick a path σ = σζ in V connecting z to ζ . let ωz be another path in Z connecting z0 to z . Necessity. for every arbitrarily small. which is ex0 .20. We still have to show that this map is also continuous. Indeed.2. y0 ) Proposition 6. evenly covered neighborhood U of f (z ) ∈ Y there exists a path connected neighborhood V of z ∈ Z such that f (V ) ⊂ U .2. Assume the spaces Y and Z are connected (and thus path connected). i. the monodromy is trivial. The map π induces a group morphism π∗ : π1 (X. Pick z ∈ Z . We have to show that Λz (1) = Az (1). x0 ) such that π (γ ) is trivial in π1 (Y. Indeed.1) we deduce that the lift Bz of f (βz ) at x0 ∈ X is a closed path. For any z ∈ Z . Sufficiency. Then f (βz ) is a loop in Y based at y0 . f admits a lift F : Z → X such that F (z0 ) = x0 if and only if f∗ (π1 (Z.2. Since f is continuous.2.e. We now have Λz (1) = Bz (1/2) = Az (0) = Az (1). choose a path γz from z0 to z .. Note that when Y is simply connected.214 and The Fundamental group and covering spaces x · ey0 = x. γ ) → x · γ ∈ x · γ ∈ π −1 (y0 ) defines a right action of π1 (Y.4 On the existence of lifts π x0 ∈ π −1 (y0 ). Let ω denote . x0 ∈ X . y0 = π (x0 ) ∈ Y . If F is such a lift then. §6. Proof. Theorem 6.1).19. the monodromy along f (βz ) is trivial. The homotopy connecting π (γ ) to ey0 lifts to a homotopy connecting γ to the unique lift of ey0 at x0 .2. z0 )) ⊂ π∗ (π1 (X. y0 ). Denote by Az the unique lift of αz starting at x0 . using the functoriality of the fundamental group construction. Then αz = f (γz ) is a path from y0 to y = f (z ). This proves that F is a well defined map. x0 )) . and set F (z ) = Az (1). (6. y0 ) on the fiber π −1 (y0 ). y0 ) (x. This implies the inclusion (6. Set λz := f (ωz ). Thus. Sketch of proof. . Corollary 6. we deduce that Ω(1) belongs to the local sheet Σ.2. .26. Fix xi ∈ Xi such that p0 (x0 ) = p(x1 ) = y0 ∈ Y .2. . Fix y0 ∈ Y . is the universal cover of T n . The map R → S 1 is the universal cover of S 1 .2. g ) be a complete Riemann manifold with non-positive sectional curvature. A covering space X → Y is said to be universal if X is simply connected. Every space admits at most one universal covering space (up to isomorphism).6. the universal covering space is contractible !!! exp . Let Y be a connected. Let Y be a connected space. It can be topologized using the metric of uniform convergence in Y . defines an equivalence relation on Py0 . Let X1 → Y (i=0. Then F (ζ ) = Ω(1). the exponential map expq : Tq M → M is a covering map. . By Cartan-Hadamard theorem. Then (Y Exercise 6. Finish the proof of the above theorem. If X0 is universal. Define p : Y → Y by p([γ ]) = γ (1) ∀γ ∈ Py0 . tn ) → (exp(2π it1 ).2. The corollary follows immediately from Theorem 6.21.2. More generally.2.2. A bundle morphism F : X0 → X1 can be viewed as a lift of the map p0 : X0 → Y to the total space of the covering space defined by p1 . COVERING SPACES 215 the path ω = γz ∗ σζ (go from z0 to z along γz . The proof is complete since the local sheets Σ form a basis of neighborhoods of F (z ).27.1) be two covering spaces of Y .23. Proof. In particular. i = −1. exp(2π itn )).20 and the unique lifting property. which homeomorphically covers U . . Assume for simplicity that Y is a metric space. containing F (z ). the universal cover of such a manifold is a linear space of the same dimension. where Ω is the unique lift of f (ω ) starting at x0 . Then Y admits an (essentially unique) universal covering space.22.2. . locally path connected space such that each of its points admits a simply connected neighborhood. We have thus proved z ∈ V ⊂ F −1 (Σ). The natural projection p : S n → RPn is the universal cover of RPn .24.2. This ˜. Let (M. We denote the space of equivalence classes by Y ˜ and we endow it with the quotient topology. . Example 6. Theorem 6. Definition 6. Two paths in Py0 are said to be homotopic rel endpoints if they we can deform one to the other while keeping the endpoints fixed.25. pi π ˜ . Let Py0 denote the collection of continuous paths in Y starting at y0 . Corollary 6. Since (f (ζ ) ∈ U . Example 6. . and then from z to ζ along σζ ). p) is an universal covering space of Y . then there exists a unique covering morphism F : X0 → X1 such that F (x0 ) = x1 . exp : Rn → T n √ (t1 . 216 The Fundamental group and covering spaces We now have another explanation why exp : Rn → T n is a universal covering space of the torus: the sectional curvature of the (flat) torus is zero. ˜ → M its Exercise 6.2.28. Let (M, g ) be a complete Riemann manifold and p : M universal covering space. ˜ has a natural structure of smooth manifold such that p is a local (a) Prove that M diffeomorphism. ˜ locally isometric (b) Prove that the pullback p∗ g defines a complete Riemann metric on M with g . Example 6.2.29. Let (M, g ) be a complete Riemann manifold such that Ric (X, X ) ≥ const.|X |2 g, (6.2.2) where const denotes a strictly positive constant. By Myers theorem M is compact. Using ˜ is a complete Riemann manifold the previous exercise we deduce that the universal cover M locally isometric with (M, g ). Hence the inequality (6.2.2) continues to hold on the covering ˜ . Myers theorem implies again that the universal cover M ˜ is compact !! In particular, M the universal cover of a semisimple, compact Lie group is compact!!! §6.2.5 The universal cover and the fundamental group p ˜→ Theorem 6.2.30. Let X X be the universal cover of a space X . Then ˜ → X ). π1 (X, pt) ∼ = Deck (X ˜ and set x0 = p(ξ0 ). There exists a bijection Proof. Fix ξ0 ∈ X ˜ ) → p−1 (x0 ), Ev : Deck (X given by the evaluation Ev (F ) := F (ξ0 ). For any ξ ∈ π −1 (x0 ), let γξ be a path connecting ξ0 to ξ . Any two such paths are homotopic ˜ is simply connected (check this). Their projections on the base X rel endpoints since X determine identical elements in π1 (X, x0 ). We thus have a natural map ˜ ) → π1 (X, x0 ) F → p(γF (ξ ) ). Ψ : Deck (X 0 The map Ψ is clearly a group morphism. (Think monodromy!). The injectivity and the surjectivity of Ψ are consequences of the lifting properties of the universal cover. Corollary 6.2.31. If the space X has a compact universal cover, then π1 (X, pt) is finite. Proof. Indeed, the fibers of the universal cover have to be both discrete and compact. Hence they must be finite. The map Ev in the above proof maps is a bijection onto Deck ˜ ). (X 6.2. COVERING SPACES 217 Corollary 6.2.32 (H. Weyl). The fundamental group of a compact semisimple group is finite. Proof. Indeed, we deduce from Example 6.2.29 that the universal cover of such a group is compact. Example 6.2.33. From Example 6.2.11 we deduce that π1 (S 1 ) ∼ = (Z, +). Exercise 6.2.34. (a)Prove that π1 (RPn , pt) ∼ = Z2 , ∀n ≥ 2. n n ∼ (b) Prove that π1 (T ) = Z . Exercise 6.2.35. Show that the natural inclusion U (n − 1) → U (n) induces an isomorphism between the fundamental groups. Conclude that the map det : U (n) → S 1 induces an isomorphism π1 (U (n), 1) ∼ = π1 (S 1 , 1) ∼ = Z. Chapter 7 Cohomology 7.1 DeRham cohomology §7.1.1 Speculations around the Poincar´ e lemma To start off, consider the following partial differential equation in the plane. Given two smooth functions P and Q, find a smooth function u such that ∂u ∂u = P, = Q. (7.1.1) ∂x ∂y As is, the formulation is still ambiguous since we have not specified the domains of the functions u, P and Q. As it will turn out, this aspect has an incredible relevance in geometry. Equation (7.1.1) has another interesting feature: it is overdetermined, i.e., it imposes too many conditions on too few unknowns. It is therefore quite natural to impose some additional restrictions on the data P , Q just like the zero determinant condition when solving overdetermined linear systems. To see what restrictions one should add it is convenient to introduce the 1-form α = P dx + Qdy . The equality (7.1.1) can be rewritten as du = α. (7.1.2) If (7.1.2) has at least one solution u, then 0 = d2 u = dα, so that a necessary condition for existence is dα = 0, (7.1.3) ∂Q ∂P = . ∂y ∂x A form satisfying (7.1.3) is said to be closed. Thus, if the equation du = α has a solution then α is necessarily closed. Is the converse also true? Let us introduce a bit more terminology. A form α such that the equation (7.1.2) has a solution is said to be exact. The motivation for this terminology comes from the fact that sometimes the differential form du is called the exact differential of u. We thus have an inclusion of vector spaces {exact forms} ⊂ {closed forms}. 218 i.e., 7.1. DERHAM COHOMOLOGY 219 Is it true that the opposite inclusion also holds? Amazingly, the answer to this question depends on the domain on which we study (7.1.2). The Poincar´ e lemma comes to raise our hopes. It says that this is always true, at least locally. Lemma 7.1.1 (Poincar´ e lemma). Let C be an open convex set in Rn and α ∈ Ωk (C ). Then the equation du = α (7.1.4) has a solution u ∈ Ωk−1 (C ) if and only if α is closed, dα = 0. Proof. The necessity is clear. We prove the sufficiency. We may as well assume that 0 ∈ C . Consider the radial vector field on C r = xi ∂ , ∂xi and denote by Φt the flow it generates. More explicitly, Φt is the linear flow Φt (x) = et x, x ∈ Rn . The flow lines of Φt are half-lines, and since C is convex, for every x ∈ C , and any t ≤ 0 we have Φt (x) ∈ C . We begin with an a priori study of (7.1.4). Let u satisfy du = α. Using the homotopy formula, Lr = dir + ir d, we get dLr u = d(dir + ir d)u = dir α ⇒ d(Lr u − ir α) = 0. This suggests looking for solutions ϕ of the equation Lr ϕ = ir α, ϕ ∈ Ωk−1 (C ). If ϕ is a solution of this equation, then Lr dϕ = dLr ϕ = dir α = Lr α − iR dα = Lr α. Hence the form ϕ also satisfies Lr (dϕ − α) = 0. Set ω := dϕ − α = I (7.1.5) ωI dxI . Using the computations in Subsection 3.1.3 we deduce Lr dxi = dxi , so that Lr ω = I (Lr ωI )dxI = 0. We deduce that Lr ωI = 0, and consequently, the coefficients ωI are constants along the flow lines of Φt which all converge at 0. Thus, ωI = cI = const. 220 CHAPTER 7. COHOMOLOGY Each monomial cI dxI is exact, i.e., there exist ηI ∈ Ωk−1 (C ) such that dηI = cI dxI . For example, when I = 1 < 2 < · · · < k dx1 ∧ dx2 ∧ · · · ∧ dxk = d(x1 dx2 ∧ · · · ∧ dxk ). Thus, the equality Lr ω = 0 implies ω is exact. Hence there exists η ∈ Ωk−1 (C ) such that d(ϕ − η ) = α, i.e., the differential form u := ϕ − η solves (7.1.4). Conclusion: any solution of (7.1.5) produces a solution of (7.1.4). We now proceed to solve (7.1.5), and to this aim, we use the flow Φt .Define 0 u= −∞ (Φt )∗ (ir α)dt. (7.1.6) Here the convexity assumption on C enters essentially since it implies that Φt (C ) ⊂ C ∀t ≤ 0, so that if the above integral is convergent, then u is a form on C . If we write (Φt )∗ (ir α) = |I |=k−1 t ηI (x)dxI , then u(x) = I 0 −∞ t ηI (x)dt dxI . (7.1.7) We have to check two things. A. The integral in (7.1.7) is well defined. To see this, we first write α= |J |=k αJ dxJ , and then we set A(x) := max |αJ (τ x)|. J ;0≤τ ≤1 Then (Φt )∗ (ir α) = et ir J αJ (et x)dxJ , so that t |ηI (x)| ≤ Cet |x|A(x) ∀t ≤ 0. This proves the integral in (7.1.7) converges. 7.1. DERHAM COHOMOLOGY B. The differential form u defined by (7.1.6) is a solution of (7.1.5). Indeed, Lr u = lim 1 ( (Φs )∗ u − u ) s →0 s 0 1 = lim (Φs )∗ (Φt )∗ (ir α)dt − s →0 s −∞ s 221 0 −∞ (Φt )∗ (ir α)dt = lim s →0 −∞ s (Φt )∗ (ir α)dt − 0 −∞ (Φt )∗ (ir α)dt = lim s →0 0 (Φt )∗ (ir α)dt = (Φ0 )∗ (ir α) = ir α. The Poincar´ e lemma is proved. The local solvability does not in any way implies global solvability. Something happens when one tries to go from local to global. Example 7.1.2. Consider the form dθ on R2 \{0} where (r, θ) denote the polar coordinates in the punctured plane. To write it in cartesian coordinates (x, y ) we use the equality tan θ = so that (1 + tan2 θ)dθ = − i.e., dθ = y x y y2 dy −ydx + xdy and (1 + dx + )dθ = , x2 x x2 x2 −ydx + xdy = α. x2 + y 2 Obviously, dα = d2 θ = 0 on R2 \ {0} so that α is closed on the punctured plane. Can we find a smooth function u on R2 \ {0} such that du = α? We know that we can always do this locally. However, we cannot achieve this globally. Indeed, if this was possible, then du = S1 S1 α= S1 dθ = 2π. On the other hand, using polar coordinates u = u(r, θ) we get du = S1 S1 ∂u dθ = ∂θ 2π 0 ∂u dθ = u(1, 2π ) − u(1, 0) = 0. ∂θ Hence on R2 \ {0} {exact forms} = {closed forms}. We see what a dramatic difference a point can make: R2 \ {point} is structurally very different from R2 . The artifice in the previous example simply increases the mystery. It is still not clear what makes it impossible to patch-up local solutions. The next subsection describes two ways to deal with this issue. 222 CHAPTER 7. COHOMOLOGY ˇ §7.1.2 Cech vs. DeRham Let us try to analyze what prevents the “spreading” of local solvability of (7.1.4) to global solvability. We will stay in the low degree range. ˇ The Cech approach. Consider a closed 1-form ω on a smooth manifold. To solve the equation du = ω , u ∈ C ∞ (M ) we first cover M by open sets (Uα ) which are geodesically convex with respect to some fixed Riemann metric. Poincar´ e lemma shows that we can solve du = ω on each open set Uα so that we can find a smooth function fα ∈ C ∞ (Uα ) such that dfα = ω . We get a global solution if and only if fαβ = fα − fβ = 0 on each Uαβ = Uα ∩ Uβ = ∅. For fixed α, the solutions of the equation du = ω on Uα differ only by additive constants, i.e., closed 0-forms. The quantities fαβ satisfy dfαβ = 0 on the (connected) overlaps Uαβ so they are constants. Clearly they satisfy the conditions fαβ + fβγ + fγα = 0 on every Uαβγ := Uα ∩ Uβ ∩ Uγ = ∅. (7.1.8) On each Uα we have, as we have seen, several choices of solutions. Altering a choice is tantamount to adding a constant fα → fα + cα . The quantities fαβ change according to fαβ → fαβ + cα − cβ . Thus, the global solvability issue leads to the following situation. Pick a collection of local solutions fα . The equation du = ω is globally solvable if we can alter each fα by a constant cα such that fαβ = (cβ − cα ) ∀α, β such that Uαβ = ∅. (7.1.9) We can start the alteration at some open set Uα , and work our way up from one such open set to its neighbors, always trying to implement (7.1.9). It may happen that in the process we might have to return to an open set whose solution was already altered. Now we are in trouble. (Try this on S 1 , and ω = dθ.) After several attempts one can point the finger to the culprit: the global topology of the manifold may force us to always return to some already altered local solution. Notice that we replaced the partial differential equation du = ω with a system of linear equations (7.1.9), where the constants fαβ are subject to the constraints (7.1.8). This is no computational progress since the complexity of the combinatorics of this system makes it impossible solve in most cases. The above considerations extend to higher degree, and one can imagine that the comˇ plexity increases considerably. This is however the approach Cech adopted in order to study the topology of manifolds, and although it may seem computationally hopeless, its theoretical insights are invaluable. The DeRham approach. This time we postpone asking why the global solvability is not always possible. Instead, for each smooth manifold M one considers the Z-graded vector spaces B • (M ) = k≥0 B k (M ), B 0 (M ) := {0} Z • (M ) := k≥0 Z k (M ), 7.1. DERHAM COHOMOLOGY where B k (M ) = and Z k (M = Clearly Bk ⊂ Zk. η ∈ Ωk (M ); dη = 0 = closed k − forms. We form the quotients, H k (M ) := Z k (M )/B k (M ). dω ∈ Ωk (M ); ω ∈ Ωk−1 (M ) = exact k − forms, 223 Intuitively, this space consists of those closed k -forms ω for which the equation du = ω has no global solution u ∈ Ωk−1 (M ). Thus, if we can somehow describe these spaces, we may get an idea “who” is responsible for the global nonsolvability. Definition 7.1.3. For any smooth manifold M the vector space H k (M ) is called the k -th DeRham cohomology group of M . Clearly H k (M ) = 0 for k > dim M . Example 7.1.4. The Poincar´ e lemma shows that H k (Rn ) = 0 for k > 0. The discussion in Example 7.1.2 shows that H 1 (R2 \ {0}) = 0. Proposition 7.1.5. For any smooth manifold M dim H 0 (M ) = number of connected components of M. Proof. Indeed H 0 (M ) = Z 0 (M ) = f ∈ C ∞ (M ) ; df = 0 Thus, H 0 (M ) coincides with the linear space of locally constant functions. These are constant on the connected components of M . We see that H 0 (M ), the simplest of the DeRham groups, already contains an important topological information. Obviously the groups H k are diffeomorphism invariants, and its is reasonable to suspect that the higher cohomology groups may contain more topological information. Thus, to any smooth manifold M we can now associate the graded vector space H • (M ) := k≥0 H k (M ). A priori, the spaces H k (M ) may be infinite dimensional. The Poincar´ e polynomial of M , denoted by PM (t), is defined by PM (t) = k ≥0 tk dim H k (M ), every time the right-hand-side expression makes sense. The number dim H k (M ) is usually denoted by bk (M ), and it is called the k -th Betti number of M . Hence PM (t) = k bk (M )tk . 224 The alternating sum χ(M ) := k CHAPTER 7. COHOMOLOGY (−1)k bk (M ), is called the Euler characteristic of M . Exercise 7.1.6. Show that PS 1 (t) = 1 + t. We will spend the remaining of this chapter trying to understand what is that these groups do and which, if any, is the connection between the two approaches outlined above. §7.1.3 Very little homological algebra At this point it is important to isolate the ˇ common algebraic skeleton on which both the DeRham and the Cech approaches are built. This requires a little terminology from homological algebra. In the sequel all rings will be assumed commutative with 1. Definition 7.1.7. (a) Let R be a ring, and let C• = n∈Z C n , D• = n∈ Z Dn be two Z-graded left R-modules. A degree k -morphism φ : C • → D• is an R-module morphism such that φ(C n ) ⊂ Dn+k ∀n ∈ Z. (b) Let C• = n∈ Z Cn be a Z-graded R-module. A boundary (respectively coboundary ) operator is a degree −1 (respectively a degree 1) endomorphism d : C • → C • such that d2 = 0. A chain (respectively cochain ) complex over R is a pair (C • , d), where C • is a Z-graded R-module, and d is a boundary (respectively a coboundary) operator. In this book we will be interested mainly in cochain complexes so in the remaining part of this subsection we will stick to this situation. In this case cochain complexes are usually described as (C • = ⊕n∈Z C n , d). Moreover, we will consider only the case C n = 0 for n < 0. Traditionally, a cochain complex is represented as a long sequence of R-modules and morphisms of R-modules (C • , d ) : n · · · → C n−1 −→ C n −→ C n+1 → · · · , dn−1 d such that range (dn−1 ) ⊂ ker (dn ), i.e., dn dn−1 = 0. Definition 7.1.8. Let n · · · → C n−1 −→ C n −→ C n+1 → · · · dn−1 d 7.1. DERHAM COHOMOLOGY be a cochain complex of R-modules. Set Z n (C ) := ker dn B n (C ) := range (dn−1 ). 225 The elements of Z n (C ) are called cocycles, and the elements of B n (C ) are called coboundaries. Two cocycles c, c ∈ Z n (C ) are said to be cohomologous if c − c ∈ B n (C ). The quotient module H n (C ) := Z n (C )/B n (C ) is called the n-th cohomology group (module) of C . It can be identified with the set of equivalence classes of cohomologous cocycles. A cochain complex complex C is said to be acyclic if H n (C ) = 0 for all n > 0. For a cochain complex (C • , d) one usually writes H • (C ) = H • (C, d) = n≥0 H n (C ). Let M be an m-dimensional smooth d Example 7.1.9. (The DeRham complex). manifold. Then the sequence d d 0 → Ω0 (M ) → Ω1 → · · · → Ωm (M ) → 0 (where d is the exterior derivative) is a cochain complex of real vector spaces called the DeRham complex. Its cohomology groups are the DeRham cohomology groups of the manifold. Example 7.1.10. Let (g, [·, ·]) be a real Lie algebra. Define d : Λk g∗ → Λk+1 g∗ , by (dω )(X0 , X1 , . . . , Xk ) := 0≤i<j ≤k ˆi, . . . , X ˆ j , . . . , Xk ), (−1)i+j ω ([Xi , Xj ], X0 , . . . , X where, as usual, the hatˆindicates a missing argument. According to the computations in Example 3.2.9 the operator d is a coboundary operator, so that (Λ• g∗ , d) is a cochain complex. Its cohomology is called the Lie algebra cohomology, and it is denoted by H • (g). Exercise 7.1.11. (a) Let g be a real Lie algebra. Show that H 1 (g) ∼ = (g/[g, g])∗ , where [g, g] = span [X, Y ] ; X, Y ∈ g . (b) Compute H 1 (gl(n, R)), where gl(n, R) denotes the Lie algebra of n × n real matrices with the bracket given by the commutator. (c) (Whitehead) Let g be a semisimple Lie algebra, i.e., its Killing pairing is nondegenerate. Prove that H 1 (g) = {0}. (Hint: Prove that [g, g]⊥ = 0, where ⊥ denotes the orthogonal complement with respect to the Killing pairing.) 14.e. Let (A• . d) and (B • .226 Proposition 7. show that for each vector field X on a smooth manifold M . . d(x · y ) = ±(dx) · y ± x · (dy ) ∀x.15. n. y ∈ C. δ ) be two cochain complexes of R-modules. (c) Two cochain complexes (A• .1.. Corollary 7. and [iX .13. Let CHAPTER 7. i. δ ) are said to be homotopic. i. if there exists a degree −1 morphism χ : {An → B n−1 } such that φ(a) − ψ (a) = ±δ ◦ χ(a) ± χ ◦ d(a). Example 7..12. An φn dn M An+1 φn+1 B Kn δn K M Bn+1 (b) Two cochain morphisms φ. i. (a) A cochain morphism. d]s = LX . and B • (C ) · B • (C ) ⊂ B • (C ). d] = 0. The commutation rules in Subsection 3. the Lie derivative along X . COHOMOLOGY n C n+1 → · · · . ψ : A• → B • are said to be cochain homotopic . (C • . such that ψ ◦ φ A .2. the diagram below is commutative for any n. The interior derivative iX is the cochain homotopy achieving this. Assume moreover that C is also a Z-graded R-algebra.1. if there exist cochain morphism φ : A → B and ψ : B → A. If dx = dy = 0. and φ ◦ ψ B .1. The DeRham cohomology of a smooth manifold has an R-algebra structure induced by the exterior multiplication of differential forms.1. It suffices to show Z • (C ) · Z • (C ) ⊂ Z • (C ).e. we deduce xy = ±(dx dy ). namely [LX . If d is a quasi-derivation. and we write this φ ψ . Proof. then d(xy ) = ±(dx)y ± x(dy ) = 0. there exists an associative multiplication such that C n · C m ⊂ C m+n ∀m. and (B • . LX : Ω• (M ) → Ω• (M ) is a cochain morphism homotopic with the trivial map (≡ 0). d). then H • (C ) inherits a structure of Z-graded R-algebra..1. since d2 = 0. If x = dx and y = dy then. or morphism of cochain complexes is a degree 0 morphism φ : A• → B • such that φ ◦ d = δ ◦ φ. Definition 7. d) : · · · → C n−1 −→ C n −→ dn−1 d be a cochain complex of R-modules. A cochain complex as in the above proposition is called a differential graded algebra or DGA.e. then (ψ ◦ φ)∗ = ψ∗ ◦ φ∗ . If two cochain complexes (A• . DERHAM COHOMOLOGY 227 Proposition 7.7. φ∗ = ψ∗ .1. ψ : A → B are cochain homotopic. These follow immediately from the definition of a cochain map.1.18.17. Let 0 → (A• . (c) Obvious. dC ) → 0 φ ψ be a short exact sequence of cochain complexes of R-modules. Corollary 7. (b) We have to show that φ(cocycle) − ψ (cocycle) = coboundary. δ ) are cochain homotopic. (a) Any cochain morphism φ : (A• .16.1. d ) −→ (A2 . d) and (B • . d ) −→ (A1 . (a) It boils down to checking the inclusions φ(Z n (A)) ⊂ Z n (B ) and φ(B n (A)) ⊂ B n (B ).1. (b) If the cochain maps φ. Then φ(a) − ψ (a) = ±δ (χ(a) ± χ(da) = δ (±χ(a)) = coboundary in B. Let da = 0. d) → (B • . φ ψ Proof. dA ) −→ (B • . and if (A• 0 . then they induce identical morphisms in cohomology. δ ) induces a degree zero morphism in cohomology φ∗ : H • (A) → H • (B ). Proposition 7. dB ) −→ (C • . 0 • 1 • 2 (c) (A )∗ = H • (A) . d ) are cochain morphisms. then their cohomology modules are isomorphic. This means that we have a commutative diagram . In other words. B B B φn+2 dA n+1 a = dn+1 φn+1 a = dn+1 dn b = 0.228 CHAPTER 7. .. We let the reader check that the correspondence Z n (C ) c → a ∈ Z n+1 (A) above induces a well .10) we deduce 0 = dC ψn (b) = ψn+1 dB (b). . i. . If we trace back the path which lead us from c ∈ Z n (C ) to a ∈ Z n+1 (A). COHOMOLOGY . . . . We believe this is one proof in homological algebra that the reader should try to produce on his/her own. there exists a ∈ An+1 such that φn+1 (a) = dB n b. Indeed. From the commutativity of the diagram (7. The cohomology class x can be represented by some cocycle c ∈ Z n (C ). Since ψn is surjective there exists b ∈ B n such that c = ψn (b).1. We will just indicate the construction of the connecting maps ∂n . This construction. i. .K 0 M An+1 K M Bn+1 K M C n+1 K M0 M0 M0 (7.e. . dB (b) ∈ ker ψn+1 = range φn+1 . ∂n−1 φ∗ ψ∗ ∂ (7. Then there exists a long exact sequence n · · · → H n−1 (C ) → H n (A) → H n (B ) → H n (C ) → H n+1 (A) → · · · .10) dA dB φn dC ψn 0 M An K M Bn K M Cn K dA dB ψn−1 dC ψn−1 0 M An−1 K M B n− 1 K M C n−1 K dA dB dC . Since φn+2 is injective we deduce dA n+1 a = 0. This description is not entirely precise since a depends on various choices.1.e. .K dA dB φn+1 .1.11) We will not include a proof of this proposition. relies on a simple technique called diagram chasing. . in which the rows are exact. We claim that a is a cocycle. . . we can write 1 −1 B −1 B a = φ− n+1 ◦ d ◦ ψn (c) = φn+1 ◦ d b.K dC ψn+1 . .. and in fact the entire proof. . Start with x ∈ H n (C ). a is a cocycle. a canonical projections We have canonical inclusions α : L• → Cyl(f ). 0 M L• α ¯ f π ¯ M C (f )•  C (f ) π δ (f ) M K [1]• M0 M0 0 MK • M Cyl(f ) β K M C (f ) K K K K• f K M L• ¯ f π (iii) Show that the connecting morphism in the long exact sequence corresponding to the short exact sequence 0 → K • −→ Cyl(f ) −→ C (f ) → 0 coincides with the morphism induced in cohomology by δ (f ) : C (f ) → K [1]• . where the rows are exact. (b) The cylinder Cyl(f ). f • • • • β : Cyl(f ) → L . β ◦ α = L and α ◦ β is cochain homotopic to (i) Prove that α. ∂n : H n (C ) → H n+1 (A) and moreover. 1 This exercise describes additional features of the long exact sequence in cohomology. dK ) of R-modules.7. DERHAM COHOMOLOGY 229 defined map in cohomology. the sequence (7. They are particularly useful in the study of derived categories. δ = δ (f ) : C (f ) → K [1] .11) is exact. Exercise 7.1.1. . (ii) Show that we have the following commutative diagram of cochain complexes.   ki dK Cyl(f )• ∼ = K • ⊕ C (f )• . 1 Suppose R is a commutative ring with 1. (iv) Prove that f induces an isomorphism in cohomology if and only if the cone of f is acyclic. dC (f ) where dC ( f ) k i+1 i C (f )• = K [1]• ⊕ L• . dK [n] = (−1)n dK .19. and π : Cyl(f ) → C (f ). dCyl(f )  k i+1  =  0 i 0  −K i+1 −dK f    0 ki 0  ·  k i+1  . δ (f ) are cochain maps. For any cochain complex (K • . We associate to any cochain map f : K • → L• two new cochain complexes: (a) The cone C (f )• . and any integer n we denote by K [n]• the complex defined by K [n]m = K n+m . β f Cyl(f ) . ¯.1. i dL ¯ : K • → Cyl(f )• . dCyl(f ) = −dK f 0 dL · k i+1 i . . Prove that if at least two of the cohomology modules H • (A). Prove that there exists a formal series R(t) ∈ Z[[t]] with non-negative coefficients such that PC • (t) = PH • (C ) (t) + (1 + t)R(t). where | · | denotes the Euclidean norm in V n . for all n. (b) Prove that ∆n c = 0 if and only if dn c = 0 and d∗ n−1 c = 0. In particular. Assume each of the vector spaces C n is finite dimensional. Assume that n+1 → V n the adjoint each V n is equipped with an Euclidean metric. (Additivity of Euler characteristic). H • (B ) and H • (C ) have finite dimension over F.1. the graded spaces C ∗ and H ∗ have identical Euler characteristics χ(C • ) = PC (−1) = PH • (C ) (−1) = χ(H • (C )). Form the Poincar´ e series PC (t) = n≥0 tn dimF C n .20. d) := n≥0 V n . Let C • = n≥0 C n be a cochain complex of vector spaces over the field F.22. (a) Prove that ⊕n≥0 ∆n = (d + d∗ )2 . Let 0 → A• → B • → C • → 0 be a short exact sequence of cochain complexes of vector spaces over the field F.1. Exercise 7. c − c ∈ B n (C ) . ker ∆n ⊂ Z n (V • ). then the same is true about the third one. and PH • (C ) (t) = n≥0 tn dimF H n (C ).1. whenever it makes sense. be a cochain complex of real vector spaces such that dim V n < ∞. Prove that there exists a unique c ∈ Z n (C ) cohomologous to c such that |c| = min |c |.230 CHAPTER 7. Exercise 7. In particular. (Finite dimensional Hodge theory). and denote by d∗ n :V of dn . dn ). ∆n := d∗ n dn + dn−1 dn−1 . We can now form the Laplacians ∗ ∆n : V n → V n . (Abstract Morse inequalities). and moreover χ(H • (B )) = χ(H • (A)) + χ(H • (C )). COHOMOLOGY Exercise 7.21. Let (V • . (c) Let c ∈ Z n (C ). . m) → Φt (m). φ∗ : H • (N ) → H • (M ).24.. (a) Two smooth maps φ0 . Thus. φ∗ induces a morphism in cohomology which we continue to denote by φ∗ . Exercise 7. (It is known as the Koszul complex.1. (c) Two smooth manifolds M and N are said to be homotopy equivalent if there exists a homotopy equivalence φ : M → N . H k Λ• V. Proposition 7. (a) Prove that k · · · −→ Λk V −→ Λk+1 V −→ · · · dk−1 d dk+1 is a cochain complex.1. Deduce from all the above that the natural map ker ∆n → H n (V • ) is a linear isomorphism.7. φ∗ (α ∧ β ) = (φ∗ α) ∧ (φ∗ β ). DERHAM COHOMOLOGY 231 (d) Prove that c determined in part (b) satisfies ∆n c = 0. and the exterior differentiation is a quasi-derivation. i. In fact. Note that the pullback is an algebra morphism φ∗ : Ω• (N ) → Ω• (M ). and v0 ∈ V .1. §7. i. if there exists a smooth map Φ : I × M → N (t. φ∗ dN = dM φ∗ . Definition 7. 1. φ1 : M → N are said to be (smoothly) homotopic.e.e. we have a more precise statement. β ∈ Ω• (N ). For any smooth map φ : M → N the pullback φ∗ : Ω• (N ) → Ω• (M ) is a cochain morphism.1. ∀k ≥ 0. and respectively N . The DeRham cohomology construction is a contravariant functor from the category of smooth manifolds and smooth maps to the category of Z-graded vector spaces with degree zero morphisms. and ψ ◦ φ sh M . where dM and respectively dN denote the exterior derivative on M . (b) A smooth map φ : M → N is said to be a (smooth) homotopy equivalence if there exists a smooth map ψ : N → M such that φ ◦ ψ sh N .1. d(v0 ) = 0. for i = 0. and we write this φ0 sh φ1 . Let M and N be two .25. so that the map it induces in cohomology will also be a ring morphism. ω −→ v0 ∧ ω. Define dk = dk (v0 ) : Λk V → Λk+1 V. ∀α. such that Φi = φi .23. Let V be a finite dimensional real vector space.4 Functorial properties of the DeRham cohomology smooth manifolds.) (b) Use the finite dimensional Hodge theory described in previous exercise to prove that the Koszul complex is acyclic if v0 = 0. Then they induce identical maps in cohomology ∗ • • φ∗ 0 = φ1 : H (N ) → H (M ). (ω.232 CHAPTER 7. (I ×M )/M Thus χ(ω ) := (I ×M )/M Φ∗ (ω ) is the sought for cochain homotopy. φ1 : M → N be two homotopic smooth maps. Let φ0 . .1. For any ω ∈ Ω• (N ) we have the equality ∗ ∗ ∗ φ∗ 1 (ω ) − φ0 (ω ) = Φ (ω ) |1×M −Φ (ω ) |0×M = Φ∗ (ω ). η ) → −ω |U ∩V +η |U ∩V . We get a cochain morphism ∗ r : Ω• (M ) → Ω• (U ) ⊕ Ω• (V ). At this point. According to the general results in homological algebra. (∂I ×M )/M We now use the homotopy formula in Theorem 3.4. Thus. and U . φ1 : Ω (N ) → Ω (M ) are cochain homotopic. and • • ı∗ V : Ω (M ) → Ω (V ). V → M ). Corollary 7.26. ω → ω |U . These induce the restriction maps • • ı∗ U : Ω (M ) → Ω (U ). ω → (ı∗ U ω. and we deduce Φ∗ (ω ) = (∂I ×M )/M (I ×M )/M dI ×M Φ∗ (ω ) − dM Φ∗ (ω ) (I ×M )/M = (I ×M )/M Φ∗ (dN ω ) − dM Φ∗ (ω ). our discussion on the fibered calculus of Subsection 3.4. There exists another cochain morphism δ : Ω• (U ) ⊕ Ω• (V ) → Ω• (U ∩ V ).5.4. Two homotopy equivalent spaces have isomorphic cohomology rings. ∀ω ∈ Ω (M ).5 will pay off. we have to produce a map χ : Ω• (N ) → Ω∗•−1 (M ). V two open subsets of M such that M = U ∪ V . Denote by ıU (respectively ıV ) the inclusions U → M (resp.27. Proof. ω → ω |V .54 of Subsection 3.1. Consider a smooth manifold M . COHOMOLOGY Proposition 7. it suffices to show that the pullbacks ∗ • • φ∗ 0 . The projection Φ : I × M → M defines an oriented ∂ -bundle with standard fiber I . such that ∗ • φ∗ 1 (ω ) − φ0 (ω ) = ±χ(dω ) ± dχω. ıV ω ). 28.e. V } is an open cover of M . The Mayer-Vietoris sequence has the following functorial property. ϕV ≤ 1. supp ϕV ⊂ V. The short Mayer-Vietoris sequence 0 → Ω• (M ) −→ Ω• (U ) ⊕ Ω• (V ) −→ Ω∗ (U ∩ V ) → 0 is exact. ϕV } ⊂ C ∞ (M ) subordinated to this cover. Let us recall that construction.1. i. ϕU ω ) = (ϕV + ϕU )ω = ω. Then there exists a long exact sequence · · · → H k (M ) → H k (U ) ⊕ H k (V ) → H k (U ∩ V ) → H k+1 (M ) → · · · . The reader can prove directly that the above definition is independent of the various choices. upon extending ϕV ω by 0 outside V . ϕU + ϕV = 1.29 (Mayer-Vietoris). Note that δ (−ϕV ω. we deduce d(ϕV ω ) = d(−ϕU ω ) on U ∩ V. Start with ω ∈ Ωk (U ∩ V ) such that dω = 0. Thus. The proof of the equality Range r = ker δ can be safely left to the reader. DERHAM COHOMOLOGY Lemma 7.1. The surjectivity of δ requires a little more effort. Note that for any ω ∈ Ω∗ (U ∩ V ) we have supp ϕV ω ⊂ supp ϕV ⊂ V. supp ϕU ⊂ U. Theorem 7.1. we can find η ∈ Ωk+1 (M ) such that η |U = d(ϕV ω ) η |V = d(−ϕU ω ). Similarly. Writing as before ω = ϕV ω + ϕU ω. Using the abstract results in homological algebra we deduce from the above lemma the following fundamental result. r δ ∂ . Obviously r is injective. r δ 233 Proof. The collection {U. Let M = U ∪ V be an open cover of the smooth manifold M.. so we can find a partition of unity {ϕU . ϕU ω ∈ Ω∗ (V ). This establishes the surjectivity of δ . we can view it as a form on U .18 in the previous subsection.1.7. 0 ≤ ϕU . called the long Mayer-Vietoris sequence. The connecting morphisms ∂ can be explicitly described using the prescriptions following Proposition 7. and thus. Then ∂ω := η. 1. 2π -periodic. We have H 0 (S 1 ) = R since S 1 is connected. The cohomology of S 1 can be easily computed using the definition of DeRham cohomology. In principle. knowing the cohomologies of its constituents. To compute the cohomology of higher dimensional spheres we use the Mayer-Vietoris theorem. and dF = η .31. S1 Indeed. Thus.1. S1 Conversely if η = f (θ)dθ. then η = F (2π ) − F (0) = 0.1. η → S1 S1 η.234 CHAPTER 7. . the map : Ω1 (S 1 ) → R. A 1-form η ∈ Ω1 (S 1 ) is automatically closed. (The cohomology of spheres). and it is exact if and only if η = 0. H k (N ) φ∗ k r M H k (U ) ⊕ H k (V ) φ∗ δ M H k (U ∩ V ) φ∗ ∂ M H k+1(N ) φ∗ H (M ) K r MH k (U ) ⊕ H (V ) K k δ MH k (U ∩ V ) K ∂ MH k+1 K (M ) Exercise 7.30. Prove the above proposition. In this subsection we will illustrate this principle on some simple examples. the diagram below is commutative. it allows one to recover the cohomology of a manifold decomposed into simpler parts.32. 0 then the function F (t) = 0 t f (s)ds is smooth.5 Some simple examples The Mayer-Vietoris theorem established in the previous subsection is a very powerful tool for computing the cohomology of manifolds.1. V = φ−1 (V ) form an open cover of M and moreover. COHOMOLOGY Proposition 7. Example 7. and we deduce PS 1 (t) = 1 + t. Let φ : M → N be a smooth map and {U. if η = dF . where f : R → R is a smooth 2π -periodic function and 2π f (θ)dθ = 0. Then U = φ−1 (U ). induces an isomorphism H 1 (S 1 ) → R. where F : R → R is a smooth 2π -periodic function. §7. V } an open cover of N . Let Σ be a surface with finite Betti numbers.33. and then compute its Euler characteristic.1.1. Note that the overlap Unorth ∩ Usouth is homotopically equivalent with the equator S n . so that we have the isomorphisms H k (S n ) ∼ = H k (Unorth ∩ Usouth ) ∼ = H k+1 (S n+1 ) k > 0. Since Q1 (t) = t we deduce Qn (t) = tn . We can rewrite the above equality as Qn+1 (t) = tQn (t) n > 0.1. From the decomposition Σ = (Σ \ disk) ∪ disk. For k > 0 the group on the left is also trivial.1. (7. V and U ∩ V are finite. The Mayer-Vietoris sequence gives H k (Unorth ) ⊕ H k (Usouth ) → H k (Unorth ∩ Usouth ) → H k+1 (S n+1 ) → 0. V } be an open cover of the smooth manifold M. Denote by Pn (t) the Poincar´ e polynomial of S n and set Qn (t) = Pn (t) − Pn (0) = Pn (t) − 1. If now Σ1 and Σ2 are two surfaces with finite Betti numbers. DERHAM COHOMOLOGY The (n + 1)-dimensional sphere S n+1 can be covered by two open sets Usouth = S n+1 \ {North pole} and Unorth = S n+1 \ {South pole}. and the Exercise 7. then Σ1 #Σ2 = (Σ1 \ disk) ∪ (Σ2 \ disk).3. Example 7.21 in Subsection 7.e.1. we deduce that all the Betti numbers of M are finite. Let {U. Since (Σ \ disk) ∩ disk is homotopic to a circle. and moreover χ(M ) = χ(U ) + χ(V ) − χ(U ∩ V ). 235 Each is diffeomorphic to Rn+1 . We assume that all the Betti numbers of U . i. .12) This resembles very much the classical inclusion-exclusion principle in combinatorics. and χ(disk) = 1.1. we deduce χ(Σ) = χ(Σ \ disk) + 1 − χ(S 1 ) = χ(Σ \ disk) + 1. we deduce (using again Exercise 7. PS n (t) = 1 + tn .7. We will use this simple observation to prove that the Betti numbers of a connected sum of g tori is finite..21) χ(Σ) = χ(Σ \ disk) + χ(disk) − χ ( (Σ \ disk) ∩ (disk) ) . The Poincar´ e lemma implies that H k+1 (Unorth ) ⊕ H k+1 (Usouth ) ∼ =0 for k ≥ 0. Using the Mayer-Vietoris short exact sequence. (c) The connected sums.37.31 of Subsection 4.2. A smooth manifold M is said to be of finite type if it can be covered by finitely many open sets U1 . Let p : E → B be a smooth vector bundle such that B is diffeomorphic to Rn .1. We conclude as in Proposition 4.31 that χ(connected sum of g tori) = 2 − 2g.36. this overlap is a disjoint union of two circles.1. it suffices to cover such a manifold by finitely many open sets which are geodesically convex with respect to some Riemann metric. We can decompose a torus as a union of two cylinders. This is a pleasant. COHOMOLOGY where the two holed surfaces intersect over an entire annulus.2.1. surprising connection with the Gauss-Bonnet theorem. then so is the total space E . which is homotopically a circle. and the direct products of finite type manifolds are finite type manifolds. Denote by F the standard fiber of E .36.236 CHAPTER 7.34.6 The Mayer-Vietoris principle We describe in this subsection a “patching” technique which is extremely versatile in establishing general homological results about arbitrary manifolds building up from elementary ones. For each ordered multi-index I := {i1 < · · · < ik } denote by UI the multiple overlap Ui1 ∩ · · · ∩ Uik . To see this.1. And the story is not over. Hence χ(torus) = 2χ(cylinder) = 2χ(circle) = 0. . In particular.1. . This equality is identical with the one proved in Proposition 4. (It suffices to see that Rn \ closed ball is of finite type). and U ⊂ M is a closed subset homeomorphic with the closed unit ball in Rdim M . Let (Ui )1≤i≤ν be a good cover of B . If the base B is of finite type.2. Proof of Proposition 7. §7. the Euler characteristic of the intersection is zero. Thus χ(Σ1 #Σ2 ) = χ(Σ1 \ disk) + χ(Σ2 \ disk) − χ(S 1 ) = χ(Σ1 ) + χ(Σ2 ) − 2.1. Definition 7. Let p : E → B be a smooth vector bundle. Using the previous .35.5. then M \ U is a finite type non-compact manifold. . The fiber F is a vector space. Example 7. (a) All compact manifolds are of finite type. . Proposition 7. (b) If M is a finite type manifold. Such a cover is said to be a good cover. Um such that any nonempty intersection Ui1 ∩ · · · ∩ Uik (k ≥ 1) is diffeomorphic to Rdim M . In the proof of this proposition we will use the following fundamental result. Lemma 7. Then p : E → B is a trivializable bundle. The intersection of these cylinders is the disjoint union of two annuli so homotopically. 37.39.38. V } is an M V -cover of M such that {U ∩ N. if N ∈ Mn is an open submanifold of N .1. Prove Lemma 7. . Fix a connection ∇ on E . x ∈ Rn \ {0}. U. V.  We denote by Mn the category finite type smooth manifolds of dimension n. V ∩ N } is an M V -cover of N . then there exist morphisms of R-modules ∂n : Fn (U ∩ V ) → Fn+1 (M ). U ∩ V ∈ Mn .e. Exercise 7. then the diagram below is commutative..1. i.7. and δ is defined by δ (x ⊕ y ) = F(ıU ∩V )(y ) − F(ıU ∩V )(x).e. such that the sequence below is exact n · · · → Fn (M ) → Fn (U ) ⊕ Fn (V ) → Fn (U ∩ V ) → Fn+1 (M ) → · · · . i. Definition 7.. with the following property. [0. (The maps ı• denote natural embeddings. to the category of Z-graded R-modules F = ⊕n∈Z Fn → GradR Mod. DERHAM COHOMOLOGY 237 lemma we deduce that each EI = E |UI is a product F × Ui . M → n Fn (M ). and then use the ∇-parallel transport along the half-lines Lx . ∞) t → tx ∈ Rn . Mi ∈ Mn . The covariant M V -functors are defined in the dual way.) Moreover. r∗ δ ∂ where r∗ is defined by r∗ = F(ıU ) ⊕ F(ıV ). and thus it is diffeomorphic with some vector space. If {U. Let R be a commutative ring with 1. by reversing the orientation of all the arrows in the above definition.1.1. Fn (U ∩ V ) ∂n M Fn+1(M ) MF n+1 F (U ∩ V ∩ N ) n K ∂n K (N ) The vertical arrows are the morphisms F(ı• ) induced by inclusions. A contravariant Mayer-Vietoris functor (or M V -functor for brevity) is a contravariant functor from the category Mn . V } is a M V -cover of M ∈ Mn . Hence (Ei ) is a good cover. The morphisms of this category are the smooth embeddings. and {U. Hint: Assume that E is a vector bundle over the unit open ball B ⊂ Rn . the one-to-one immersions M1 → M2 . G be two (contravariant) MayerVietoris functors on Mn and φ : F → G a correspondence. The theorem is clearly true for r = 1 by hypothesis. A correspondence 2 between these functors is a collection of R-module morphisms φM = n∈Z φn M : Fn (M ) −→ Gn (M ). where Mr n is the collection of all smooth manifolds which admit a good cover consisting of at most r open sets. G : Mn → GradR Mod. for any embedding M1 → M2 . Ur } of M . the diagram below is commutative F(ϕ) Fn (M2 ) Fn (M1 ) M φM2 Gn (M2 ) K φM1 G(ϕ) M Gn(M1) M Fn+1(M ) φM K . We thus get a commutative diagram 2 Our notion of correspondence corresponds to the categorical notion of morphism of functors. If k n k n φk Rn : F ( R ) → G ( R ) is an isomorphism for any k ∈ Z.41 (Mayer-Vietoris principle). the diagram below is commutative.40. G be two contravariant M V -functors. such that. Fn (U ∩ V ) φU ∩V n ∂n G (U ∩ V ) K ∂n MG n+1 K (M ) The correspondence is said to be a natural equivalence if all the morphisms φM are isomorphisms. We will prove the theorem using an induction over r.1.1. . and. Let F. The family of finite type manifolds Mn has a natural filtration 2 r M1 n ⊂ Mn ⊂ · · · ⊂ Mn ⊂ · · · . . COHOMOLOGY Definition 7. ϕ one morphism for each M ∈ Mn . . Proof. for any M ∈ Mn and any M V -cover {U. . F. Let F. Ur } is an M V -cover of M . then φ is a natural equivalence. V } of M . Theorem 7. Assume φk M is an isomorphism r r − 1 for all M ∈ Mn . Then {U = U1 ∪ · · · ∪ Ur−1 . . Let M ∈ Mn and consider a good cover {U1 .238 CHAPTER 7. (c) Assume R is a field.45. i. A−2 A−1 A0 A1 A2 M M M M f−2 B−2 K f−1 M B−1 K f0 M B0 K f1 M B1 K K M B2 f2 If fi is an isomorphism for any i = 0. Exercise 7. .1.1. The proof is obtained by reversing the orientation of the horizontal arrows in the above proof. §7. (The five lemma.1. Corollary 7.44. Then. each of the R-module morphisms φM are in fact morphisms of R-algebras. if φRn are isomorphisms of Z-graded R-algebras . then so is f0 . DERHAM COHOMOLOGY 239 Fn (U ) ⊕ Fn (Ur ) Gn (U ) ⊕ Gn (Ur ) K M Fn(U ∩ Ur ) K M Gn(U ∩ Ur ) ∂ ∂ M Fn+1(M ) K M Gn+1(M ) M Fn+1(U ) ⊕ Gn+1(Ur ) K n+1 n+1 MF (U ) ⊕ G (Ur ) The vertical arrows are defined by the correspondence φ.1. Assume that F and G are functors from Mn to the category of Z-graded R-algebras. We will now use the Mayer-Vietoris principle to compute the cohomology of products of manifolds. (b) The Mayer-Vietoris principle can be refined a little bit.1. The proof of the Mayer-Vietoris principle shows that if F is a MV-functor and dimR F∗ (Rn ) < ∞ then dim F∗ (M ) < ∞ for all M ∈ Mn .) Consider the following commutative diagram of R-modules. Note the inductive assumption implies that in the above infinite sequence only the morphisms φM may not be isomorphisms. and φ : F → G is a correspondence compatible with the multiplicative structures.7 The K¨ unneth formula We learned in principle how to compute the cohomology of a “union of manifolds”. Any finite type manifold has finite Betti numbers. (a) The Mayer-Vietoris principle is true for covariant M V -functors as well.. At this point we invoke the following technical result.42. then so are the φM ’s. (d) The Mayer-Vietoris principle is a baby case of the very general technique in algebraic topology called the acyclic models principle. Remark 7.1. Lemma 7. for any M ∈ Mn .7. Prove the five lemma.e.43. The five lemma applied to our situation shows that the morphisms φM must be isomorphisms. 1. define φM : F(M ) → G(N ) by ∗ ∗ φM (ω ⊗ η ) = ω × η := πM ω ∧ πN η ( ω ∈ H • (M ). (b) φRm is an isomorphism. Lemma 7. The operation × : H • (M ) ⊗ H • (N ) → H • (M × N ) (ω ⊗ η ) → ω × η. Let M ∈ Mm and N ∈ Mn . F and G are contravariant M V -functors. ∀f : M1 → M2 . . and G:M → r≥0 Gr (M ) = r ≥0 H r (M × N ). ∀f : M1 → M2 .46 (K¨ unneth formula). and G(f ) = r≥0 (f × N )∗ |H r (M2 ×N ) . In particular. For M ∈ MM . Proof. G : Mm → GradR Alg. Exercise 7.1. is called the cross product. (a) φ is a correspondence of MV-functors. We let the reader check the following elementary fact.47. Then there exists a natural isomorphism of graded R-algebras H • (M × N ) ∼ = H • (M ) ⊗ H ∗ (N ) = n≥0 p+q =n H p (M ) ⊗ H q (N ) .48. We construct two functors F. η ∈ H • (M ) ).240 CHAPTER 7. where πM (respectively πN ) are the canonical projections M × N → M (respectively M × N → N ). we deduce PM ×N (t) = PM (t) · PN (t). F:M → r≥0 Fr (M ) = r≥0 p+q =r H p (M ) ⊗ H q (N ) .1. COHOMOLOGY Theorem 7. The K¨ unneth formula is a consequence of the following lemma. where F(f ) = r ≥0 p+q =r f ∗ |H p (M2 ) ⊗H q (N ) . By writing it as a direct product of n circles we deduce from K¨ unneth formula that PT n (t) = {PS 1 (t)}n = (1 + t)n . We have an isomorphism of R-algebras H • (T n ) ∼ = Λ• Rn . Example 7. x) is a homotopy equivalence.1. The only nontrivial thing to prove is that for any MV-cover {U. ∗ ϕ and ψ = π ∗ ϕ form a partition of unity subordinated to the the functions ψU = πM U V M V cover U × N. V } of M ∈ Mm the diagram below is commutative. . DERHAM COHOMOLOGY 241 Proof. Thus. Then. ϕV } subordinated to the cover {U. they form a basis of H ∗ (T n ). and ∂ (ω × η ) = ω ˆ × η . non-exact form dθI . by the homotopy invariance of the DeRham cohomology we deduce G(Rm ) ∼ = H • (N ). with πN a homotopy inverse. These monomials are linearly independent (over R) and there are 2n of them. Choose angular coordinates (θ1 . V }.49. where ω ˆ |U = −d(ϕV ω ) ω ˆ = d(ϕU ω ). ∂ (ω ⊗ η ) = ω ˆ ⊗ η.7. Hence. This proves (a). One considers a partition of unity {ϕU . T n . φU ∩V (ω ⊗ η ) = ω × η . dim H ∗ (T n ) = 2n . For each ordered multi-index I = (1 ≤ i1 < · · · < ik ≤ n) we have a closed. note that the inclusion  : N → Rm × N. . θn ) on n T . One can easily describe a basis of H ∗ (T n ). Consider the n-dimensional torus. ⊕p+q=r H p (U ∩ V ) ⊗ H q (N ) φU ∩V r ∂ M ⊕p+q=r H p+1(M ) ⊗ H q (N ) φM ∂ H ((U × N ) ∩ (V × N )) K MH r+1 (M × N ) K We briefly recall the construction of the connecting morphisms ∂ and ∂ . Thus bk (T n ) = n . . x → (0. . k and χ(T n ) = 0. If ω ⊗ η ∈ H • (U ∩ V ) ⊗ H • (N ). V × N of M × N . one can read the multiplicative structure using this basis. Using the Poincar´ e lemma and the above isomorphism we can identify the morphism φRm with Rm . To establish (b). In fact. On the other hand. .1. (ω0 × η0 ) ∧ (ω1 × η1 ) = (−1)deg η0 deg ω1 (ω0 ∧ ω1 ) × (η0 ∧ η1 ). . Its cohomology is denoted by Hcpt ham cohomology with compact supports. 7. η ∈ H • (E ). ηj ∈ H ∗ (N ) (i. we can identify M with an open subset of N . (Leray-Hirsch).1 Cohomology with compact supports Let M be a smooth n-dimensional manifold.242 CHAPTER 7. (b) There exist cohomology classes e1 . As we will see.1) the following equality holds.2. er .50. then the pull-back φ ω may not have compact support. . The projection p induces a H • (M )-module structure on H • (E ) by ω · η = p∗ ω ∧ η ∀ω ∈ H • (M ). so this new construction is no longer a contravariant functor from the category of smooth manifolds and smooth maps.1. Note that when M is compact this cohomology coincides with the usual DeRham cohomology.51. • is a covariant functor from the category M . Let p : E → M be smooth bundle with standard fiber F .j=0. and φ is an embedding. In terms of our category Mn we see that Hcpt n to the category of graded real vector spaces. The most visible one is that if φ : M → N is a smooth map. We assume the following: (a) Both M and F are of finite type. if dim M = dim N . there are many important differences. Denote by Ωk cpt (M ) the space of smooth compactly supported k -forms. d d One can verify easily that φ∗ is a cochain map so that it induces a morphism • • φ∗ : Hcpt (M ) → Hcpt (N ). . and it is called the DeRis a cochain complex. and then any η ∈ Ω• cpt (M ) can be extended by 0 outside M ⊂ N . ∗ and ω ∈ Ω• cpt (N ). Show that for any ωi ∈ H ∗ (M ).1. On the other hand. Then n 0 → Ω0 cpt (M ) → · · · → Ωcpt (M ) → 0 • (M ). Exercise 7. er ∈ H • (E ) such that their restrictions to any fiber generate the cohomology algebra of that fiber.2 The Poincar´ e duality §7. Show that H • (E ) is a free H • (M )-module with generators e1 . . Although it looks very similar to the usual DeRham cohomology. This extension by zero defines a push-forward map • φ∗ : Ω• cpt (M ) → Ωcpt (N ). . to the category of graded vector spaces. . COHOMOLOGY Exercise 7. . Let M ∈ Mm and N ∈ Mn . . it is a rather nice functor. . k<n . v0 + t(v1 − v0 )). Proof. . Consider the ∂ -bundle π : E = I × (E ⊕ E ) → E.4. We first prove the Poincar´ e lemma for compact supports. Note that ∂ E = ( {0} × (E ⊕ E ) ) Define π t : E ⊕ E → E as the composition π E⊕E ∼ = {t} × E ⊕ E → I × (E ⊕ E ) → E. k=n The last assertion of this theorem is usually called the Poincar´ e lemma for compact supports. β ∈ Ω• cpt (E ⊕ E ) by α (Verify that the support of α we have the equalities 1 p∗ p∗ α ∧ β = π∗ (α β := (π 0 )∗ α ∧ (π 1 )∗ β. and moreover Theorem 7. R . v0 . β ) ∈ Ω• cpt (E ) × Ωcpt (E ) define α π1.2. dβ ).) For α ∈ Ωi cpt (E ). β ) = p∗ p∗ α ∧ β − α ∧ p∗ p∗ β = π∗ (α 0 β ) − π∗ (α β) = ∂ E/E α β. THE POINCARE • is a covariant M V -functor. Observe that ∂π = π |∂ E = (−π 0 ) • For (α.2.2. p∗ p∗ α ∧ β − α ∧ p∗ p∗ β = (−1)r d(m(α. v1 ) → (t. Lemma 7. and β ∈ Ωcpt (E ) i+j −r β ) ∈ Ωcpt (E ). p Then there exists a smooth bilinear map j i+j −r−1 m : Ωi (E ) cpt (E ) × Ωcpt (E ) → Ωcpt such that. β )) − m(dα. β ) + (−1)deg α m(α. j β is indeed compact.5.´ DUALITY 7. Denote by p∗ the integration-along-fibers map •−r p∗ : Ω • cpt (E ) → Ωcpt (B ). π : (t.2. The crucial step is the following technical result that we borrowed from [11]. Let E → B be a rank r real vector bundle which is orientable in the sense described in Subsection 3.1. ({1} × (E ⊕ E )) . α ∧ p∗ p∗ β = π∗ (α cpt Hence 1 D(α. Hcpt k Hcpt (Rn ) = 243 0 . +j −r 0 β ) ∈ Ωi (E ). 244 We now use the fibered Stokes formula to get D(α. The hat ˆ denotes the extension by zero outside the support.1 we must construct a Mayer-Vietoris sequence. n p∗ : Ω• cpt (R ) → R. Rn If now ω is a closed. To finish the proof of Theorem 7.2 we deduce ω − τ ∧ p∗ p∗ ω = (−1)n dm(τ. compactly supported form ω on Rn is cohomologous to τ ∧ p∗ p∗ ω . V → M. The latter is always zero if deg ω < n. 0 ≤ δ ≤ 1. ω → p∗ ω = 0 . deg ω < n ω . ω ).2 in which E is the rank n bundle over a point. where T is the natural projection E = I × (E ⊕ E ) → E ⊕ E . When deg ω = n we deduce that ω is cohomologous to ( ω )τ . E = p {pt} × Rn → {pt}. that δ (x)dx = 1. COHOMOLOGY dE T ∗ (α β ) + (−1)r dE T ∗ (α E/E β ).2. compactly supported form on Rn . deg ω = n. i j where i(ω ) = (ˆ ω. i. The lemma holds with m(α. ω ˆ ).2. β ) = E/E T ∗ (α β ). closed. ∞ (Rn ) such Consider δ ∈ C0 Proof of the Poincar´ e lemma for compact supports. compactly supported n-form τ := δ (x)dx1 ∧ · · · ∧ dxn . Thus any closed. This completes the proof of the Poincar´ e lemma.. Let M be a smooth manifold decomposed as an union of two open sets M = U ∪ V . . Using Lemma 7. η ) = η ˆ−ω ˆ. we have ω = (p∗ p∗ τ ) ∧ ω.2. This sequence is called the Mayer-Vietoris short sequence for compact supports. j (ω. β ) = E/E CHAPTER 7. We want to use Lemma 7. The integration along fibers is simply the integration map. induces a short sequence • • • 0 → Ω• cpt (U ∩ V ) → Ωcpt (U ) ⊕ Ωcpt (V ) → Ωcpt (M ) → 0. The sequence of inclusions U ∩ V → U.e. Rn Define the. 2. THE POINCARE Lemma 7. Then. Moreover. Corollary 7. V }. we almost tautologically get a commutative diagram k Hcpt (N ) φ∗ δ k+1 −1 M Hcpt (φ (U ∩ V )) k Hcpt (M ) K δ K k+1 M Hcpt (U ∩ V ) φ∗ • is a covariant Mayer-Vietoris sequence. and any k ≤ n we have dim Hcpt . η = j (−ϕU η. For any M ∈ Mn . ϕV ) a partition of unity subordinated to the cover {U. If φ : N → M is a morphism of Mn then for any MV-cover of {U. k ≤0 k≤0 and correspondingly the associated homology ˜ • := H −• . We have to prove that j is surjective. the since the connecting morphism δ goes in the right direction H simplicity of the original notation is worth the small formal ambiguity. However. The above Mayer-Vietoris sequence is exact. This proves Hcpt Remark 7. Clearly.3. so we stick to our upper indices. φ−1 (V )} is an MV-cover of N . The reader can check immediately that the cohomology class of δω is independent of all the choices made. Range (i) = ker(j ). If ω ∈ Ωk cpt (M ) is a closed form then d(ϕU ω ) = d(−ϕV ω ) on U ∩ V.and its homology H ˜• → H ˜ •−1 .5. 245 Proof. In particular.2. H cpt This makes the sequence ˜ −n ← · · · ← Ω ˜ −1 ← Ω ˜0 ← 0 0←Ω ˜ • is a bona-fide covariant Mayer-Vietoris functor a chain complex . From the proof of the Mayer-Vietoris principle we deduce the following. Let (ϕU . We set δω := d(ϕU ω ). which shows that j is surjective. To be perfectly honest (from a categorical point of view) we should have considered the chain complex k ˜k = Ω− Ω cpt .2.´ DUALITY 7. V } of M {φ−1 (U ). ϕV η ). We get a long exact sequence called the long Mayer-Vietoris sequence for compact supports. for any η ∈ Ω∗ cpt (M ). k (M ) < ∞. The morphism i is obviously injective. we have • ϕU η ∈ Ω• cpt (U ) and ϕV η ∈ Ωcpt (V ).4.2. k+1 k k k k · · · → Hcpt (U ∩ V ) → Hcpt (U ) ⊕ Hcpt (V ) → Hcpt (M ) → Hcpt → ··· δ The connecting homomorphism can be explicitly described as follows. Denote by M+ n the category of n-dimensional. •. • : V ∗ × V → R. Hence D descends to a map in cohomology n−k D : H k (M ) → (Hcpt (M ))∗ . deg ω + deg η = n . We can extend this pairing to any (ω. If n− k moreover ω is exact. Above. Given M ∈ M+ n . η = M ω ∧ dη Stokes = ± M dω ∧ η = 0. η κ . ω ∧ η . compactly supported n−k n−k−1 (n − k )-forms vanishes on the subspace Bcpt = dΩcpt (M ). The morphisms are the embeddings of such manifold. Equivalently. the linear functional D(ω ) defines an element of (Hcpt (M ))∗ . η κ κ −k : Ωk (M ) × Ωn cpt (M ) → R. •. η ∈ n−k−1 Ωcpt (M ) then D(ω ).2.246 §7.2. COHOMOLOGY Definition 7. η κ = 0 . oriented manifolds. • κ n−k : H k (M ) × Hcpt (M ) → R. deg ω + deg η = n M The Kronecker pairing induces maps −k ∗ D = Dk : Ωk (M ) → (Ωn cpt (M )) . • defined by ω. . Theorem 7.2 The Poincar´ e duality CHAPTER 7. η ) ∈ Ω• × Ω• cpt as ω.7 (Poincar´ e duality). D(ω ). Indeed. this means that the Kronecker pairing descends to a pairing in cohomology. This pairing is called the Kronecker pairing. if ω is closed. the restriction of D(ω ) to the space Zcpt of closed. The M V functors on M+ n are defined exactly as for Mn . • denotes the natural pairing between a vector space V and its dual V ∗ . η = ω.6. •. n−k Thus. The Kronecker pairing in cohomology is a duality for + all M ∈ Mn . if η = dη . n−k If ω is closed. a computation as above shows that D(ω ) = 0 ∈ (Hcpt (M ))∗ . finite type. there is a natural pairing •. := M ω ∧ η.2. η = (φ∗ )† DN (ω ). Then the diagram below is commutative. THE POINCARE Proof. For purely formal reasons which will become apparent in a little while. (The exactness of the Mayer-Vietoris sequence is preserved by transposition.´ DUALITY 7. φ∗ η φ = N ω ∧ φ∗ η = M →N ω |M ∧η = M φ∗ ω ∧ η = DM (φ∗ ω ). Proof of Fact A. The Poincar´ e lemma for compact supports can be rephrased δ ˜ k (Rn ) = H The Kronecker pairing induces linear maps R . H H n−k−1 n−k to be (−1)k δ † . k=0 . Proof. η . H k (N ) DN φ∗ ˜ k (N ) H K ˜∗ φ M H k (M ) D K . Fact A. then the diagram bellow is commutative H k (U ∩ V ) DU ∩V ∂ M H k+1(M ) DM ˜ k (U ∩ V ) H K (−1)k δ † ˜ k+1 (M ) MH K .2. Let M → N be a morphism in M+ n . We have to check two facts. If {U. ˜ k (M ) MH M Fact B. DM : H k (M ) → H Lemma 7. k>0 ˜ k (M ). φ k Dk is a correspondence of M V functors. 0 . we re-define the connecting morphism ˜ d ˜ k (U ∩ V ) → ˜ k+1 (U ∪ V ). . and δ † denotes its transpose.8. where Hcpt (U ∩ V ) → Hcpt (U ∪ V ) denotes the connecting morphism in the DeRham cohomology with compact supports. Let ω ∈ H k (N ). The functor M+ n → Graded Vector Spaces defined by M→ k 247 ˜ k (M ) = H k n−k Hcpt (M ) ∗ is a contravariant M V -functor. Denoting by •. This is where the fact that all the cohomology groups are finite dimensional vector spaces plays a very important role). V } is an M V -cover of M ∈ M+ n . η = DN (ω ).2. • the natural duality between a n−k vector space and its dual we deduce that for any η ∈ Hcpt (M ) we have ˜∗ ◦ DN (ω ). and the vanishing now follows from Stokes formula. d(ϕU ω ) on V −k−1 Choose η ∈ Ωn (M ) such that dη = 0. Then the connecting morphism in usual DeRham cohomology acts as ∂ω = d(−ϕV ω ) on U . The second term is dealt with in a similar fashion. Corollary 7. q := dim N − dim M = n − m) defined by the composition N M H • (M ) → H •+n−m (N ). over U we have the equality d(ϕV ω ) ∧ η = d ( ϕV ω ∧ η ) . Using the Poincar´ e duality we can associate to any smooth map f : M → N between compact oriented manifolds of dimensions m and respectively n a natural pushforward or Gysin map f∗ : H • (M ) → H •+q (N ). we have d(ϕV ω ) ∧ η = dϕV ∧ ω ∧ η = (−1)deg ω ω ∧ (dϕV ∧ η ) U ∩V U ∩V U ∩V = (−1)k U ∩V ω ∧ δη = (−1)k DU ∩V ω. Indeed. This concludes the proof of Fact B. . If M ∈ M+ n then • Hcpt (M ) ∼ = (H n−k (M ))• . δη = (−1)k δ † DU ∩V ω. Hence φ CHAPTER 7. η = =− U ∂ω ∧ η = M U ∂ω ∧ η + V ∂ω ∧ η − U ∩V ∂ω ∧ η d(ϕV ω ) ∧ η + V dϕU ω ) ∧ η + U ∩V d(ϕV ω ) ∧ η. (H m−• (M ))∗ → (H m−• (N ))∗ → D (f ∗ )† D −1 where (f ∗ )† denotes the transpose of the pullback morphism. η . Remark 7.9.2. As for the last term. We have cpt DM ∂ω.2.248 ˜∗ ◦ DN = DM ◦ φ∗ . Consider a closed k -form ω ∈ Ω (U ∩ V ). ϕV be a partition of unity subordinated to the M V -cover k {U. Let ϕU . Note that the first two integrals vanish. V } of M ∈ M+ n . COHOMOLOGY Proof of Fact B. The Poincar´ e duality now follows from the MayerVietoris principle.10. connected.2. Show that if P satisfies the symmetry condition (7. Assume P (0) = 1 and P satisfies (7. oriented manifold M such that PM (t) = P (t). n-dimensional manifold. In particular. We have shown that χ(Σg ) = b0 − b1 + b2 = 2 − 2g. The bilinear form I is called the cohomological intersection form of M . connected.2. Its signature is called the signature of M . for any finite dimensional vector space there exists a natural isomorphism V ∗∗ ∼ = V. oriented.´ DUALITY 7. H 2k+1 (M ) must be an even dimensional space. I is skew-symmetric. Then the pairing H k (M ) × H n−k (M ) → R (ω. Corollary 7. ∀k . Let M be a compact.1) there exists a compact. When k is even (so that n is divisible by 4) I is a symmetric form. In particular. The symmetry of Betti numbers can be translated in the language of Poincar´ e polynomials as 1 (7. Corollary 7.1) tn PM ( ) = PM (t).14. If the coefficient of tk is even then there exists a compact connected manifold M ∈ M+ 2k such that PM (t) = P (t). t Example 7. THE POINCARE k (M ) is finite dimensional.2.1). If M is connected H 0 (M ) ∼ = H n (M ) ∼ = R so that H n (M ) is generated by any volume form defining the orientation.11. PΣg (t) = 1 + 2gt + t2 . bk (M ) = bn−k (M ). When k is odd. and it is denoted by σ (M ). the transpose Proof.2. Hence b1 = 2g i.2. Combine this fact with the K¨ unneth formula. Let Σg denote the connected sum of g tori. For any compact manifold M ∈ M+ 4k+2 the middle Betti number b2k+1 (M ) is even.e. Since Hcpt n−k ∗∗ Dt → (H k (M ))∗ M : (Hcpt (M )) 249 is an isomorphism. it is a symplectic form. η ) = M ω ∧ η.2. (a) Let P ∈ Z[t] be an odd degree polynomial with non-negative integer coefficients such that P (0) = 1. Exercise 7. Consider now a compact oriented smooth manifold such that dim M = 2k .2.12. the Poincar´ e duality implies b2 = b0 = 1.13. ..e. On the other hand. Since Σg is connected. η ) → M ω∧η is a duality. Hint: Describe the Poincar´ e polynomial of a connect sum in terms of the polynomials of its constituents.2. (b) Let P ∈ Z[t] be a polynomial of degree 2k with non-negative integer coefficients. The Kronecker pairing induces a non-degenerate bilinear form I : H k (M ) × H k (M ) → R I(ω. i. 250 CHAPTER 7. COHOMOLOGY Remark 7.2.15. The result in the above exercise is sharp. Using his intersection theorem F. Hirzebruch showed that there exist no smooth manifolds M of dimension 12 or 20 with Poincar´ e polynomials 1 + t6 + t12 and respectively 1 + t10 + t20 . Note that in each of these cases both middle Betti numbers are odd. For details we refer to J. P. Serre, “Travaux de Hirzebruch sur la topologie des vari´ et´ es”, Seminaire Bourbaki 1953/54,n◦ 88. 7.3 §7.3.1 Cycles and their duals Intersection theory Suppose M is a smooth manifold. Definition 7.3.1. A k -dimensional cycle in M is a pair (S, φ), where S is a compact, oriented k -dimensional manifold without boundary, and φ : S → M is a smooth map. We denote by Ck (M ) the set of k -dimensional cycles in M . Definition 7.3.2. (a) Two cycles (S0 , φ0 ), (S1 , φ1 ) ∈ Ck (M ) are said to be cobordant, and we write this (S0 , φ0 ) ∼c (S1 , φ1 ), if there exists a compact, oriented manifold with boundary Σ, and a smooth map Φ : Σ → M such that the following hold. (a1) ∂ Σ = (−S0 ) S1 where −S0 denotes the oriented manifold S0 equipped with the opposite orientation, and “ ” denotes the disjoint union. (a2) Φ |Si = φi , i = 0, 1. (b) A cycle (S, φ) ∈ Ck (M ) is called trivial if there exists a (k + 1)-dimensional, oriented manifold Σ with (oriented) boundary S , and a smooth map Φ : Σ → M such that Φ|∂ Σ = φ. We denote by Tk (M ) the set of trivial cycles. (c) A cycle (S, φ) ∈ Ck (M ) is said to be degenerate if it is cobordant to a constant cycle, i.e., a cycle (S , φ ) such that φ is map constant on the components of S . We denote by Dk (M ) the set of degenerate cycles. Exercise 7.3.3. Let (S0 , φ0 ) ∼c (S1 , φ1 ). Prove that (−S0 S1 , φ0 φ1 ) is a trivial cycle. The cobordism relation on Ck (M ) is an equivalence relation3 , and we denote by Zk (M ) the set of equivalence classes. For any cycle (S, φ) ∈ Ck (M ) we denote by [S, φ] its image in Zk (M ). Since M is connected, all the trivial cycles are cobordant, and they define an element in Zk (M ) which we denote by [0]. Given [S0 , φ0 ], [S1 , φ1 ] ∈ Zk (M ) we define [S0 , φ0 ] + [S1 , φ1 ] := [S0 where “ ” denotes the disjoint union. Proposition 7.3.4. Suppose M is a smooth manifold. (a) Let [Si , φi ], [Si , φi ] ∈ Ck (M ), i = 0, 1. If [Si , φi ] ∼c [Si , φi ], for i = 0, 1, then [S0 S1 , φ0 φ1 ] ∼c [S0 S1 , φ0 φ1 ] S1 , φ0 φ1 ], so that the above map + : Zk (M ) × Zk (M ) → Zk (M ) is well defined. 3 We urge the reader to supply a proof of this fact. 7.3. INTERSECTION THEORY 251 S0 S 1 Figure 7.1: A cobordism in R3 (b) The binary operation + induces a structure of Abelian group on Zk (M ). The trivial element is represented by the trivial cycles. Moreover, −[S, φ] = [−S, φ] ∈ Hk (M ). Exercise 7.3.5. Prove the above proposition. We denote by Hk (M ) the quotient of Zk (M ) modulo the subgroup generated by the degenerate cycles. Let us point out that any trivial cycle is degenerate, but the converse is not necessarily true. k Suppose M ∈ M+ n . Any k -cycle (S, φ) defines a linear map H (M ) → R given by H k (M ) ω→ S φ∗ ω. Stokes formula shows that this map is well defined, i.e., it is independent of the closed form representing a cohomology class. Indeed, if ω is exact, i.e., ω = dω , then φ∗ dω = S S dφ∗ ω = 0. In other words, each cycle defines an element in ( H k (M ) )∗ . Via the Poincar´ e duality we n−k n−k k ∗ identify the vector space ( H (M ) ) with Hcpt (M ). Thus, there exists δS ∈ Hcpt (M ) such that ω ∧ δS = φ∗ ω ∀ω ∈ H k (M ). M S The compactly supported cohomology class δS is called the Poincar´ e dual of (S, φ). −k There exist many closed forms η ∈ Ωn ( M ) representing δ . When there is no risk S cpt of confusion, we continue denote any such representative by δS . 252 CHAPTER 7. COHOMOLOGY Figure 7.2: The dual of a point is Dirac’s distribution Example 7.3.6. Let M = Rn , and S is a point, S = {pt} ⊂ Rn . pt is canonically a 0-cycle. Its Poincar´ e dual is a compactly supported n-form ω such that for any constant λ (i.e. closed 0-form) λω = Rn pt λ = λ, i.e., ω = 1. Rn Thus δpt can be represented by any compactly supported n-form with integral 1. In particular, we can choose representatives with arbitrarily small supports. Their “profiles” look like in Figure 7.2. “At limit” they approach Dirac’s delta distribution. Example 7.3.7. Consider an n-dimensional, compact, connected, oriented manifold M . We denote by [M ] the cycle (M, M ). Then δ[M ] = 1 ∈ H 0 (M ).  For any differential form ω , we set (for typographical reasons), |ω | := deg ω . + Example 7.3.8. Consider the manifolds M ∈ M+ m and N ∈ Mn . (The manifolds M and N need not be compact.) To any pair of cycles (S, φ) ∈ Cp (M ), and (T, ψ ) ∈ Cq (N ) we can associate the cycle (S × T, φ × ψ ) ∈ Cp+q (M × N ). We denote by πM (respectively πN ) the natural projection M × N → M (respectively M × N → N ). We want to prove the equality δS ×T = (−1)(m−p)q δS × δT , (7.3.1) where ∗ ∗ ω × η := πM ω ∧ πN η, ∀(ω, η ) ∈ Ω• (M ) × Ω• (N ). Pick (ω, η ) ∈ Ω• (M ) × Ω• (N ) such that, deg ω + deg η = (m + n) − (p + q ). Then, using Exercise 7.1.50 (ω × η ) ∧ (δS × δT ) = (−1)(m−p)|η| (ω ∧ δS ) × (η ∧ δT ). M ×N M ×N 7.3. INTERSECTION THEORY 253 The above integral should be understood in the generalized sense of Kronecker pairing. The only time when this pairing does not vanish is when |ω | = p and |η | = q . In this case the last term equals (−1)q(m−p) S ω ∧ δS T η ∧ δT = (−1)q(m−p) S φ∗ ω T ψ∗η = (−1)q(m−p) S ×T (φ × ψ )∗ (ω ∧ η ). This establishes the equality (7.3.1). • Example 7.3.9. Consider a compact manifold M ∈ M+ n . Fix a basis (ωi ) of H (M ) such i that each ωi is homogeneous of degree |ωi | = di , and denote by ω the basis of H • (M ) dual to (ωi ) with respect to the Kronecker pairing, i.e., ω i , ωj κ = (−1)|ω i |·|ω j| ωj , ω i κ i = δj . In M × M there exists a remarkable cycle, the diagonal ∆ = ∆M : M → M × M, x → (x, x). We claim that the Poincar´ e dual of this cycle is δ∆ = δ M = i (−1)|ω | ω i × ωi . i (7.3.2) Indeed, for any homogeneous forms α, β ∈ Ω• (M ) such that |α| + |β | = n, we have (α × β ) ∧ δ M = i i (−1)|ω i| M ×M M ×M i| (α × β ) ∧ (ω i × ωi ) = i (−1)|ω | (−1)|β |·|ω (−1)|ω | (−1)|β |·|ω i i i| M ×M (α ∧ ω i ) × (β ∧ ωi ) β ∧ ωi . = α ∧ ωi M M The i-th summand is nontrivial only when |β | = |ω i |, and |α| = |ωi |. Using the equality |ω i | + |ω i |2 ≡ 0(mod 2) we deduce (α × β ) ∧ δ M = i α ∧ ωi M M M ×M β ∧ ωi = i α, ω i κ β, ωi κ . From the equalities α= i ωi ω i , α κ and β = j β, ωj , κω j 254 we conclude ∆∗ (α × β ) = M M CHAPTER 7. COHOMOLOGY α∧β = M ( i ωi ω i , α κ ) ∧ ( j β, ωj κ ω j ) = M i,j ωi, α κ β, ωj κ ωi ∧ ω j = i,j ωi, α κ β, ωj κ ωi , ω j κ = i (−1)|ωi |(n−|ωi |) α, ω i κ (−1)|ωi |(n−|ωi |) β, ωi κ = i α, ω i κ β, ωi κ . Equality (7.3.2) is proved. Proposition 7.3.10. Let M ∈ M+ n be a manifold, and suppose that (Si , φi ) ∈ Ck (M ) (i = 0, 1) are two k -cycles in M . n−k (a) If (S0 , φ0 ) ∼c (S1 , φ1 ), then δS0 = δS1 in Hcpt (M ). n−k (b) If (S0 , φ0 ) is trivial, then δS0 = 0 in Hcpt (M ). (c) δS0 S1 n−k = δS0 + δS1 in Hcpt (M ). n− k (d) δ−S0 = −δS0 in Hcpt (M ). Proof. (a) Consider a compact manifold Σ with boundary ∂ Σ = −S0 S1 and a smooth map Φ : Σ → M such that Φ |∂ Σ = φ0 φ1 . For any closed k -form ω ∈ Ωk (M ) we have 0= Σ Φ∗ (dω ) = Σ d Φ∗ ω Stokes = ω= ∂Σ S1 φ∗ 1ω − S0 φ∗ 0 ω. Part (b) is left to the reader. Part (c) is obvious. To prove (d) consider Σ = [0, 1] × S0 and Φ : [0, 1] × S0 → M, Φ(t, x) = φ0 (x), ∀(t, x) ∈ Σ. Note that ∂ Σ = (−S0 ) S0 so that δ−S0 + δS0 = δ−S0 S0 = δ∂ Σ = 0 . The above proposition shows that the correspondence Ck (M ) descends to a map n−k δ : Hk (M ) → Hcpt (M ). n−k (S, φ) → δS ∈ Hcpt (M ) This is usually called the homological Poincar´ e duality. We are not claiming that δ is an isomorphism. 7.3. INTERSECTION THEORY 255 2 1 C 1 C 2 -1 1 1 Figure 7.3: The intersection number of the two cycles on T 2 is 1 §7.3.2 Intersection theory Consider M ∈ M+ n , and a k -dimensional compact oriented submanifold S of M . We denote by ı : S → M inclusion map so that (S, ı) is a k -cycle. Definition 7.3.11. A smooth map φ : T → M from an (n − k )-dimensional, oriented manifold T is said to be transversal to S , and we write this S φ, if the following hold. (a) φ−1 (S ) is a finite subset of T; (b) for every x ∈ φ−1 (S ) we have φ∗ (Tx T ) + Tφ(x) S = Tφ(x) M (direct sum). If S Φ, then for each x ∈ φ−1 (S ) we define the local intersection number at x to be (or = orientation) ix (S, T ) = 1 , or (Tφ(x) S ) ∧ or (φ∗ Tx T ) = or (Tφ(x) M ) . −1 , or (Tφ(x) S ) ∧ or (φ∗ Tx T ) = −or (Tφ(x) M ) Finally, we define the intersection number of S with T to be S · T := x∈φ−1 (S ) ix (S, T ). Our next result offers a different description of the intersection number indicating how one can drop the transversality assumption from the original definition. Proposition 7.3.12. Let M ∈ M+ n . Consider a compact, oriented, k -dimensional submanifold S → M , and (T, φ) ∈ Cn−k (M ) a (n − k )-dimensional cycle intersecting S transversally, i.e., S φ. Then S·T = M δS ∧ δT , (7.3.3) 256 where δ• denotes the Poincar´ e dual of •. CHAPTER 7. COHOMOLOGY The proof of the proposition relies on a couple of technical lemmata of independent interest. Lemma 7.3.13 (Localization lemma). Let M ∈ M+ n and (S, φ) ∈ Ck (M ). Then, for any N ∈ Ωn−k (M ) such that open neighborhood N of φ(S ) in M there exists δS cpt n−k N represents the Poincar´ (a) δS e dual δS ∈ Hcpt (M ); N ⊂ N. (b) supp δS Proof. Fix a Riemann metric on M . Each point p ∈ φ(S ) has a geodesically convex open neighborhood entirely contained in N. Cover φ(S ) by finitely many such neighborhoods, + and denote by N their union. Then N ∈ M+ n , and (S, φ) ∈ Ck (N ). N Denote by δS the Poincar´ e dual of S in N . It can be represented by a closed form in n−k N . If we pick a closed form ω ∈ Ωk (M ), then Ωcpt (N ) which we continue to denote by δS ω |N is also closed, and M N δS ∧ω = N N δS ∧ω = φ∗ ω. S n−k N represents the Poincar´ N ⊂ N. Hence, δS e dual of S in Hcpt (M ), and moreover, supp δS Definition 7.3.14. Let M ∈ M+ n , and S → M a compact, k -dimensional, oriented submanifold of M. A local transversal at p ∈ S is an embedding φ : B ⊂ Rn−k → M, B = open ball centered at 0 ∈ Rm−k , such that S φ and φ−1 (S ) = {0}. Lemma 7.3.15. Let M ∈ M+ n , S → M a compact, k -dimensional, oriented submanifold of M and (B, φ) a local transversal at p ∈ S . Then for any sufficiently “thin” closed neighborhood N of S ⊂ M we have S · (B, φ) = B N φ∗ δS . Proof. Using the transversality S φ, the implicit function theorem, and eventually restricting φ to a smaller ball, we deduce that, for some sufficiently “thin” neighborhood N of S , there exist local coordinates (x1 , . . . , xn ) defined on some neighborhood U of p ∈ M diffeomorphic with the cube {|xi | < 1, ∀i} such that the following hold. (i) S ∩ U = {xk+1 = · · · = xn = 0}, p = (0, . . . , 0). (ii) The orientation of S ∩ U is defined by dx1 ∧ · · · ∧ dxk . −k (iii) The map φ : B ⊂ Rn → M is expressed in these coordinates as (y 1 ,...,y n−k ) x1 = 0, . . . , xk = 0, xk+1 = y 1 , . . . , xn = y n−k . 7.3. INTERSECTION THEORY 257 (iv) N ∩ U = {|xj | ≤ 1/2 ; j = 1, . . . , n}. Let = ±1 such that dx1 ∧ . . . ∧ dxn defines the orientation of T M . In other words, = S · (B, φ). For each ξ = (x1 , . . . , xk ) ∈ S ∩ U denote by Pξ the (n − k )-“plane” Pξ = {(ξ ; xk+1 , . . . , xn ) ; |xj | < 1 j > k }. We orient each Pξ using the (n − k )-form dxk+1 ∧ · · · ∧ dxn , and set v (ξ ) := Pξ N δS . Equivalently, v (ξ ) = B N φ∗ ξ δS , where φξ : B → M is defined by φξ (y 1 , . . . , y n−k ) = (ξ ; y 1 , . . . , y n−k ). To any function ϕ = ϕ(ξ ) ∈ C ∞ (S ∩ U ) such that supp ϕ ⊂ {|xi | ≤ 1/2 ; i ≤ k }, we associate the k -form ωϕ := ϕdx1 ∧ · · · ∧ dxk = ϕdξ ∈ Ωk cpt (S ∩ U ). Extend the functions x1 , . . . , xk ∈ C ∞ (U ∩ N) to smooth compactly supported functions ∞ x ˜ i ∈ C0 (M ) → [0, 1]. The form ωϕ is then the restriction to U ∩ S of the closed compactly supported form ω ˜ ϕ = ϕ(˜ x1 , . . . , x ˜k )dx ˜1 ∧ · · · ∧ dx ˜k . We have M N ω ˜ ϕ ∧ δS = U N ωϕ ∧ δS = S ωϕ = ϕ(ξ )dξ. Rk (7.3.4) The integral over U can be evaluated using the Fubini theorem. Write N δS = f dxk+1 ∧ · · · ∧ dxn + , where is an (n − k )-form not containing the monomial dxk+1 ∧ · · · ∧ dxn . Then N ωϕ ∧ δS = f ϕdx1 ∧ · · · ∧ dxn U U = U f ϕ|dx1 ∧ · · · ∧ dxn | (|dx1 ∧ · · · ∧ dxn | = Lebesgue density) 258 F ubini CHAPTER 7. COHOMOLOGY f |dxk+1 ∧ · · · ∧ dxn | |dξ | Pξ N δS = ϕ(ξ ) S ∩U = S ϕ(ξ ) Pξ |dξ | = S ϕ(ξ )v (ξ )|dξ |. Comparing with (7.3.4), and taking into account that ϕ was chosen arbitrarily, we deduce that B N φ∗ δS = v (0) = . The local transversal lemma is proved. Proof of Proposition 7.3.12 Let φ−1 (S ) = {p1 , . . . , pm }. The transversality assumption implies that each pi has an open neighborhood Bi diffeomorphic to an open ball such that φi = φ |Bi is a local transversal at yi = φ(xi ). Moreover, we can choose the neighborhoods Bi to be mutually disjoint. Then S·T = i S · (Bi , φi ). The compact set K := φ(T \ ∪Di ) does not intersect S so that we can find a “thin”, closed N is compactly supported in neighborhood” N of S → M such that K ∩ N = ∅. Then, φ∗ δS the union of the Di ’s and N δS ∧ δT = N φ∗ δS = i N φ∗ i δS . M T Di From the local transversal lemma we get N φ∗ δS = (−1)k(n−k) S · (Bi , φ) = ipi (S, T ). Di Equality (7.3.3) has a remarkable feature. Its right-hand-side is an integer which is defined only for cycles S , T such that S is embedded and S T . The left-hand-side makes sense for any cycles of complementary dimensions, but a priori it may not be an integer. In any event, we have a remarkable consequence. Corollary 7.3.16. Let (Si , φi ) ∈ Ck (M ) and ((Ti , ψi ) ∈ Cn−k (M ) where M ∈ M+ n, i = 0, 1. If (a) S0 ∼c S1 , T0 ∼c T1 , (b) the cycles Si are embedded, and (c) Si Ti , 7.3. INTERSECTION THEORY then S0 · T0 = S1 · T1 . Definition 7.3.17. The homological intersection pairing is the Z-bilinear map I = IM : Hk (M ) × Hn−k (M ) → R, (M ∈ M+ n ) defined by I(S, T ) = M 259 δS ∧ δT . We have proved that in some special instances I(S, T ) ∈ Z. We want to prove that, when M is compact, this is always the case. Theorem 7.3.18. Let M ∈ M+ n be compact manifold. Then, for any (S, T ) ∈ Hk (M ) × Hn−k (M ), the intersection number I(S, T ) is an integer. The theorem will follow from two lemmata. The first one will show that it suffices to consider only the situation when one of the two cycles is embedded. The second one will show that, if one of the cycles is embedded, then the second cycle can be deformed so that it intersects the former transversally. (This is called a general position result.) Lemma 7.3.19 (Reduction-to-diagonal trick). Let M , S and T as in Theorem 7.3.18. Then I(S, T ) = (−1)n−k δS × T ∧ δ∆ , M ×M where ∆ is the diagonal cycle ∆ : M → M × M , x → (x, x). (It is here where the compactness of M is essential, since otherwise ∆ would not be a cycle). Proof. We will use the equality (7.3.1) δS ×T = (−1)n−k δS × δT . Then (−1)n−k = M ×M δS ×T ∧ δ∆ = M ×M (δS × δT ) ∧ δ∆ = M ∆∗ (δS × δT ) = M δS ∧ δT . Lemma 7.3.20 (Moving lemma). Let S ∈ Ck (M ), and T ∈ Cn−k (M ) be two cycles in ˜ ˜. T M ∈ M+ n . If S is embedded, then T is cobordant to a cycle T such that S The proof of this result relies on Sard’s theorem. For details we refer to [45], Chapter 3. Proof of Theorem 7.3.18 Let (S, T ) ∈ Ck (M ) × Cn−k (M ). Then, I(S, T ) = (−1)n−k I(S × T, ∆). Since ∆ is embedded we may assume by the moving lemma that (S × T ) I(S × T, ∆) ∈ Z. ∆, so that 260 CHAPTER 7. COHOMOLOGY §7.3.3 The topological degree Consider two compact, connected, oriented smooth manifolds M , N having the same dimension n. Any smooth map F : M → N canonically defines an n-dimensional cycle in M × N ΓF : M → M × N x → (x, F (x)). ΓF is called the graph of F. Any point y ∈ N defines an n-dimensional cycle M × {y }. Since N is connected all these cycles are cobordant so that the integer ΓF · (M × {y }) is independent of y . Definition 7.3.21. The topological degree of the map F is defined by deg F := (M × {y }) · ΓF . Note that the intersections of ΓF with M × {y } correspond to the solutions of the equation F (x) = y . Thus the topological degree counts these solutions (with sign). Proposition 7.3.22. Let F : M → N as above. Then for any n-form ω ∈ Ωn (N ) F ∗ ω = deg F M N ω. Remark 7.3.23. The map F induces a morphism R∼ = H n (N ) → H n (M ) ∼ =R which can be identified with a real number. The above proposition guarantees that this number is an integer. Proof of the proposition Note that if ω ∈ Ωn (N ) is exact, then ω= N M F∗ F ∗ ω = 0. Thus, to prove the proposition it suffices to check it for any particular form which generates H n (N ). Our candidate will be the Poincar´ e dual δy of a point y ∈ N . We have δy = 1, N while equality (7.3.1) gives δM ×{y} = δM × δy = 1 × δy . We can then compute the degree of F using Theorem 7.3.18 deg F = deg F N δy = M ×N (1 × δy ) ∧ δΓF = M Γ∗ F (1 × δy ) = M F ∗ δy . Corollary 7.3.24 (Gauss-Bonnet). Consider a connected sum of g -tori Σ = Σg embedded in R3 and let GΣ : Σ → S 2 be its Gauss map. Then deg GΣ = χ(Σ) = 2 − 2g. In particular. Exercise 7.3.25. Exercise 7. R) to a diagonal matrix.3. 1. M1 .e. i = 0.3.7. N ∈ M+ n and the smooth maps Fi : Mi → N . N are smooth. INTERSECTION THEORY 261 This corollary follows immediately from the considerations at the end of Subsection 4. Dx F preserves orientations −1 . Exercise 7. x).3. |Ax| Prove that deg FA = sign det A.3. Show that if F0 is cobordant to F1 then deg F0 = deg F1 . Consider the compact. connected manifolds M0 . Exercise 7. and define the Lefschetz number λ(F ) of F as the supertrace of the pull back F ∗ : H • (M ) → H • (M ). For an elementary approach to degree theory.. based only on Sard’s theorem.3. and deduce from this the Lefschetz fixed point theorem: λ(F ) = 0 ⇒ F has a fixed point. It defines a smooth map FA : S n−1 → S n−1 .4. [74]. Assume y ∈ N is a regular value of F . Remark 7. For x ∈ F −1 (y ) define deg(F.28. Prove that λ(F ) = ∆ · ΓF . we refer to the beautiful book of Milnor. and consider a smooth map F : M → M Regard H • (M ) as a superspace with the obvious Z2 -grading H • (M ) = H even (M ) ⊕ H odd (M ). Let M denote a compact oriented manifold. Let M → N be a smooth map (M . .2. homotopic maps have the same degree. i. Hint: Use the polar decomposition A = P · O (where P is a positive symmetric matrix and O is an orthogonal one) to deform A inside GL(n.27.26.29. otherwise deg(F. for all x ∈ F −1 (y ) the derivative Dx F : Tx M → Ty N is invertible. Let A be a nonsingular n × n real matrix. compact oriented of dimension n). x) = Prove that deg F = F (x)=y F 1 . x → Ax . The natural projection M × N → N allows us to regard M × N as a trivial fiber bundle over N . p∗ (η ) k If we denote by DM : → the Poincar´ e duality isomorphism on a compact oriented smooth manifold M . η = DE ω.3.45) with standard fiber F and compact. Exercise 7.9 can be equivalently defined by f∗ = π∗ ◦ (if )∗ .4. The integration along fibers E/B •−r = p∗ : Ω• (B ) cpt (E ) → Ω satisfies p∗ dE = (−1)r dB p∗ . is the pushforward morphism defined by the embedding if .4 Thom isomorphism theorem Let p : E → B be an orientable fiber bundle (in the sense of Definition 3. oriented basis B . Remark 7. Using Proposition 3.262 CHAPTER 7.9. Consider a smooth map f : M → N between compact. Show that the push-forward map f∗ : H • (M ) → H •+n−m (N ) defined in Remark 7. we have p∗ (ω ). COHOMOLOGY §7. so that it induces a map in cohomology • p∗ : Hcpt (E ) → H •−r (B ). If the standard fiber F is compact. Denote by if the embedding of M in M × N as the graph of f M x → (x. then we can rewrite the above equality as DB p∗ (ω ).2.48 we deduce that for any ω ∈ Ω• (E ). η =⇒ DB p∗ (ω ) = (p∗ )† DE ω.30. p∗ η = (p∗ )† DE ω.3. then the total space E is also compact. Ωdim M −• (M )∗ so that.4. oriented manifolds M . f (x)) ∈ M × N.31. and any η ∈ Ω• (B ) such that deg ω + deg η = dim E. Hence 1 ∗ † p∗ = D− B (p ) DE . the Gysin map coincides with the pushforward (Gysin) map defined in Remark 7. In this subsection we will extensively use the techniques of fibered calculus described in Subsection 3. This induced map in cohomology is sometimes called the Gysin map. in this case. .5. The total space E is a compact orientable manifold which we equip with the fiber-first orientation. η Ω• (M ) κ = ω. N of dimensions m and respectively n. where π∗ : H • (M × N ) → H •−m (N ) denotes the integration along fibers while (if )∗ : H • (M ) → H •+n (M × N ). Let dim B = m and dim F = r.3.2.4. Hence p∗ δS = 0. Denote by δσ its Poincar´ e dual in r (E ). Then for any ω ∈ Ω (B ) we have (p∗ δS ) ∧ ω = δ S ∧ p∗ ω = ± φ∗ p∗ ω = D S (p ◦ φ)∗ ω = 0. INTERSECTION THEORY 263 Let us return to the fiber bundle p : E → B . B E since p ◦ φ is constant. and we conclude that that if F carries nontrivial cycles ker p∗ may not be trivial. B E E = (−1)rm B σ ∗ p∗ ω = (−1)rm B (pσ )∗ ω = (−1)rm Hence p∗ δσ = E/B δσ = (−1)rm ∈ Ω0 (B ). b → 0 ∈ Eb . The Thom class of E . Note r (E ). Denote by τσ the map •+r τσ : H • (B ) → Hcpt (E ) ω → (−1)rm δσ ∧ p∗ ω = p∗ ω ∧ δσ . Denote by δS its Poincar´ e m+r−k m − k dual in Hcpt (E ). Definition 7. then the Gysin map • p∗ : Hcpt (E ) → H •−r (B ) is surjective. Hcpt Using the properties of the integration along fibers we deduce that.3.7. Indeed ω = (−1)rm p∗ δσ ∧ ω = (−1)rm p∗ (δσ ∧ p∗ ω ) = p∗ (τσ ω ). if (S. for any ω ∈ Ωm (B ). Any smooth section σ : B → E defines an embedded cycle in E of dimension m = dim B . If it admits at least one section. Let p : E → B be an orientable vector bundle over the compact oriented manifold B (dim B = m. The map p∗ is not injective in general.3.3. Then τσ is a right inverse for p∗ . φ) is a k -cycle in F .32. we have E δ σ ∧ p∗ ω = B E/B δσ ω. Proposition 7. then it defines a cycle in any fiber π −1 (b). and consequently in E . denoted by τE is the Poincar´ e dual of the cycle defined by the zero section ζ0 : B → E . that τE ∈ Hcpt . On the other hand. Let p : E → B a bundle as above. The simplest example of standard fiber with only trivial cycles is a vector space. Proof.33. For example. by Poincar´ e duality we get δσ ∧ p∗ ω = (−1)rm p∗ ω ∧ δσ ω. rank (E ) = r). Then the map •+r τ : H • (B ) → Hcpt (E ) ω → τE ∧ p∗ ω is an isomorphism called the Thom isomorphism.264 CHAPTER 7. Proof.e. oriented. ζ ) in NS . β ) ∈ Ω• cpt (E ). . Since p p∗ τE = (−1) • (−1)rm β = (τE ∧ p∗ (p∗ β ) + exact form ⇒ (−1)rm β = τE ◦ p∗ (β ) in Hcpt (E ). The exponential map defined by the metric g defines a smooth map exp : NS → M. Fix a Riemann metric g on M .34 (Thom isomorphism).5) Exercise 7.3. Suppose M ∈ M+ n . Using the identification between O and N we obtain a natural submersive projection π : N → S corresponding to the natural projection NS → S . Remark 7. where ζ : S → NS denotes N is a compactly supported form representing the the zero section. H • (S ) N ω −→ exp∗ δS ∧ π ∗ ω ∈ Hcpt •+(n−k) •+(n−k) (M ). we will use Lemma 7. •+dim M Exercise 7. where i : S → M denotes the canonical inclusion. ∗ rm . This shows that exp∗ δS Thom class of the normal bundle NS . Show that τE = ζ∗ 1. N is the Poincar´ Then exp∗ δS e dual of the cycle (S.3.2. k -dimensional manifold. we deduce where m(τE . and denote by NS the normal bundle of the embedding S → M . For β ∈ Ω• cpt (E ) we have (p∗ p∗ τE ) ∧ β − τE ∧ (p∗ p∗ β ) = (−1)r d(m(τE .3. where ζ• : H • (M ) → Hcpt (E ) is the pushforward map defined by a section ζ : M → E . the orthogonal complement of T N in (T M )|S .3. i).5). Prove the equality (7. τ p∗ = (−1)rm .2. In more intuitive terms. Fix a closed form δS support contained in N.37. Let p : E → B as in the above definition.2 of Subsection 7. To prove the reverse equality. (M ). i.1. and S → M is a compact.3. Its inverse is the Gysin map • (−1)rm p∗ : Hcpt (E ) → H •−r (B ).36. to an open N ∈ Ωn−k (M ) with compact (tubular) neighborhood N of S in M . and representing the Poincar´ e dual of (S. π associates to each x ∈ N the unique point in S which closest to x.35. COHOMOLOGY Theorem 7. β )). which induces a diffeomorphism from an open neighborhood O of S in NS .. (7. One can prove that the Gysin map i∗ : H • (S ) → Hcpt is given by.3. We have already established that p∗ τ = (−1)rm . Moreover. Let σ0 .5 Gauss-Bonnet revisited We now examine a very special type of vector bundle: the tangent bundle of a compact. e(T M ) is called the Euler class of M .3. M n (T M ) is the Thom Proof. independent of the two sections. oriented. Then M e(M ) = σ0 · σ1 . Exercise 7. Denote by τE ∈ Hcpt E .3. n (E ) the Thom class of oriented. Then the integral of e(M ) over M is equal to the Euler characteristic of M .41. INTERSECTION THEORY 265 §7. σ1 : M → T M be two sections of T M . M Theorem 7. σ1 : M → T M determine cycles of complementary dimension.7. smooth manifold M . where ζ0 : M → E denotes the zero section. Definition 7. It is a number reflecting the topological structure of the manifold. n-dimensional. and denote by e(M ) ∈ H n (M ) its Euler class. Hence σ0 · σ1 = = TM TM δσ0 ∧ δσ1 = TM τM ∧ τM e(M ). and their Poincar´ e dual in Hcpt class τM . affine along the fibers of T M .3. The section σ0 . smooth manifold M . M τM ∧ δζ0 = M ∗ ζ0 τM = If dim M is odd then M e(M ) = σ0 · σ1 = −σ1 · σ0 = − e(M ). any such section σ : M → T M tautologically defines an n-dimensional cycle in T M . any two such cycles are homotopic: try a homotopy. The Euler class of E is defined by ∗ e(E ) := ζ0 τE ∈ H n (M ).39. n e(M ) = χ(M ) = M k=0 (−1)k bk (M ). Note first the following fact. Proposition 7. In particular.40.38. Let M be a compact oriented n-dimensional manifold. . and it is denoted by e(M ).3. Prove that M is orientable if and only if T M is orientable as a bundle. σ1 are cobordant. Let E → M be a real orientable vector bundle over the compact.3. Any two sections σ0 .3. and in fact. and thus the intersection number σ0 · σ1 is a well defined integer. Note that the sections of T M are precisely the vector fields on M . if dim M is odd then e(M ) = 0. taking into account that |ωj | + |ω j | · |ωj | ≡ n|ωj | (mod 2). ωj ωi . Then χ(M ) = ∆ · ∆. (−1)|ω | = χ(M ). ∀i. |ω i | + n|ωi | + |ωi |2 + |ωi | · |ω i | ≡ |ω i | mod 2. ωj According to (7.1.266 CHAPTER 7. Lemma 7. we have δM = i κ i = δj .41 The tangent bundle of M × M restricts to the diagonal ∆ as a rank 2n vector bundle.50 we deduce ∆·∆= M ×M  (−1)|ω | ω i × ωi i i  (−1)n|ωj | ωj × ω j  j δM ∧ δM =  ∧  M ×M = M ×M  i.42. i Similarly. If we choose a Riemann metric on M × M then we get an orthogonal splitting T (M × M ) |∆ = N∆ ⊕ T ∆. Denote by ∆ the diagonal cycle ∆ ∈ Hn (M × M ). COHOMOLOGY In the proof we will use an equivalent description of χ(M ). i Proof of theorem 7. . i.j i |+n·|ω j| (−1)|ωi |·|ωj | ω i .3. ω j and the congruences n≡0 to conclude that ∆·∆= ωi κ j = (−1)|ωi |·|ω | δi . ω i .2).3. j mod 2. Using Exercise 7.3. ω j κ . we also have δM = i (−1)n·|ωj | ωj × ω j .e. if we start first with the basis (ω i ). n (M × M ) the Poincar´ Proof.j (−1)|ω (−1)|ω i |+n·|ω j| (−1)|ωi |·|ωj | ω i ∧ ωj  × ωi × ω j κ = i. Consider a basis (ωj ) of • H (M ) consisting of homogeneous elements. j. The diagonal map M → M × M identifies M with ∆ so that T ∆ ∼ = T M . so. We now have the following remarkable result. Denote by δ M ∈ Hcpt e dual of ∆. then its dual basis is (−1)|ω i |·|ω i| ωi .. We denote by (ω i ) the dual basis. In the last expression we now use the duality equations ωi . (−1)|ω | ω i × ωi . 267 (7. Lemma 7.3. Then there exists an open neighborhood U of ∆ ⊂ N∆ ⊂ T (M × M ) such that exp |U : U → M × M is an embedding. 2π 4π . δ∆ ∈ Hcpt (N). and by δ N the Poincar´ Denote by δ∆ e dual of ∆ in U.44. Let U be a neighborhood of ∆ ⊂ N∆ as in the above lemma. Then U U δ∆ ∧ δ∆ = N N δ∆ ∧ δ∆ = U N M ×M δ M ∧ δ M = χ(M ). From this lemma we immediately deduce the equality of Thom classes τN∆ = τM . M If M is a connected sum of g tori then we can rephrase the Gauss-Bonnet theorem as follows.6) means that δ∆ = τN∆ = τM .43. For any Riemann metric h on a connected sum of g -tori Σg we have 1 1 ε(h) = sh dvh = e(Σg ) in H ∗ (Σg ).7.3. M U ∆ ∆ M Hence e(M ) = χ(M ). N∆ ∼ = TM.45. U is the Thom class of the bundle N The cohomology class δ∆ ∆ → ∆. which in view of U (7.3.6) At this point we want to invoke a technical result whose proof is left to the reader as an exercise in Riemann geometry. Corollary 7. Regard ∆ as a submanifold in N∆ via the embedding given by the zero section. U the Poincar´ U ∈ H n (U).3. We get U U δ∆ ∧ δ∆ = U δ∆ = ∗ ζ0 τN∆ = ∗ ζ0 τM = e(M ). Denote by exp the exponential map of a Riemann metric g on M × M . INTERSECTION THEORY Lemma 7.3.3. and set N := exp(U). δ∆ e dual of cpt ∆ N n ∆ in N. Proof Use the isomorphisms T (M × M ) |∆ ∼ = T ∆ ⊕ N∆ ∼ = T M ⊕ N∆ and T (M × M ) |∆ = T M ⊕ T M. . The Euler characteristic is then the intersection number χ(M ) = ΓX · M. then χ(M ) = j i(X. Prove that the local intersection number of ΓX with M at x0 is given by ix0 (ΓX .268 CHAPTER 7. For example if χ(M ) = 0 this means that any vector field on M must have a zero ! We have thus proved the following result. is an n-dimensional submanifold of T M . . Given a smooth vector field X on M . ∗ M is given by for some local coordinates (xi ) near x0 such that the orientation of Tx 0 dx1 ∧ · · · ∧ dxn . M ) = sign det ∂Xi ∂xj |x=x0 . From the above exercise we deduce the following celebrated result. . Let X be a vector field over the compact oriented manifold M . oriented. Exercise 7. even-dimensional Riemann manifold.47. and we will have more to say about it in the next two chapters.48 (Poincar´ e-Hopf). A point x ∈ M is said to be a non-degenerate zero of X if X (0) = 0 and det ∂Xi ∂xj |x=x0 = 0. ΓX = {(x. If X is a vector field along a compact. the Euler characteristic counts (with sign) the zeroes of the vector fields on M . xj ). We now have a new interpretation of the Euler characteristic of a compact oriented manifold M . Using the notations of that subsection we can write v ( S 2 n ) = 0. Corollary 7. If χ(M ) = 0 then the tangent bundle T M is nontrivial. and it is denoted by i(X. the result is known as Gauss-Bonnet-Chern. .1. x0 ). COHOMOLOGY The remarkable feature of the Gauss-Bonnet theorem is that.46. This is sometimes called the local index of X at x0 . We have thus solved “half” the vector field problem. its “graph” in T M . . The same is true for any compact.4. The equality χ(S 2n ) = 2 is particularly relevant in the vector field problem discussed in Subsection 2. where we regard M as a submanifold in T M via the embedding given by the zero section. x ∈ M }. oriented manifold M . we can explicitly describe a representative of the Euler class in terms of the Riemann curvature.3. X (x)) ∈ Tx M . xk . In other words. In this generality. once we choose a metric. with only non-degenerated zeros x1 .3. Corollary 7.3. To prove (b). For any point x of a homogeneous space M we define the isotropy group at x by Ix = {g ∈ G . the map Ψm : G g →g·m∈M is surjective. Hint: Deform X to a linear vector field. It is worth mentioning some fundamental results in the theory of Lie groups which will shed a new light on the considerations of this section. choose g ∈ G such that y = g · x.29). g · x = x}. |X (x)| Prove that i(X.3. (b) Consider Fr : Sr → S n−1 defined by Fr (x) = 1 X (x). for any m ∈ M . They are smooth submanifolds of the symmetry group. on a smooth manifold M G × M → M (g. Then (a) Ix is a closed subgroup of G.4. §7.4.4. Then note that Iy = g Ix g −1 . y ∈ M . Remark 7.2.2. A homogeneous space is a smooth manifold M acted transitively by a Lie group G called the symmetry group. (b) Ix ∼ = Iy . (a) prove that for all r > 0 sufficiently small X has no zeros on Sr = {|x| = r}. Proof.4 Symmetry and topology The symmetry properties of a manifold have a great impact on its global (topological) structure. Any closed subgroup of a Lie group is also a Lie group (see Remark 1. In particular. . SYMMETRY AND TOPOLOGY 269 Exercise 7. We devote this section to a more in depth investigation of the correlation symmetry-topology.4.3. is called transitive if.7. 95].1. Let M be a homogeneous space with symmetry group G and x. Lemma 7. 0) = deg Fr for all r > 0 sufficiently small.49. Fact 1. m) → g · m.4. Let X be a vector field on Rn and having a non-degenerate zero at the origin. the isotropy groups Ix of a homogeneous space are all Lie groups. For proofs we refer to [44. (a) is immediate. 7. Recall that a smooth left action of a Lie group G.1 Symmetric spaces Definition 7. (d1) ∀m ∈ M . G. a Riemann manifold (M. g ) → im g is a smooth map such that the following hold. and certainly it is not the minimal one. h. g2 H ) → (g1 g2 ) · H is smooth. (d2) im ◦ im = 1G . A symmetric space is a collection of data (M. g ) ∈ M × G. Definition 7. (c) σ : M × M → M is a smooth map (m1 . i. m) → g · m ∈ M. All the isotropy groups subgroups of G conjugate to H .4. G/H := g · H. i) satisfying the following conditions. Then the space of left cosets. (b) G is a connected Lie group acting isometrically and transitively on M G×M (g. (c4) σgm = gσm g −1 . σ. We refer to [44. and H a closed subgroup.. We will be mainly interested in a very special class of homogeneous spaces. (m. g ∈ G . Our definition has one academic advantage: it lists all the properties we need to establish the topological results of this section. g ) is a symmetric space if and only if there exists a smooth map σ : M × M → M satisfying the conditions (c1). (d) i : M × G → G. ∀m. . can be given a smooth structure such that the map G × (G/H ) → G/H (g1 . (a) (M. (c3) Dσm |Tm M = −Tm M . then M is equivariantly diffeomorphic to G/Ix . im : G → G is a homomorphism of G. such that φ(g · y ) = g · φ(y ).4.270 Cohomology Fact 2. (d3) igm = g im g −1 .e. m2 ) → σm1 (m2 ) such that the following hold. 50] for an extensive presentation of this subject. −1 (x) = σ gσ (x) = i (g ) · x. Fact 3. x ∈ M . g ∈ G. The manifold G/H becomes a homogeneous space with symmetry group G. If M is a homogeneous space with symmetry group G. and σm (m) = m. (e) σm gσm m m m Remark 7. Let G be a Lie group. and x ∈ M . there exists a diffeomorphism φ : M → G/Ix .(c2) and (c3). ∀(m. including a proof of the equivalence of the two descriptions. (c2) σm ◦ σm = M . (c1) ∀m ∈ M σm : M → M is an isometry. As a matter of fact. h) is a Riemann manifold.4.5. This may not be the most elegant definition of a symmetric space. it describes the geometric significance of the family of involutions σm .4. For each m ∈ S n+1 we denote by σm the orthogonal reflection through −1 . Denote by ∇ the Levi-Civita connection. Define σ : G × G → G σg h = gh−1 g −1 and i : G × (G × G) → G × G ig (g1 .7. via the map which associates to each complex subspace S . g2 ) = (gg1 g −1 . 1 Re tr (AB ∗ ) that restricts The linear space End+ (Cn ) has a natural metric g0 (A.4. Exercise 7. Thus U (n) acts transitively. U (n) × End+ (Cn ) (T. .4. and g0 is U (n)-invariant. Consider the complex Grassmannian M = Grk (Cn ). Example 7. and + 2 = 2. Note that U (n) M = M . Example 7. (b) Fix m ∈ M . and let γ (t) be a geodesic of M such that γ (0) = m. the 1-dimensional space determined by the radius Om. The operator RS is the orthogonal reflection through S ⊥ .8. Show that σm γ (t) = γ (−t).20 we described Grk (Cn ) as a submanifold of End+ (V ) – the linear space of selfadjoint n × n complex matrices. m). B ) = 2 to a Riemann metric g on M . We let the reader check that σ and i satisfy all the required axioms. This action is clearly transitive and since m is bi-invariant its is also isometric.Symmetry and topology 271 The next exercise offers the reader a feeling of what symmetric spaces are all about. The symmetry group is G × G. (a) Prove that ∇R = 0. and isometrically on M . PS2 ) → RS1 PS2 . We leave the reader to check that these data do indeed define a symmetric space structure on (G. Note that RS ∈ U (n). h) be a symmetric space. Recall that in Example 1. the group of orientation preserving “rotations” of Rn+1 . A) −→ T A := T AT ∗ . gg2 g −1 ). Let G be a connected Lie group and m a bi-invariant Riemann metric on G. Example 7. The map A → R n RS S A is an involution of End (C ).6. the orthogononal projection PS : Cn → Cn onto S . (PS1 . The symmetry group is SO(n + 1). The unitary group U (n) acts on End+ (Cn ) by conjugation. For each subspace S ∈ M define RS := PS − PS ⊥ = 2PS − 1. and by R the Riemann curvature tensor. It descends to an involution of M . We thus get an entire family of involutions σ : M × M → M.4.9. Perhaps the most popular example of symmetric space is the round sphere S n ⊂ Rn+1 .2. We then set im (T ) = σm T σm ∀T ∈ SO(n + 1). g2 ) · h = g1 hg2 . In particular. Let (M. The direct product G × G acts on G by −1 (g1 . . where we denoted by ∗ • • g : Ω (M ) → Ω (M ) the pullback defined by the left action by g : m → g · m. Proposition 7. Bj := gj Br . . . The form ω is an invariant form on M . i. A differential form ω ∈ Ω∗ (M ) is said to be (left) invariant if ∗ g ω = ω ∀g ∈ G. Lemma 7. Choose r > 0 sufficiently small so that the map exp : Dr = |X |m = r . §7. Consider a bi-invariant Riemann metric m on G. Denote by dVG (g ) the normalized bi-invariant volume form on G. . the exponential map exp : LG → G X → exp(tX ) is surjective. .272 Define i : M × U (n) → U (n). The proposition is a consequence of the following result. We can select finitely many g1 .e. gm ∈ G such that m G= j =1 Bj .11. Then any cohomology class of M can be represented by a (not necessarily unique) invariant form.12. Let M be a compact homogeneous space with compact. Proof. For any form ω ∈ Ω∗ (M ) we define its G-average by ω= G ∗ g ω dVG (g ).4.4. Cohomology We leave the reader to check that the above collection of data defines a symmetric space structure on Grk (Cn ). iS T = RS T RS . j αj = 1. Let M be a homogeneous space with symmetry group G. Since G is connected. The form ω is obviously closed so we only need to prove it is cohomologous to ω .2 Symmetry and cohomology Definition 7. . Exercise 7. Fill in the details left out in the above example. If ω is a closed form on M then ω is closed and cohomologous to ω .13.10.4. supp αj ⊂ Bj . X ∈ LG →G is an embedding. Now pick a partition of unity (αj ) ⊂ C ∞ (G) subordinated to the cover (Bj ).4. Proof. 0 ≤ αj ≤ 1. connected symmetry group G.4. Set Br = exp Dr . for all g ∈ G. Tj = aj H ∗ (M ) . Define f : I × M → M ft (m) := exp(tX ) · m = exp(tX ) m. ∗ g =  on H ∗ (M ).t (t) ω dVG (g ) −1 gj exp−1 t·− exp gj G = Tj φ∗ j. The proof of the lemma will be completed in several steps. To verify this claim set t := es . consider φj.t ω.4.0 is cochain homotopic to Tj. .t : Bj → G defined as the composition Bj −→ Br −→ Dr −→ Dtr −→ Btr → G. m define Tj : Ω∗ (M ) → Ω∗ (M ) by Tj ω := Note that ω= j aj = 1. Define Tj. For G αj (g ) ∗ g ω dVG (g ). m) → Ψs (m) = gs (m) defines a local flow on M . . It induces a morphism in cohomology which we continue to denote by Tj . Then Us ω = Tj. and j αj = 1. . 1]. αj (gj g ) = Br α(gj g ) For each g ∈ Br the map (s. (7.es ω = def Br ∗ gj gs ω ∗ ∗ gs gj ω dVG (g ). . −∞ < s ≤ 0 and gs = exp(es exp−1 (g )) ∀g ∈ Br . j Since the volume of G is normalized to 1. For t ∈ [0. Step 1. ∗. we deduce that any j = 1. Then d (Us ω ) = ds = Br Br αj (gj g )LXg ∗ ∗ gs gj ω dVG (g ) αj (gj g )(dM iXg + iXg dM )( ∗ ∗ gs gj ω ) dVG (g ).t : Ω∗ (M ) → Ω∗ (M ) by Tj. g This concludes Step 1. Tj ω and dTj ω = Tj dω.1 . We denote by Xg its infinitesimal generator. Let X ∈ LG such that g = exp X . The map f is a homotopy connecting M with Step 2.t ω = αj (g ) ∗ φj.1) We claim that Tj.Symmetry and topology Set aj := G 273 αj dVG (g ). . Each Tj is thus a cochain morphism. 0 = aj · .14. Indeed. . we deduce Tj. h) be an. dω ˆ = (−1)k dω on M . ∗ dω = dω . ∗ ω . Fix m0 ∈ M and set ω ˆ = σm ˆ 0 is invariant.0 ω − Tj.274 Consequently. we have ω ˆ = (−1)k ω . Let (M.4. at m0 ∈ M . (a) Every invariant form on M is closed. then the only invariant form cohomologous to zero is the trivial one. Tj. oriented symmetric space with symmetry group G. Taking into account Step 1.1 ω = U−∞ ω − U0 ω 0 ∗ ∗ gs gj ω ) dVG (g ) Cohomology =− −∞ Br αj (gj g )(diXg + iXg d)( ds. In particular. 0 Since Dσm0 |Tm0 M = −Tm0 M . Step 2 is completed. ∀g ∈ G ∗ ˆ gω = ∗ ∗ g σm0 ω ∗ = (σm0 g )∗ ω = (gig σm0 g )∗ ω = σm 0 ∗ i(g ) ω ∗ = σm ω=ω ˆ. Invoking the transitivity of the G-action we conclude that ω ˆ = (−1)k ω on M. (b) If moreover M is compact. arguing as above.0 ω = Bj αj (g )dVG (g ) = aj ∗ gj ω. (An argument entirely similar to the one we used in the proof of the Poincar´ e lemma shows that the above improper integral is pointwise convergent). Then the following are true. We claim ω Proof. Both ω and ω ˆ are G-invariant. (a) Consider an invariant k -form ω . and dω are both invariant and. while Tj.1 ω = Tj ω . The last two inequalities imply dω = 0. The lemma and hence the proposition follow from the equality H ∗ (M ) = j aj H ∗ (M ) = j Tj = G − average. Proposition 7. From the above formula we immediately read a cochain homotopy χ : Ω• (M ) → Ω•−1 (M ) connecting U−∞ to U0 . More precisely 0 {χ(ω )} |x∈M := − Now notice that −∞ Br αj (gj g ) iXg ∗ ∗ gs gj ω |x dVG (g ) ds. The proposition we have just proved has a greater impact when M is a symmetric space. The (k + 1)-forms dω ˆ = σm 0 we deduce dω ˆ = dω = (−1)k+1 dω. and we deduce that the above equality holds at any point m = g · m0 . Tj. we deduce that. . and satisfies the following conditions. X2 ].4. X1 ]. • Corollary 7.3 The cohomology of compact Lie groups Consider a compact. Denote by ∗ the Hodge ∗-operator corresponding to the invariant metric h. This forces ω ≡ 0.11) LG = [LG . From Proposition 7. i. while H (LG ) denotes the Lie algebra cohomology introduced in Example 7.16. X2 ∈ LG . We deduce the following result. dω = 0 ⇐⇒ ω ([X0 .1. ω = dα. oriented symmetric space with Corollary 7. since H 1 (LG ) = 0 we deduce (see Exercise 7. If G is a compact semisimple Lie group then H 1 (G) = 0. X1 . On the other hand. so that dη = 0.10. H • (G) ∼ = H • (LG ) ∼ = Λ• inv LG . Then the cohomology algebra H ∗ (M ) of M is isomorphic with the graded algebra Ω∗ inv (M ) of invariant forms on M .4.1. Proof. connected symmetry group G. §7.12. X1 ]. Let (M. h) be a compact.4. A closed bi-invariant 2-form ω on G is uniquely defined by its restriction to LG . This can be identified with the exterior algebra Λ• L∗ G. it suffices to restrict our considerations to the subcomplex consisting of left invariant forms. and denote by LG its Lie algebra. Thus ω ([X0 . in computing its cohomology.15 (E. X1 ) = 0. X1 ) + ω ([X1 . LG ]. Corollary 7. Proposition 7. X0 ) = 0. Y ∈ LG .4. In the coming subsections we will apply this result to the symmetric spaces discussed in the previous subsection: the Lie groups and the complex Grassmannians. X2 ]. X2 ) = 0 ∀X0 ∈ LG ⇐⇒ ω ([X0 .4.4.11 we deduce the following consequence.1. X1 ]. We can now integrate (M is compact). Since G acts by isometries. X2 ) − ω ([X0 . Using the Exercise 7. so that the last equality can be rephrased as ω (X. connected Lie group G. Let G be a compact semisimple Lie group. X2 ]. and (right-invariance) (LX0 ω )(X1 .e.12 and the above theorem we deduce the following celebrated ´ Cartan ([20]) result of Elie ´ Cartan).4. Then H 2 (G) = 0. X2 ) = 0 ∀X0 . X2 ) − ω ([X0 . Y ) = 0 ∀X. According to Proposition 7.Symmetry and topology 275 (b) Let ω be an invariant form cohomologous to zero. where Λinv LG denotes the algebra of • bi-invariant forms on G. the form η = ∗ω is also invariant.18. compact. and use Stokes theorem to get ω ∧ ∗ω = M M dα∧ = ± M α ∧ dη = 0.17. Let G be a compact. As we have seen in the previous subsection.1. Exercise 7.4. Pay very much attention to the various constants. Compute α and SU (2) SO(3) α. Y.13). Proposition 7. (c) Prove that the eigenspaces of A are ideals of LG .276 Cohomology Definition 7. The proof of the proposition is contained in the following sequence of exercises.25. . T ) ∀X. Moreover.4. Z ) = ηX ([Y.61 and the double cover SU (2) → SO(3) described in the Subsection 6. Exercise 7. Z.1. T ]) = ω ([X. Exercise 7.4. Then ω (X. Exercise 3.4. §7. It is a complex manifold so that it is orientable (cf. simple Lie group.20. (a) Prove that for any X ∈ LG there exists a unique left-invariant form ηX ∈ Ω1 (G) such that (iX ω )(Y.4. Y ]. bi-invariant 3-form on a Lie group G. Use this to prove Proposition 7. where α denotes the Cartan form. [Z. Z ).) Hint: Use the computation in the Exercise 4. Y. the correspondence X → ηX is linear.4. Hint: Use H 1 (G) = H 2 (G) = 0. Alternatively.21. Let ω be a closed.2. Z. Exercise 7. A Lie algebra is called simple if it has no nontrivial ideals. Y ]. Z ]). bi-invariant 3-form on a compact. where κ denotes the Killing pairing. A Lie group is called simple if its Lie algebra is simple.4. Set = n − k .2 to compute the Poincar´ e polynomial of the complex Grassmannian Grk (Cn ).4.4 Invariant forms on Grassmannians and Weyl’s integral formula We will use the results of Subsection 7.19. Prove that SU (n) and SO(m) are simple. Let ω be a closed.24. the orientability of Grk (Cn ) is a consequence of the following fact.4. Z ) = κ([X.4. (b) Denote by A the linear operator LG → LG defined by κ(AX. Y ) = ηX (Y ). Prove that a simple Lie algebra is necessarily semi-simple. (These groups are oriented by their Cartan forms. the Grassmannian Grk (Cn ) is a symmetric space with symmetry group U (n). Prove that A is selfadjoint with respect to the Killing metric. Then H 3 (G) ∼ = R.23.4. H 3 (G) is generated by the Cartan form α(X.22. Moreover. Exercise 7. semisimple Lie group. Y. T ∈ LG .21. (7.4. any H -invariant element of Λ• V0∗ extends via the transitive action of U (n) to an invariant form on Grk (Cn ). ϕ(ghg −1 ) = ϕ(h).4. At this point.26. where Tk = diag (eiθ1 .. We choose this point to correspond to the subspace S0 determined by the canonical inclusion Ck → Cn . . The group H acts linearly on the tangent space V0 = TS0 Grk (Cn ). We have thus established the following result.e.e. . multilinear map V0 × · · · × V0 → R.4 we deduce Pk. it can be dramatically simplified using a truly remarkable idea of H. . . i.Symmetry and topology 277 Exercise 7. (t) = | det(V0 + tTh )|2 dVH (h). . These forms are completely determined by their values at a particular point in the Grassmannian.2) H where dVH denotes the normalized bi-invariant volume form on H . . there exists g ∈ H such that ghg −1 ∈ T.27.4. Each h ∈ U (k ) × U ( ) is conjugate to diagonal unitary matrix. then its restriction to V0 is an H -invariant skew-symmetric. . Weyl. h) → C g (h) = ghg −1 . Proposition 7. h ∈ H . (t) = tj dimC Λj inv ⊗ C = PGrk (Cn ) (t). We can rephrase this fact in terms of the conjugation action of H on itself C : H × H → H (g.4. Using the equality (3. Λ• inv We want to determine the Poincar´ e polynomial of the complexified Z-graded space.10) of Subsection 3. ⊗C Pk. If M is a homogeneous space with connected isotropy groups. eiθk ) ∈ U (k ) . Conversely. Denote by Λ• inv the space of H -invariant elements of Λ• V0∗ . If ω is an U (n)-invariant form. We have to describe the U (n)-invariant forms on Grk (Cn ). The isotropy of S0 is the group H = U (k ) × U ( ). the above formula may look hopelessly complicated. . eiφ ) ∈ U ( )}.4. Note first that the function H h → ϕ(h) = | det(V0 + tTh )|2 is a class function. Inside H sits the maximal torus T = Tk × T . . There exists an isomorphism of graded R-algebras: H • (Grk (Cn )) ∼ = Λ• inv . j Denote the action of H on V0 by H h → Th ∈ Aut (V0 ). and T = {diag (eiφ1 . then M is orientable.. Fortunately. i. ∀g. . and by V the volume of G. Then 1 V ϕ(g )dg = G 1 s k ! vol( T) j =1 j 1 s T ϕ(θ 1 . k while dθ j denotes the bi-invariant volume on Tkj . θk ) = 1≤i<j ≤n (eiθ − eiθ ). Consider a class function ϕ on the group G = U (k1 ) × · · · × U (ks ). Define ∆k : Tk → C by ∆k (θ1 . In other words. and by Vk the total volume Vk := dvk . . = 1 s kj (2 j =1 π ) kj ! T ϕ(θ 1 . Y ) = Re i. .28 (Weyl’s integration formula). denote by dg the volume form dg = dvk1 ∧ · · · dvks . . On the unitary group U (k ) we fix the bi-invariant metric m = mk such that at T1 U (k ) we have m(X. s. i = i j √ −1. and each such orbit intersects the maximal torus T. a class function is completely determined by its restriction to the maximal torus. . . . θ s ) j =1 s |∆kj (θ j )|2 dθ 1 ∧ · · · ∧ dθ s . then m(X. Hence. . it is reasonable to expect that we ought to be able to describe the integral in (7. . This is achieved in a very explicit manner by the next result. we denoted by θ j the angular coordinates on Tkj . . . .278 Cohomology The class functions are constant along the orbits of this conjugation action. U (k) Proposition 7. 1 θ j := (θj .j xij y ¯ij . for every j = 1.4. . θ s ) j =1 |∆kj (θ j )|2 dθ 1 ∧ · · · ∧ dθ s Above. . . Y as k × k matrices.2) as an integral over T.4. k . . V = Vk1 · · · Vks . If we think of X. Y ) := Re tr(XY ∗ ). . . . . θj j ). 1 dθ j := dθj ∧ · · · ∧ dθj j . We denote by dvk the volume form induced by this metric. −1 −1 Note that if g1 T = g2 T. . . and go directly to Subsection 7. Moreover. In this concrete case. In this case. if ω ∈ det Tx G/T.28 To keep the main ideas as transparent as possible. and in particular it is connected. X ∈ u(k ). We denote by Ad : G → Aut(LG ) the adjoint representation of G. Thus. Ad can be given the explicit description Adg (X ) = gXg −1 .Symmetry and topology 279 The remainder of this subsection is devoted to the proof of this proposition. The orientation on LG/T will be determined by the condition or(LG ) = or (LT ) ∧ or (LG/T ). The reader may skip this part at the first lecture. Step 1. .4. ∀g ∈ U (k ). see Example 3. For this reason we will write LG/T instead of L⊥ . Hence. we can orient G/T by fixing an orientation on LG/T . an orientation in one of the tangent spaces of G/T “spreads” via the action of G to an orientation of the entire manifold. We have a m-orthogonal splitting of the Lie algebra LG LG = LT ⊕ L⊥ G. In other words.4. The tangent space to G/T at 1 · T ∈ G/T can be identified with L⊥ T . we will consider only the case s = 1. the volume form dg is the volume form associated to metric mk . .30. Consider the homogeneous space G/T. Thus. so the map q is well defined. then Lg ω ∈ det Ty G/T and Lh ω ∈ det Ty G/T define the same orientation of Ty G/T. and the smooth map q : T × G/T → G (t. This stabilizer is isomorphic to T. The general situation is entirely similar. then Lg and Lh differ by an element in the stabilizer of x ∈ G/T. θk ). Any g ∈ G defines a linear map Lg : Tx G/T → Tgx G/T. We fix an orientation on LG . . then g1 tg1 = g1 tg2 . The restriction of the metric m to T is described in these coordinates by m|T = (dθ1 )2 + · · · + (dθk )2 . The proof of Weyl’s integration formula will be carried out in three steps. and we orient LT using the form dθ = dθ1 ∧ · · · ∧ dθk . if gx = hx = y . T Fix x ∈ G/T. gT ) = gtg −1 . G = U (k ) and T = Tk . Proof of Proposition 7. G ωϕ = 1 k! T×G/T q ∗ ωϕ ∀ω.4. To any class function ϕ : G → C we associate the complex valued differential form ωϕ := ϕ(g )dg.5 where this formula is used to produce an explicit description of the Poincar´ e polynomial of a complex Grassmannian. Denote the angular coordinates θ on T by θ = (θ1 . Geometrically. exp(iα )) ∈ T . ∀n ∈ Z. . N (T) = and then form the Weyl group W := N (T)/T. Set τ := (exp(iα ).280 Cohomology Step 2.4.29. g Tg −1 ⊂ T . This action descends to an action on the quotient W so that W ⊂ Sk . we defer the proof of this lemma to the end of this subsection. g ∈ N (T) .. .30. .4. We prove that there exists a positive constant Ck such that for any class function ϕ on G we have k! G ωϕ = T×G/T q ∗ ωϕ = Ck T ϕ(θ )|∆k (θ )|2 dθ . Let τ ∈ Tk ⊂ G be a generator of Tk .) 1 k g ∈ G . α 2π . this corresponds to a reordering of an orthonormal basis. any permutation of the entries of a diagonal matrix can be achieved by a conjugation. Denote by N (T) the normalizer of T in G. gT ) ∈ T × G/T . Let α1 . . . 2π are linearly independent 1 k k over Q.e. α Lemma 7. . multiplicities included. . The adjoint action of N (T) on T= diagonal unitary matrices simply permutes the entries of a diagonal unitary matrix. For the sake of clarity. The Weyl group W is isomorphic to the group Sk of permutations of k symbols. Lemma 7. . . This is a pompous rephrasing of the classical statement in linear algebra that two unitary matrices are similar if and only if they have the same spectrum. q (s. We use the equality q ∗ ωϕ = deg q ωϕ . . In particular. gT ) = τ ⇐⇒ gsg −1 = τ ⇐⇒ gτ g −1 = s ∈ T. (The element τ is said to be a generator of Tk . T×G/T G so it suffices to compute the degree of q . Lemma 7.4. Step 3. Then the sequence (τ n )n∈Z is dense in Tk . We prove that Ck = Vk Vk = vol (T) (2π )k Step 1. gτ n g −1 = sn ∈ T.31. . Then q −1 (τ ) ⊂ T × G/T consists of |W| = k ! points. we deduce gT g −1 ⊂ T ⇒ g ∈ N (T). Proof. . Since (τ n ) is dense in T. i. Conversely. αk ) ∈ Rk such that 1. Proof. Hence q −1 (τ ) = (g −1 τ g. . Observe that the volume form on T induced by the metric m is precisely dθ = θ1 ∧ · · · ∧ dθk . in block form. we deduce 1 −1 −1 2 h− 0 hs = g0 t0 (1 + sY )t0 (1 + sX )(1 − sY )g0 + O (s ) 1 −1 −1 −1 = 1 + s g0 t− + O(s2 ). 0 Y t0 g0 + g0 Xg0 − g0 Y g0 Hence. Now observe that LG/T = L⊥ T is equal to X ∈ u(k ) . Xjj = 0 ∀j = 1. . 0 or. Proof. since G = U (k ) is connected. Dx0 q = Adg0 LT 0 Adt−1 0 0 − LG/T . and exp(sY ) = 1 + sY + O(s2 ). so that det Adg = ±1. In particular. On the other hand. We want to describe d 1 |s=0 h− 0 hs ∈ T1 G = LG . g0 T) ∈ T × G/T. ∀σ ∈ q −1 (τ ). . ds Using the Taylor expansions exp(sX ) = 1 + sX + O(s2 ). Fix X ∈ LT . the differential Dx0 : Tx0 (T × G/T) ∼ = LT ⊕ LG/T → LT ⊕ LG/T ∼ = LG . det Dx0 q = det(Adt−1 − LG/T ). 281 The metric m on LG/T extends to a G-invariant metric on G/T. can be rewritten as Dx0 q (X ⊕ Y ) = Adg0 (Adt−1 − )Y + Adg0 X.Symmetry and topology and thus q −1 (τ ) has the same cardinality as the Weyl group W. . q ∗ dg = |∆k (θ)|2 dθ ∧ dµ. Fix a point x0 = (t0 . Hence. . Y ∈ LG/T . g0 exp(sY )T = g0 exp(sY )t0 exp(sX ) exp(−sY )g0 ∈ G. det Adg = det Ad1 = 1. We can identify Tx0 (T × G/T) with LT ⊕ LG/T using the isometric action of T × G on T × G/T. k ⊂ u(k ) = LG .4. For every real number s consider −1 hs = q t0 exp(sX ) . Lemma 7.32. The linear operator Adg is an m-orthogonal endomorphism of LG . . any generator τ of Tk ⊂ G is a regular value of q since ∆k (σ ) = 0. It defines a leftinvariant volume form dµ on G/T. k } we denoted by (σ ) its signature. . . . Ck k! ∆k (θ )dθ . they are exp(−i(θi − θj )) . . to find the constant Ck .31 and Exercise 7. .3. . . . . .4. Using Lemma 7. More precisely. To complete Step 3. More precisely. . . where. θ ) = 1 k k iσ (j )θj j =1 e k iθ j j =1 e . 1 ≤ i = j ≤ k . . we can explicitly compute the eigenvalues of Adt−1 acting on LG/T .32. . Consequently det Dx0 q = det(Adt−1 − 1) = |∆k (t)|2 . that is. we apply the above equality in the special case ϕ = 1.27 we deduce deg q = |W| = k !. We deduce Vk = so that Ck = Thus. . . Lemma 7. for any permutation σ of {1. .4.32 shows that q is an orientation preserving map. . Step 2 follows immediately from Lemma 7. . Tk ∆k (θ )dθ Tk To compute the last integral we observe that ∆k (θ) can be expressed as a Vandermonde determinant 1 1 ··· 1 1 2 k i θ i θ i e e ··· eθ 1 2 k e2iθ e2iθ ··· e2iθ ∆k (θ) = . exp(iθk )) ∈ Tk ⊂ U (k ). we deduce ωϕ = 1 k! dµ G/T =:Ck Tk G ϕ(θ)∆k (θ)dθ.282 Cohomology Given t = diag (exp(iθ1 ). 2. and by eσ (θ ) the trigonometric monomial eσ (θ ) = eσ (θ . . . . . . so that ωϕ = dvk . we have to show that ∆k (θ )dθ = k !vol (T) = (2π )k k !.4. 1 2 i ( k − 1) θ i ( k − 1) θ i ( k − 1)θk e e ··· e This shows that we can write ∆k as a trigonometric polynomial ∆k (θ ) = σ ∈Sk (σ )eσ (θ ). Step 1 is completed. Tk Vk k ! . where S is a subset of X spanning a dense subspace. non-negative function supported in U (f ≡ 0) then. . We compute easily 1 Ln (eζ ) = n+1 n ejζ (α) = j =0 1 eζ (α)n+1 − 1 . We let S be the subset consisting of the trigonometric monomials eζ (θ1 .4. . . θk ) = exp(iζ1 θ1 ) · · · exp(iζk θk ). lim Ln (f ) = L(f ) f ∈ X. Hence n→∞ lim Ln (eζ ) = 0 = T eζ dt = L(eζ ). i. and L(f ) = j =0 T f dt. . (7. L : X → C Ln (f ) = We have to prove that n→∞ 1 n+1 n f (τ j ).e. Lemma 7. Let X = C (T.30 We follow Weyl’s original approach ([96.30 is proved. We will prove that 1 n→∞ n + 1 lim n f (τ j ) = j =0 T f dt. . . 97]) in a modern presentation. Proof of Lemma 7. . 2π αk are linearly independent over Q we deduce that eζ (α) = 1 for all k ζ ∈ Z .3) If U ⊂ T is an open subset and f is a continuous. .4. . .Symmetry and topology 283 The monomials eσ are orthogonal with respect to the L2 -metric on Tk .4.4. ∀f ∈ X. τ j ∈ U for some j . . for very large n.4) for any f ∈ S. 1 n+1 n f (τ j ) ≈ j =0 T f dt = 0.. (7. The Weierstrass approximation theorem guarantees that this S spans a dense subspace. This completes the proof of Weyl’s integration formula. n + 1 eζ (α) − 1 1 Since 1. . This means that f (τ j ) = 0. and we deduce |∆k (θ )|2 dθ = σ ∈Sk k T Tk |eσ (θ )|2 dθ = (2π )k k !. ζk ) ∈ Zk . ζ = (ζ1 . 21 π α1 .4) It suffices to establish (7.3) consider the continuous linear functionals Ln . .4.4. C) denote the Banach space of continuous complex valued functions on T. To prove the equality (7. x. (t) = |1 + tεα ej |2 |∆k (ε(τ ))|2 |∆ (e(θ ))|2 dτ ∧ dθ k ! ! T k ×T α. e) ∈ T k × T viewed as a linear operator on E has eigenvalues {εα ej . T ∈ U (k ).j (εα + tej )∆k (ε)∆ (e). The normalized measure on T k is then dτ = dτ 1 ∧ · · · ∧ dτ k . (7. Set Ik.4. . . so that Pk. e ). εα = e2πiτ .284 Cohomology §7. The isotropy group at S0 is H = U (k ) × U ( ). Consider the maximal torus T k × T ⊂ H formed by the diagonal unitary matrices.j α = 1 k! ! T k ×T |εα + tej |2 |∆k (ε(τ ))|2 |∆ (e(θ ))|2 dτ ∧ dθ . We will denote the elements of T k by ε := (ε1 . .j (xα + tyj ). ej = exp(2π iθj ). (t. 1 ≤ α ≤ k 1 ≤ j ≤ }. Using the Weyl integration formula we deduce that the Poincar´ e polynomial of Grk (Cn ) is 1 Pk. .j We definitely need to analyze the integrand in the above formula. S ∈ U ( ). = n − k .33. εk ). (t) := α. . The tangent space of Grk (Cn ) at S0 can be identified with the linear space E of complex linear maps Ck → C . α. Exercise 7. Prove that the isotropy group H acts on E = {L : Ck → C } by (T. . S ) · L = SLT ∗ ∀L ∈ E. . . and the elements of T by e := (e1 . .4.5) We will study in great detail the formal expression Jk. k n Let S0 denote the canonical subspace C → C . The element (ε. (t)dτ ∧ dθ .4. and the normalized measure on T is dθ = dθ1 ∧ · · · ∧ dθ . y ) = α. (t) = 1 k! ! T k ×T Ik.5 The Poincar´ e polynomial of a complex Grassmannian After this rather long detour we can continue our search for the Poincar´ e polynomial of Grk (Cn ). (t)Ik. namely the Schur polynomials. Traditionally. 2. .1) is (5.3. 0. . ˆ defined by Any partition λ has a conjugate λ ˆ n := #{j ≥ 0 . n − 2. x. Denote by δ = δn ∈ P∗ length n − 1 ≤ L(λ) ≤ n. . λj ≥ n}. This will require a short trip in the beautiful subject of symmetric polynomials. λn = 0}. λ2 . . 1.1) The Weyl group W = Sk × S acts on the variables (x. Denote by Pn the set of partitions of length ≤ n.Symmetry and topology 6 5 5 3 1 5 4 4 3 3 1 285 Figure 7. . Given a partition (λ1 ≥ · · · ≥ λn ). . σ (x).5. ϕ(y )) = Jk. y ) by separately permuting the x-components and the y -components. An extensive presentation of this topic is contained in the monograph [66]. . λn ). A partition is a compactly supported. its Young diagram will have λ1 boxes on the first row.4). where λ1 ≥ · · · ≥ λn ≥ λn+1 = 0. 2.). To understand the nature of these polynomials we need to introduce a very useful class of symmetric polynomials. ϕ) ∈ W then Jk. as a sum Jk. .3. We will describe a partition by an ordered finite collection (λ1 . (t.4).4. y ). . . . Thus we can write Jk. . . . and by P∗ n the set of strict partitions λ of ⊂ P . one can visualize a partition using Young diagrams. If (σ.4. Clearly P∗ n n the partition n (n − 1. The Young diagram of λ A strict partition is a partition which is strictly decreasing on its support. . The weight of a partition λ is the number |λ| := n≥1 λn .}. . decreasing function λ : {1. .4: The conjugate of (6.5.3. The length of a partition λ is the number L(λ) := max{n . λ ˆ is the transpose of the Young diagram of λ (see Figure 7. A Young diagram is an array of boxes arranged in left justified rows (see Figure 7. 1. . (t) = d≥0 td Qd (x)Rd (y ) where Qd (x) and Rd (y ) are symmetric polynomials in x and respectively y . (t. λ2 boxes on the second row etc.} → {0. 3. so that the polynomial aλ+δ (x) is divisible by each of the differences (xi − xj ) and consequently. . 1. we denoted by λ the complementary partition λ = (k − λ . Note that each Schur polynomial Sλ is homogeneous of degree |λ|.4. it is divisible by aδ . so that aλ+δ is well defined and nontrivial. . . 4. 3. The polynomial Sλ (x) is called the Schur polynomial corresponding to the partition λ. 6. 0) ∈ P7. It is a symmetric polynomial since each of the quantities aλ+δ and aδ is skew-symmetric in its arguments. xn ) := det(xλ j )= i (σ )xλ σ (i) . xn ) = det(xn )= i i<j (xi − xj ) = ∆n (x). λ1 ≤ k L(λ) ≤ . We have the following remarkable result. Note that aλ+δ vanishes when xi = xj . . . The correspondence Pn λ → λ + δn ∈ P ∗ n . Hence Sλ (x) := aλ+δ (x) aδ (x) is a well defined polynomial. (t) = λ∈Pk. 1) ∈ P6. . is a bijection. 5.4.5: The complementary of (6. . . t|λ| Sλ ˆ (x)Sλ (y ). Jk. . . k − λ1 ). σ ∈ Sn Note that aδn is the Vandermonde determinant −1−i aδ (x1 . To any λ ∈ P∗ n we can associate a skew-symmetric polynomial i aλ (x1 . . 4. .286 Cohomology 6 4 4 2 1 0 7 6 5 3 3 1 Figure 7.7 Remark 7. where Pk. := λ . k − λ −1 . For each λ ∈ Pn we have λ + δ ∈ P∗ n . For each λ ∈ Pk.34. .35.6 is (7. Lemma 7. 2. Section 6. the partitions in Pk. Example 5.4. Similarly.4. Tk and T |aλ+δ (e)|2 dθ = !. For a proof of Lemma 7.4. µ ∈ Pk.4.6) that Pk.5). . λ∈Pk. 1/2 In other words.35. Chapter VI. (7.4. the terms 1 k! ! aλ ˆ +δ (ε)aλ+δ (e) form an orthonormal system in the space of trigonometric (Fourier) polynomials endowed with the L2 inner product. then the terms aλ ˆ +δ (ε) and aµ ˆ+δ (ε) have no monomials in common. are precisely those partitions whose Young diagrams fit inside a × k rectangle. The true essence of the Schur polynomials is however representation theoretic.5). T aλ+δ (e)aµ+δ (e)dθ = 0. 1 (t) = k! ! T k ×T t|λ| aλ ˆ +δk (ε)aλ+δ (e) dτ ∧ dθ . (t) = λ∈Pk. Theorem V.Symmetry and topology 287 Geometrically. and the definition of the Schur polynomials.k . We deduce immediately from (7. a simple computation shows that 2 |aλ ˆ +δ (ε)| dτ = k !. Note that if λ. Section I. including the one in Lemma 7.6) rk 1 The integrand in (7.4. λ→λ∈P . are distinct partitions.35. If λ is such a partition then the Young diagram of the complementary of λ is (up to a 180◦ rotation) the complementary of the diagram of λ in the × k rectangle (see Figure 7.4. Hence Tk aλ ˆ +δ (ε)aµ ˆ+δ (ε) dτ = 0. Lemma 7. where the r -s and s-s are nonnegative integers.7) The map Pk. On the other hand. for a very exciting presentation of the Schur polynomials. and a reader with a little more representation theoretic background may want to consult the classical reference [65]. Using (7. (7. and the various identities they satisfy.6) is a linear combination of trigonometric monomials εr 1 · · · εk · s 1 es 1 · · · e . if λ = µ.35 we refer to [66].4. we can describe the Poincar´ e polynomial of Grk (Cn ) as 2 Pk.4.4. t2|λ| . Hence k Pk.4. bk. +1 (w ) =# +1 diagrams of weight w which fit inside Rk+1 = # diagrams which fit inside Rk+1 +# diagrams which do not fit inside Rk+1 .36. we get a diagram of weight w − − 1 which fits inside the . = # diagrams which fit inside Rk+1 .4. bk+1. +1 (w ) + bk+1.k with weight |λ| = w.6: The rectangles R is a bijection so that Pk. and rewrite the Poincar´ e polynomial as a “fake” rational function. with a given weight is a very complicated combinatorial problem. and currently there are no exact general formulæ. bk+1. (t) = w=1 bk.k k+1 +1 +1 and Rk+1 are “framed” inside Rk+1 t2|λ| = P . We will achieve the next best thing. i.288 Cohomology R l+ 1 R k+1 l+ 1 R Figure 7. If a diagram does not fit inside Rk+1 this means that its first line consists of + 1 boxes. (t) = λ∈P . On the other hand.e. (w) is the number of Young diagrams of weight w which fit inside a k × rectangle. the number of partitions in Pk. When we drop this line.8) Computing the Betti numbers. Alternatively. Lemma 7. (7.k (t). Look at the (k + 1) × rectangle Rk+1 inside the (k + 1) × (l + 1)-rectangle Rk+1 (see Figure 7.6). +1 Proof. (w)t2w . Then bk+1. +1 (w ) = bk. (w − − 1).. Denote by bk. (w) the number of partitions λ ∈ P . Brauer.m (t) = · · · · . His method was then extended to arbitrary compact. +1 (w ) is The result in the above lemma can be reformulated as Pk+1. +1 +1 (t) +1) Pk.4. Exercise 7. Q1. 1 − t2 (1 − t2(m−k+1) ) · · · (1 − t2m ) (1 − t2 ) · · · (1 − t2k ) (1 − t2 ) · · · (1 − t2m ) . where x is a formal variable of degree 2 while (xn+1 ) denotes the ideal generated by xn+1 .C) (t). Pk.m (t) = = 1 − t2m .m−d (t) = PGd (m. In turn.Symmetry and topology k × ( + 1) rectangle R bk. (1 − t2 ) · · · (1 − t2k )(1 − t2 ) · · · (1 − t2(m−k) ) Remark 7. +1 (w − − 1). i. +1 (t) = Pk+1. We refer also to [92] for an explicit description of the invariant forms on Grassmannians.e. (a) The invariant theoretic approach in computing the cohomology of Grk (Cn ) was used successfully for the first time by C.m (t) = Pd.4. 1 − t4 1 − t2k 1 − t2(k+1) Now we can check easily that b1. (b) We borrowed the idea of using the Weyl’s integration formula from Weyl’s classical monograph [100].4. Thus.4.8)) we also have +1 (t) + t2(k+1) Pk+1.m (t) = 1 + t2 + t4 + · · · + t2(m−1) = Hence PGk (m. Our strategy is however quite different from Weyl’s.38. These two equality together yield +1 (1 − t2( ) = Pk+1. (1 − t2(k+1) ). The upper estimate is then used to established that these are the only ones. +1 (t).m (t) = Qk. Weyl uses an equality similar to (7.6. we followed a different avenue which did not require Cartan’s maximal weight theory. Weyl attributes this line of attack to R. . Iwamoto [49]. (t). Let m = k + + 1 and set Qd. Show that the cohomology algebra of CPn is isomorphic to the truncated ring of polynomials R[x]/(xn+1 ).37.m · 1 − t2(k+1) so that 1 − t2(m−k) 1 − t2(m−k+1) 1 − t2(n−1) Qk+1. (t) + t2( = Pk. oriented symmetric spaces by H. Qk+1. n Hint: Describe Λ• inv CP explicitly. Ehresmann[31]. The last equality can be rephrased as 1 − tm−k .C) (t) = Qk. the second contribution to bk+1.m−1 (w) = 1. are symmetric (cf. (7. However. +1 289 of Figure 7. Because the roles of k and Pk+1.5) to produce an upper estimate for the Betti numbers (of U (n) in his case) and then produces by hand sufficiently many invariant forms.. We will see that these are essentially two equivalent facets of the same phenomenon. The topology TX on X . For each open set U ⊂ X we denote by C (U ) the space of continuous functions U → R.1. §7. ∀α. and to each U : S(V ) → S(U ) such that. G) determine a presheaf called the constant G-presheaf which is denoted by GX .5.. The second part is a fast ˇ paced introduction to Cech cohomology. . the collection of open subsets.5 ˇ Cech cohomology Cohomology In this last section we return to the problem formulated in the beginning of this chapter: ˇ what is the relationship between the Cech and the DeRham approach. For a modern presentation. then inclusion U → V . the concept of sheaf. Definition 7.1 Sheaves and presheaves Consider a topological space X . then the G-valued continuous functions C (U. If f ∈ S(U ) then we define dom f := U .5. For a very detailed presentation of the classical aspects of subject we refer to [37]. modules. The maps rV U C (U ). i. In other words. The presheaves of rings. and f. If s ∈ S(V ) then. 53]. then the E -valued differential forms of degree k can be organized as a presheaf of vector spaces k U → Ωk E (U ) = Ω (E |U ). If X is a smooth manifold we get another presheaf U → C ∞ (U ). A presheaf of Abelian groups on X is a contravariant functor S : TX → Ab. Understanding this equivalence requires the introduction of a new and very versatile concept namely. (a) If (Uα ) is an open cover of the open set U . g ∈ S(U ) satisfy f |Uα = g |Uα .5. This is done in the first part of the section. More generally. A concise yet very clear presentation of these topics can be found in [46].e. A presheaf S on a topological space X is said to be a sheaf if the following hold. from the point of view of derived categories we refer to the monographs [48. we set U s |U := rV (s) ∈ S(U ).290 7. for any U → V . a group morphism rV U U V rW = rV ◦ rW .2. vector spaces are defined in an obvious fashion. the differential forms of degree k can be organized in a presheaf Ωk (•). Example 7. If E is a smooth vector bundle. then f = g . If G is an Abelian group equipped with the discrete topology. The assignment U → C (U ) defines a U are determined by the restrictions | : C (V ) → presheaf of R-algebras on X . if U → V → W . S associates to each open set an Abelian group S(U ). Let X be a topological space. The morphisms are the inclusions U → V . can be organized as a category. v ∈ Ex we have r · u ∈ Ex .5.3. The reason behind this violation is that in the definition of this presheaf we included a global condition namely the boundedness assumption.1 are sheaves. the set Ex := π −1 (x). CECH COHOMOLOGY (b) If (Uα ) is an open cover of the open set U . u + v ∈ Ex .6. Definition 7. (b) A section of a space of germs π : E → X over a subset Y ⊂ X is a continuous function s : Y → E such that s(y ) ∈ Ey . and + : (u. bounded functions f : U → R. ·.5. ∀x ∈ X. Let S be a presheaf of Abelian groups over a topological space X . ∀y ∈ Y . v ) ∈ E × E.5.5. together with a continuous map π : E → X satisfying the following conditions. Example 7. U runs through the open neighborhoods of X. For every x ∈ X . and with respect to the operations +. U x → E. For each x ∈ X define an equivalence relation ∼x on S(U ). each point e ∈ E has a neighborhood U such that π |U is a homeomorphism onto the open subset π (U ) ⊂ X .4. ∀x ∈ X . (The space of germs associated to a presheaf ). Example 7. ∀u.5. π (u) = π (v ) such that ∀r ∈ R.5. the stalk Ex is an R-module.5. and R a commutative ring with 1. All the presheaves discussed in Example 7. Consider the presheaf S over R defined by S(U ) := continuous. (a2) There exist continuous maps · : R × E → E. We let the reader verify this is not a sheaf since the condition (b) is violated. The spaces of sections defined over Y will be denoted by E(Y ). Let X be a topological space. . f |Uα = fα . Example 7. ∀α. (a1) The map π is a local homeomorphism that is.ˇ 7. by f ∼x g ⇐⇒ ∃ open U x such thatf |U = g |U . is called the stalk at x. We equip R with the discrete topology (a) A space of germs over X (“espace ´ etal´ e” in the French literature) is a topological space E. and fα ∈ S(Uα ) satisfy fα |Uα ∩Uβ = fβ |Uα ∩Uβ ∀Uα ∩ Uβ = ∅ 291 then there exists f ∈ S(U ) such that. Consider a space of π germs E → X over the topological space X . (b) A presheaf S over the topological space X is a sheaf if and only if S = S. u ∈ U ⊂ S. x∈X There exists a natural projection π : S → X which maps [f ]x to x. The correspondence U → E(U ) clearly a sheaf. and S := Sx . i. When the covering is trivial. it is isomorphic to a product X × discrete set .e. A basis of open neighborhoods of [f ]x ∈ S is given by the collection Ug . (The sheaf associated to a space of germs). Set Sx := {[f ]x . Definition 7.. Prove the above proposition. and it is called the germ of f at x. then the sheaf is really the constant sheaf associated to a discrete Abelian group. we say that S is a sheaf of locally constant functions (valued in some discrete Abelian group). each f ∈ S(U ) defines a continuous section of π over U [f ] : U u → [f ]u ∈ Su . Then E = E.292 Cohomology The equivalence class of f ∈ U x S(U ) is denoted by [f ]x .8. then the sheaf S is called the sheaf associated to S.5. Proposition 7. With this topology. Exercise 7. or the sheafification of S. We let the reader check that this collection of sets satisfies the axioms of a basis of neighborhoods as discussed e. it follows that π : S → X is a space of germs. dom f x}.9. (a) Let E → X be a space of germs.10. Note that each fiber Sx has a well defined structure of Abelian group [f ]x + [g ]x = [(f + g ) |U ]x U x is open and U ⊂ dom f ∩ dom g. For each open subset U ⊂ X . Any f ∈ S(U ) defines a subset Uf = [f ]u .5.) Since π : f (U ) → U is a homeomorphism. If the space of germs associated to a sheaf S is a covering space. g ∈ S(U ) [g ]x = [f ]x .g. (Check that this addition is independent of the various choices.5. we denote by E(U ) the space of continuous sections U → E.7. If S is a presheaf over the topological space X . in [57]. U x. Example 7. π . E is called the sheaf associated to the space of germs.5. We can define a topology in S by indicating a basis of neighborhoods. It is called the space of germs associated to the presheaf S. The “fibers” of this map are π −1 (x) = Sx – the germs at x ∈ X . and let h : S → T be a morphism.14. and x ∈ U .12. The proof of this proposition is left to the reader as an exercise. ı). such Abelian groups (modules etc.13. Exercise 7.5. ∀x ∈ X .5.) hU : S(U ) → S V V V denotes the restriction that. i.16.5. then h ˆ ([f ]x ) = [h(f )]x . −1 −1 (a2) For any x ∈ X . Then h(E0 ) → X is a space of germs over X called the image of h. ). . The morphism ı is called the canonical inclusion of the sub-presheaf. A morphism h is morphisms of S. It is a sub-presheaf of S. (a) Let Ei → X (i = 0. h induces a morphism between the associated spaces of germs h : S ˆ should be obvious. (a) Let A morphism between the (pre)sheaves of Abelian groups ˜ over the topological space X is a collection of morphisms of (modules etc. i Definition 7. (a) Let h : E0 → E1 be a morphism between two spaces of germs π1 over X . The correspondence U → ker hU ⊂ S(U ).11. and that it is a continuous map S described in Example 7. If both S and T are sheaves. A subspace of germs is a pair (F. Let h : S → T be a morphism of presheaves. rU V denotes the restriction morphisms of S ˜. Then ˆ → T.ˇ 7. and ı : T → S is an injective morphism.5. It is denoted by Im h.6. ˆ is a subsheaf of T called the image of The sheaf associated to the space of germs Im h h. and denoted by Im h. where F is a space of germs over X and  : F → E is an injective morphism.5. Prove the above proposition. If f ∈ SU . Proposition 7. The definition of h ˆ where h(f ) is now an element of T (U ). we have hV ◦ rU = r ˜U ◦ hU . π (b) Let E → X be a space of germs.5. 1) be two spaces of germs over the same topological space X . Let h : S → T be a morphism of presheaves. A sub-presheaf of S is a pair (T . h(π0 (x)) ⊂ π1 (x).15. when V ⊂ U . and it is a subspace of E1 .. defines a presheaf called the kernel of the morphism h. then so is the kernel of h. (b) Let S be a presheaf over the topological space X . Proposition 7.5. π −1 −1 (a1) π1 ◦ h = π0 .e. Consider two sheaves S and T . The morphism h is called injective if each hx is injective.) S and S ˜(U ). Above. while the r ˜U said to be injective if each hU is injective. We let the reader check that h is independent of ˆ→T ˆ with respect to the topologies the various choices. one for each open set U ⊂ X .5. where T is a presheaf over X . the induced map hx : π0 (x) → π1 (x) is a morphism of Abelian groups (modules etc. CECH COHOMOLOGY 293 Definition 7. Lemma 7. A morphism of spaces of germs is a continuous map h : E0 → E1 satisfying the following conditions.). 294 Cohomology Exercise 7. Show that a section g ∈ T (U ) belongs to (Im h)(U ) if an only if. such that g |Vx = h(f ). there exists an open neighborhood Vx ⊂ U . Let U ⊂ X be an open set. Consider a morphism of sheaves over X . for some f ∈ S(Vx ). h : S → T . for every x ∈ X . .5.17. (a) Prove that U → S(U ) is a presheaf. Definition 7. Let M be a smooth n-dimensional manifold.18. and it is called the nerve of the cover. Consider a short exact sequence of sheaves 0 → S−1 → S0 → S1 → 0. αq ⊂ A. Nq (U) q is denoted by N(U). ∀n.19.5. . For each open set U define S(U ) = S0 (U )/S−1 (U ). Due to Proposition 7. (a) A sequence of sheaves and morphisms of sheaves. ¯ ˆ = the sheaf associated to the presheaf S.5. . in the sequel we will make no distinction between sheaves and spaces of germs. A simplex associated to U is a nonempty subset A = α0 . h hn+1 is said to be exact if Im hn = ker hn+1 . (b) Prove that S1 ∼ =S Example 7. We define dim A to be one less the cardinality of A.5. The set of all q -dimensional simplices is denoted by Nq (U). Consider an open cover U = (Uα )α∈A of X . such that q UA := 0 Uαi = ∅. Ωk M denotes the sheaf of k -forms on M while d denotes the exterior differentiation. . Using the Poincar´ e lemma and the Exercise 7. . dim A = |A| − 1. (b) Consider a sheaf S over the space X .5.5. Their union. ˇ §7.17 we deduce immediately that the sequence d d 1 d n 0 → RM → Ω0 M → ΩM → · · · → ΩM → 0 is a resolution of the constant sheaf RM . .2 Cech cohomology Let U → S(U ) be a presheaf of Abelian groups over a topological space X .20. (The DeRham resolution).8(a). n · · · → Sn → Sn+1 → Sn+2 → · · · .5. A resolution of S is a long exact sequence 0 1 n 0 → S → S0 → S1 → · · · → Sn → Sn+1 → · · · . ı d d d Exercise 7. 22. . and we will use the symbol (α0 . .ˇ 7. . We will denote by Nq (U) the set of ordered q -simplices. U) := σ ∈Nq (U) ˇ The elements of C q (S.5. Example 7. . and where a hat “ ˆ ” indicates a missing entry. ˇ The cohomology of this cochain complex is called the Cech cohomology of the cover U with coefficients in the pre-sheaf S. Prove that ∂q −1 q q −1 q We can now define an operator δ : C q−1 (S. . αq ). q }. αq ) ∈ U(q−1) . Observe that if S is a sheaf. Sσ := S(Uσ(∆q ) ). δ δ δ δ .5.23. U) → · · · is a cochain complex. ∂ j σ |Uσ . and we define an ordered q -simplex to be a map σ : ∆q → A with the property that σ (∆q ) is a simplex of N(U). . we have cβ − cα = 0. It is a cocycle if. Let U and S as above. for all j > i. U) are called Cech q -cochains subordinated to the cover U. β ) such that Uα ∩ β = ∅. . . δ 2 = 0 so that 0 → C 0 (S. Lemma 7. where for a ordered q -simplex σ = (α0 . . αq ) to denote an ordered q -simplex σ : ∆q → A such that σ (k ) = αk . . . . . A 0-cochain is a correspondence which associates to each open set Uα ∈ U an element cα ∈ S(Uα ). . . which assigns to each (q − 1)-cochain c. . Using Exercise 7. . U) → C 1 (S. In other words. i ∂ j = ∂ j −1 ∂ i . then the collection (cα ) determines a unique section of S. possibly of dimension < q . .21. C q (S. j = 0. U) → C q (S.5. for any pair (α. We have face operators ∂ j = ∂ j : Nq (U) → Nq−1 (U).5. CECH COHOMOLOGY 295 For every nonnegative integer q we set ∆q := {0. . Define Sσ . a q -cochain c associates to each ordered q -simplex σ an element c. a q -cochain δc whose value on an ordered q -simplex σ is given by q δc. α ˆ j . Exercise 7.5.21 above we deduce immediately the following result. q. . we set j ∂ j σ = ∂q σ = (α0 . . . U). σ ∈ Sσ . . U) → · · · → C q (S. . σ := j =0 (−1)j c. (c) If U ≺ V ≺ W.25. and V = (Vβ )β ∈B of the same topological space X . In particular. then W V ıW U = ıV ◦ ıU . Then the following are true. this cocycle is a coboundary. such that Vβ ⊂ U The map (β ) ∀β ∈ B. and we write this U ≺ V. then we can associate to ˇ each closed 1-form ω ∈ Ω1 (M ) a Cech 1-cocycle valued in RX as follows. γ ) we have cβγ − cαγ + cαβ = 0. Cohomology This correspondence is a cocycle if. β ) such that Uαβ = ∅ an element cαβ ∈ S(Uαβ ). Consider two open covers U = (Uα )α∈A . RX . S).5. if ω is exact. For example. Moreover. . we deduce there exist constants cαβ such that fα − fβ = cαβ . Obviously this is a cocycle. if there exists a map : B → A.5. We will see later this is an isomorphism. and it is easy to see that its cohomology class is independent of the initial selection of local solutions fα . β. S) → H (V. We say V is finer than U. In other words we have a natural map H 1 (X ) → H 1 N(U). if X is a smooth manifold.296 A 1-cochain associates to each pair (α. is said to be a refinement map.24. (b) If r : A → B is another refinement map. Fix a sheaf of Abelian groups S. Consider two open covers U = (Uα )α∈A and V = (Vβ )β ∈B of the same topological space X such that U ≺ V. Proposition 7. S) → q C q (V. and U is a good cover. Definition 7. for any ordered 2-simplex (α. (a) Any refinement map induces a cochain morphism ∗ : q C q (U. select for each Uα a solution fα ∈ C ∞ (Uα ) of dfα = ω. S). First. any relation U ≺ V defines a unique morphism in cohomology • • ıV U : H (U. Since d(fα − fβ ) ≡ 0 on Uαβ . then ∗ is cochain homotopic to r∗ . . . . . (−1)j δc. . σ := c. (b) We define hj : Nq−1 (V) → Nq (U) by hj (β0 . . λj . hj (σ ) |Vσ ∀c ∈ C q (U) ∀σ ∈ V(q−1) . . (λ0 . . Now define χ := χq : C q (U) → C q−1 (V) by q −1 χq (c). . . . . ∂q +1 hj (σ ) |Vσ ) = j =0 q j (−1)j ( k=0 = j =0 (−1)j k=0 q ˆ j . . . µ ˆk . . . . . The reader should check that hj (σ ) is indeed an ordered simplex of U for any ordered simplex σ of V. (λ0 . . . so that. . . by definition. CECH COHOMOLOGY Proof. where. . . µj . . σ := j =0 (−1)j c. . . hj (σ ) |Vσ k (−1)k c. . βq−1 ) = ( (β0 ). µq ) |V (−1)k c. µq ) ∈ U(q) . . . µq ) |Vσ  + =j q (−1) +1 j −1 = j =0 (−1) j k=0 ˆ j . . (σ ) |Vσ ∀c ∈ C q (U) σ ∈ V(q)) . λj . . . . . . . . (σ ) ∈ Aq is the ordered q -simplex ( (β0 ). . and set (σ ) := (λ0 . (λ0 . µj . . µj . λj . . µq ) |V (−1)k c. . hj (σ ) = (λ0 . (βj ). µq ). r(σ ) = (µ0 . . . (βq )). Then χ ◦ δ (c). . . . λ σ  c. λq ). λ σ . . . . . · · · . . . r(βj ). . .5. Note that Vσ ⊂ Uhj (σ) ∀j .ˇ 7. . σ = q q +1 ∗ (c) − r∗ (c) ∀c ∈ C q (U). µj . (a) We define S(Vσ ) ∗ 297 : C q (U) → C q (V) by ∗ (c). βq ) ∈ Nq (V). The fact that ∗ is a cochain morphism follows immediately from the obvious equality j j ◦ ∂q = ∂q ◦ . We will show that δ ◦ χq (c) + χq+1 ◦ δ (σ ) = Let σ = (β0 . . . . λj . . . . . . r(βq−1 )). . . . . . µq ) |Vσ . . . W W V ıU U =  and ıU = ıV ◦ ıU . . . S) / ∼ . . S) . λj −1 . . Part (c) is left to the reader as an exercise. . µj . σ − r∗ c. S). . . U ≺ V . S). (λ0 . Then lim H • (U) = U U H • (U. such that. U by H • (U) f ∼ g ∈ H • (V) ⇐⇒ ∃W W U. . ıU : H • (U. We now have a collection of graded groups H • (U. We denote the equivalence class of f by f . . S) := lim H • (U. . . λj . µj +1 . µj . . U − open cover of X . λj . We can thus define the inductive (direct) limit H • (X. . . . One defines an equivalence relation on the disjoint union H • (U. . . . Note that we have canonical morphisms. (λ0 . µq ) |Vσ  + j =0 (−1)j c. . S). . . . . (µ0 . µq ) |Vσ = ∗ c. µq ) |Vσ + c. whenever U ≺ V ≺ W. Let us briefly recall the definition of the direct limit.298 q Cohomology  (−1) =j +1 q +1 + c. . (λ0 . V : ıW U f = ıV g. (λ0 . . S) → H (V. . The last term is a telescopic sum which is equal to c. . . . . σ . . S). µ ˆk . If we change the order of summation in the first two term we recover the term −δχc. U ˇ The group H • (X. . S) → H • (X. S) is called the Cech cohomology of the space X with coefficients in the pre-sheaf S. λq |Vσ − c. σ . and morphisms • • ıV U : H (U. . . for each open cover U we have a short exact sequence 0 → C q (U. For any open cover U = (Uα ). According to the properties of a sheaf. S0 ) → H q (X.26. S0 ) → C q (U. S−1 ) → · · · . S−1 ) → C q (U. Any morphism of pre-sheaves h : S0 → S1 over X induces a morphism in cohomology h∗ : H • (X. CECH COHOMOLOGY 299 Example 7. S) → H q+1 (U. S−1 ) → · · · .  π . S0 ) → C q (U.5.19). We obtain a long exact sequence in cohomology · · · → H q (U. Hence. Its associated sheaf is isomorphic with S1 (see Exercise 7. S−1 ) → H q (U. S) ∼ = S(X ). σ = hU ( c.5. Define h∗ : C q (U. The proposition follows by passing to direct limits. σ ) ∀c ∈ C q (U. by h∗ c. Its proof can be found in [88]. S0 ) σ ∈ U( q ). S−1 ) → H q (X. S1 ) → H q+1 (X. S1 ). Let S be a sheaf over the space X .27. S0 ) → H • (X. a 0-cycle subordinated to U is a collection of sections fα ∈ S(Uα ) such that.5.5. S−1 ) → H q (X. Then there exists a natural long exact sequence ∗ · · · → H q (X. Then the correspondence U → S(U ) defines a pre-sheaf on X . X ) → · · · . For each open set U ⊂ X define S(U ) := S0 (U )/S−1 (U ). such a collection defines a unique global section f ∈ S(X ). we have fα |Uαβ = fβ |Uαβ . Sketch of proof. H 0 (X. Proposition 7. S0 ) → H q (U. every time Uα ∩ Uβ = ∅. S) → 0. Thus. The reader can check easily that h∗ is a cochain map so it induces a map in cohomology • • hU ∗ : H (U. Theorem 7. Let U be an open cover of X . be an exact sequence of sheaves over a paracompact space X . ∗ p∗ δ Sketch of proof. Passing to direct limits we get a long exact sequence · · · → H q (X.ˇ 7. S1 ) which commutes with the refinements ıV U . To conclude the proof of the proposition we invoke the following technical result. S0 ) → H q (X. S0 ) → H (U.28. S) → H q+1 (X.5. Let  p 0 → S−1 → S0 → S1 → 0. S1 ). More generally.31. S). σ := tα (f ). A fine resolution is a resolution 0 → S → S0 → S1 → · · · . α The above sum is well defined since the cover U is locally finite.300 Cohomology Lemma 7.5. for any locally finite open cover U = (Uα )α∈A there exist morphisms hα : S → S with the following properties. the section hα (f |V ) can be extended by zero to a section tα (f ).5. (7. if E is a smooth vector bundle over X then the space Ωk E of E -valued k -forms is fine. S). σ ∈ Nq (U).30. Now. We let the reader check that χq satisfies (7. If two pre-sheaves S.29. Proof. Example 7.5. S) → C q−1 (U. For every α ∈ A. Let X be a smooth manifold.5.1). We thus need to produce a map χq : C q (U. σ . S) are trivial for q ≥ 1. according to the axioms of a sheaf. the cohomology groups H q (U. S) → C q (U. Then H q (X. S over a paracompact topological space X have isomorphic associated sheaves then H • (X. Using partitions of unity we deduce that the sheaf Ωk X of smooth k -forms is fine. for every f ∈ C q (U. where Sx denotes the stalk of S at x ∈ X . Note that hα f |V ∩W = 0 and. and every f ∈ C q (U. Definition 7. Consider the open cover of Uσ V = Uα ∩ Uσ . it suffices to show that. Proposition 7. S) ∼ = H • (X. (a) For any α ∈ A there exists a closed set Cα ⊂ Uα . (supp hα ⊂ Cα ).32. S) ∼ = 0 for q ≥ 1. and hα (Sx ) = 0 for x ∈ Cα . σ ∈ S(Uσ ) as follows.33. S) by χq (f ). Let S be a sheaf over a space X .1) Consider the morphisms hα : S → S associated to the cover U postulated by the definition of a fine sheaf. Definition 7. for each locally finite open cover U = (Uα )α∈A . W := Uσ \ Cα . Thus. This sum is well defined since the cover U is locally finite.5. such that χq+1 δ q + δ q−1 χq = . Let S be a fine sheaf over a paracompact space X . define χq f ∈ C q−1 (U.5. d d . such that each of the sheaves Sj is fine. We will achieve this by showing that the identity map C q (U. σ ∈ S(Uσ ). S ). (b) α hα = S . we construct tα (f ). S). Because X is paracompact. any open cover admits a locally finite refinement.5. A sheaf S is said to be fine if. S) is cochain homotopic with the trivial map. since Sq is a fine sheaf. we get an exact sequence H 0 (X. = Zm (X )/dm ∗ dm−1 (7.5. Zm−1 ) ∗ → H 0 (X.5. For q ≥ 1. We set for uniformity Z0 := S. and there exists a natural isomorphism H q (X.34 (Abstract DeRham theorem). denote by Zq the kernel of the sheaf morphism dq . Corollary 7.5. CECH COHOMOLOGY Theorem 7. (7. Zm−1 ) ∼ Sm−1 (X ) . The first statement in the theorem can be safely left to the reader.ˇ 7.5. Using again the long sequence associated to (7. Then 0 1 0 → S0 (X ) → S1 (X ) → ··· 301 d0 d1 d d is a cochain complex. RM ) and H 2 (M ) ∼ .2) We use the associated long exact sequence in which H k (X.5. RM ). for k ≥ 1. We deduce inductively that H m (X. The manifold M is paracompact.5. This yields the isomorphisms H k−1 (X. Zm−1 ) m ≥ 1. S) ∼ = H q (Sq (X )). We conclude using the fine resolution 1 0 → RM → Ω0 M → ΩM → · · · . Let M be a smooth manifold. Then H • (M.2). Z0 ) ∼ = H 1 (X. Sq ) = 0. Describe explicitly the isomorphisms H 1 (M ) ∼ = H 2 (M.35. Zm ) → H 1 (X. Zm−1 ) → 0.3) This is precisely the content of the theorem.26. = H 1 (M. d Exercise 7. and we get −1 H 1 (X.5.5.36. Proof. We apply the computation in Example 7. Proof. We get a short exact sequence of sheaves 0 → Zq → Sq → Zq+1 → 0 q ≥ 0. RM ) ∼ = H • (M ). Zq+1 ) ∼ = H k (X. Let 0 → S → S0 → S1 → · · · be a fine resolution of the sheaf S over the paracompact space X . Zq ) k ≥ 2. . Proof. Thm. d Using the associated long exact sequence and the fact that Ωk−1 is a fine sheaf. G) = 0 q ≥ 1. S) = 0. Uσ ∼ = Rdim M .1. even if the manifolds may not be diffeomorphic. We thus have a short exact sequence ∗ 0 → C q (U.302 Cohomology ˇ Remark 7. The above corollary has a surprising implication. Zq )/d∗ H 0 (U. S. The above result is a special case of a theorem of Leray: if S is a sheaf on a paracompact space X . Freedman were awarded Fields medals for their contributions. Such exotic situations do exist. 5. . then H • (U.5. Donaldson and M. For a proof we refer to [37].39. so must be the DeRham cohomology which is defined in terms of a smooth structure. Ωq−1 ) ∼ = Zq (M )/dΩq−1 (M ) ∼ = H • (M ). Hence if two smooth manifolds are homeomorphic they must have isomorphic DeRham groups. RM ) ∼ = H q−1 (U. When GM is a constant sheaf. J.37. H q (Rn . Using the Poincar´ e lemma we deduce Zk (Uσ ) = dΩk−1 (Uσ ). we deduce as in the proof of the abstract DeRham theorem that H q (U. (This is possible only for 4-dimensional vector spaces!) These three mathematicians. Hence. GM ). S) = H • (X. Zq−1 ) ∼ = H 0 (U. Ωk−1 ) → C q (U. i.5. Let Zk denote the sheaf of closed k -forms on M . where G is an arbitrary Abelian group. Zk ) → C q (U. Chapter IX. and U is an open cover such that H q (UA . Zk ) → 0. In a celebrated paper [70]. ∀q ≥ 1 A ∈ N(U). GM ) = H • (U. Then H • (U. RM ) ∼ = H • (M ). Remark 7. Theorem 7. Since the Cech cohomology is obviously a topological invariant. Milnor. we have a Poincar´ e lemma (see [32]. (Leray) Let M be a smooth manifold and U = (Uα )α∈A a good cover of M . More recently. Z1 ) ∼ = ··· ∼ = H 1 (U.5. the work of Simon Donaldson in gauge theory was used by Michael Freedman to construct a smooth manifold homeomorphic to R4 but not diffeomorphic with R4 equipped with the natural smooth structure. John Milnor has constructed a family of nondiffeomorphic manifolds all homeomorphic to the sphere S 7 .).e. for any good cover U H • (M.38. S). β ) ∈ A2 such that Uαβ = ∅ satisfying fαβ + fβγ + fγα = 0. Thus. whenever Uαβγ = ∅. A ∈ K.e. the nerve of a good cover contains all the homotopy information about the manifold. Since the DeRham cohomology is homotopy invariant. We topologize RV by declaring a subset C ⊂ RV closed if the intersection of C with any finite dimensional subspace of RV is a closed subset with respect to the Euclidean topology of that finite dimensional subspace. with vertex set V .5. A combinatorial simplicial complex. We will say that |K| is the geometric realization of the simplicial scheme K. CECH COHOMOLOGY 303 Remark 7. For any face F ⊂ K we denote by ∆F ⊂ RV the convex hull of the set δv . albeit very large. Denote by RV the vector space of all maps f : V → R with the property that f (v ) = 0. or a simplicial scheme. In other words. The nerve of an open cover is an example of simplicial scheme.5. we should remark that any compact manifold admits finite good covers.40. Note that ∆F is a simplex of dimension |F | − 1. A result of Borsuk-Weil (see [14]) states that if U = (Uα )α∈A is a good cover of a compact manifold M . The collection is a coboundary if there exist the constants fα such that fαβ = fβ − fα . Example 7. Note that RV has a canonical basis given by the Dirac functions δv : V → R. Let M be a smooth manifold and U = (Uα )α∈A a good cover of M . then the geometric realization of the nerve N(U) is homotopy equivalent to M . While at this point. for all but finitely many v ’s. is a collection K of nonempty. This is precisely the situation encountered in Subsection 7.2. i.41. δv (u) = 1 u=v 0 u = v. The set V is called the vertex set of K. What is remarkable is the explicit way in which one can extract the cohomological information from the combinatorics of the nerve.1.one for each pair (α. finite subsets of a set V with the property that any nonempty subset B of a subset A ∈ K must also belong to the collection K. B ⊂ A. . B = ∅ =⇒ B ∈ K. Leray’s theorem comes as no suprise.5. To any simplicial scheme K. This shows that the homotopy type is determined by a finite.. Now set |K| = F ∈K ∆F . RV = v ∈V R. A 1-cocycle of RM is a collection of real numbers fαβ . v ∈ F .ˇ 7. The ˇ abstract DeRham theorem explains why the Cech approach is equivalent with the DeRham approach. we can associate a topological space |K| in the following way. set of data. while the subsets in K are called the (open) faces) of the simplicial scheme. 42. αϕ(q) ) . . (α0 .5. Exercise 7. f0 f0 f1 f0 f1 . S−1 = S. . Show that if S0 is a flabby sheaf. . k ≥ 0. ˇ One can then define a “skew-symmetric” Cech cohomology following the same strategy. there exists a sequence of flabby sheaves over X .5. (b) Suppose 0 → S0 → S1 → S2 → 0 is a short exact sequence of sheaves on X . The resulting cohomology coincides with the cohomology described in this subsection. 1. the restriction map S(X ) → S(U ) is surjective.43. k = 0. A sheaf of Abelian groups S on a paracompact space X is called soft if for every closed subset C ⊂ X the restriction map S(X ) → S(C ) is surjective. i. (a) Prove that any flabby sheaf is also soft. For a proof of this fact we refer to [88]. Exercise 7.5. αq ). in concrete applications it is convenient to work with skewˇ symmetric Cech cochains. . . U) is skew-symmetric if for any ordered q -simplex σ = (α0 . . such that the sequence below is exacft 0 → S −→ S0 −→ S1 −→ · · · . ..304 Cohomology Remark 7. for any open subset U ⊂ X . S) are isomorphic to the cohomology groups of the cochain complex S0 (X ) −→ S1 (X ) −→ · · · . and morphisms of sheaves fk : Sk−1 → Sk .44. . . then the cohomology groups H k (X. . then the sequence of Abelian groups 0 → S0 (X ) → S1 (X ) → S2 (X ) → 0 is exact. αq ) = (ϕ) c. . A cochain c ∈ C q (S. . (a) Prove that S admits a flabby resolution. (αϕ(0) . Sk . . Often.e. . The sheaf S is called flabby if. and for any permutation ϕ of ∆q we have c. . (b) Prove that if 0 → S −→ S0 −→ S1 −→ · · · is a flabby resolution of S. Suppose X is a topological space and S is a sheaf of Abelian groups on X. . according to a nontrivial result (Peter-Weyl theorem).1. where Ad(g )X := gXg −1 . We stick with this assumption since most proofs are easier to “swallow” in this context. . only topological techniques yield the best results. ∀X ∈ g. This restriction is neither severe. The Lie algebra g of a matrix Lie group G is a Lie algebra of matrices in which the bracket is the usual commutator. We denote by Ad(P ) the vector bundle with standard fiber g associated to P via the adjoint representation. any compact Lie group is isomorphic with a matrix Lie group. All the Lie groups we will consider will be assumed to be matrix Lie groups.e. K) = GL(Kn ). Let M be a smooth manifold. in the context of smooth manifolds there are powerful differential geometric methods which will solve a large part of this problem.1 Chern-Weil theory §8. g ∈ G. The Lie group G operates on its Lie algebra g via the adjoint action Ad : G → GL(g).1 Connections in principal G-bundles In this subsection we will describe how to take into account the possible symmetries of a vector bundle when describing a connection. and these groups are sufficient for most applications in geometry. nor necessary. In other words. as we will see. 8. Recall that a principal G-bundle P over M can be defined by an open cover (Uα ) of M and a gluing cocycle gαβ : Uαβ → G. Ultimately.Chapter 8 Characteristic classes We now have sufficient background to approach a problem formulated in Chapter 2: find a way to measure the “extent of nontriviality” of a given vector bundle. Lie subgroups of a general linear group GL(n. Ad(P ) is the vector bundle defined by the open 305 g → Ad(g ) ∈ GL(g). i. This is essentially a topological issue but.. It is not necessary since all the results of this subsection are true for any Lie group. It is not severe since. 1. The bracket operation in the fibers of Ad(P ) induces a bilinear map [·. If this happens. η ]. g•• ) the set of connections defined by the open cover U and the gluing cocycle g•• : U•• → G.1. . Prove that for any ω.306 cover (Uα ). Y ]. defined by [ω k ⊗ X. such that hβα (x) = Tβ gβα (x)Tα (x)−1 . and X. [ω.3. hβα : Uαβ → G. 2 Remark 8. η ] + (−1)|ω|·|φ| [ω. ·] : Ωk Ad(P ) × Ω Ad(P ) → Ωk+ Ad(P ) . and the gluing cocycle gβα : Uαβ → G is a collection Aα ∈ Ω1 (Uα ) ⊗ g. Ω• Ad(P ) . Exercise 8. φ] ]. η. for all ω k ∈ Ωk (M ). φ] = [ [ω. ∀x ∈ Uαβ . (8. ] is a super Lie algebra. Using Proposition 3. and gluing cocycle CHAPTER 8. [η. Given an open cover (Uα ) of M .1) (8.5 as inspiration we introduce the following fundamental concept. then two gluing cocycles gβα . [ [ω.1.1. η ] = −(−1)|ω|·|η| [η. η ∈ Ω (M ).1. φ].1. (a) A connection on the principal bundle P defined by the open cover U = (Uα )α∈A . η ⊗ Y ] := (ω k ∧ η ) ⊗ [X. we say that the cocycles g•• and h•• are cohomologous. φ ∈ Ω∗ (Ad(P )) the following hold.1. Y ∈ Ω0 Ad(P ) . Definition 8. [ . define isomorphic principal bundles if and only if there exists smooth maps Tα : Uα → G. (b) The curvature of a connection A ∈ A(U. Aα ].2. g•• ) is defined as the collection Fα ∈ Ω2 (Uα )⊗g where 1 Fα = dAα + [Aα . ∀x ∈ Uαβ . ω ].3. CHARACTERISTIC CLASSES Ad(gαβ ) : Uαβ → GL(g). .2) (8. ∀α. β. In other words. satisfying the transition rules −1 −1 Aβ (x) = gαβ (x)dgαβ (x) + gαβ (x)Aα (x)gαβ (x) −1 −1 = −( dgβα (x) )gβα (x) + gβα (x)Aα (x)gβα (x).3) We will denote by A(U. g•• ) −→ A(W. and the cocycle gij . ∀i ∈ I. For simplicity. Finally. finer that both U ¯ •• : W•• → G refining g•• and respectively and W. g•• ). CHERN-WEIL THEORY 307 Suppose (gαβ ). we will denote by A(P ). The correspondence ϕ TVU : A(U. (U. (hαβ are two such cohomologous cocycles. such that the induced maps ¯ •• ) ←− A(V. A → Aϕ is also a bijection. and gβα : Uαβ → G is a gluing cocycle. h are bijections. T The correspondence T is a bijection. h•• ). . Suppose that the open cover (Vi )i∈I is finer that the cover (Uα )α∈A . gluing cocycle). We say that the connection Aϕ is the V-refinement of the connection A. g•• ) → A(V. Then the maps Tα induce a well defined correspondence T : A(U. If (Aα ) is a connection defined by the open cover (Uα ) and the cocycle (gβα ). given two pairs (open cover. x ∈ Vij . j ∈ I. g•• ) −1 −1 (Aα ) −→ Bα := −(dTα )Tα + Tα Aα Tα ∈ A(U. h h•• . g•• ) and (V.e. then it induces a connection Aϕ i = Aϕ(i) |Vi defined by the open cover (Vi ). both describing the principal bundle P → M . any of these isomorphic spaces A(U.1.. ∀i. there exists a map ϕ : I → A such that Vi ⊂ Uϕ(i) . h•• ). i. ϕ We obtain a new gluing cocycle gij : Vij → G given by ϕ gij (x) = gϕ(i)ϕ(j ) (x). Suppose U = (Uα )α∈A and V = (Vi )i∈I are open covers. there exists an open cover W. g•• ). and cohomologous gluing cocycles g ¯•• . g•• ) → A(U. h•• ) given by A(U. This new cocycle defines a principal bundle isomorphic to the principal bundle defined by the cocycle (gβα ). h•• ) A(U. g ¯•• ) −→ A(W.8. . We are convinced that the reader can easily supply the obvious. and repetitious missing details. All the constructions that we will perform in the remainder of this section are compatible with the above isomorphisms but. in our proofs we will not keep track of these isomorphisms. in order to keep the presentation as transparent as possible. ∀α. the collection (Fα ) defines a global Ad(P )valued 2-form.4. satisfying the gluing rules −1 ωβ |Uαβ = gβα ωα |Uαβ gβα . Conversely. (8. (a) The set A(P ) of connections on P is an affine space modelled by Ω1 ( Ad(P ) ). such that supp uα ⊂ Uα . (b) For any connection A = (Aα ) ∈ A(U. Suppose P → M is a principal G-bundle described by an open cover U = (Uα )α∈A .4) Proof. if (Aα ) ∈ A(P ). This proves that if A(P ) is nonempty. Bα = − γ −1 uγ (dgαγ )gαγ . The collection Aα := Aα + ωα is then a connection on P . .1. (Bα )α∈A ∈ A(P ). Fα ] = 0.) dFα + [Aα . then their difference Cα = Aα − Bα satisfies the gluing rules −1 Cβ = gβα Cα gβα . and uα = 1. (c) (The Bianchi identity. (a) If (Aα )α∈A . then ω is described by a collection of g-valued 1-forms ωα ∈ Ω1 (Uα ) ⊗ g. and ω ∈ Ω1 Ad(P ) .1. on the overlap Uαβ . ∀α. We will denote it by FA or F (A). Then. g•• ). and we will refer to it as the curvature of the connection A.308 CHAPTER 8. α For every α ∈ A we set Bα := γ −1 we deduce g −1 dg −1 Since gγα = gαγ γα = −(dgαγ )gαγ . so that −1 (dgβγ )gβγ = (dgβγ )gγβ = (dgβα )gαβ + gβα (dgαγ )gγβ . To prove that A(P ) = ∅ we consider a partition of unity subordinated to the cover Uα . then it is an affine space modelled by Ω1 ( Ad(P ) ). so that it defines an element of Ω1 (Ad(P )). so that γα −1 uγ gγα dgγα ∈ Ω1 (Uα ) ⊗ g. α ∈ A. we have the equality −1 Bβ − gβα Bα gβα =− γ −1 uγ (dgβγ )gβγ + γ −1 −1 uγ gβα (dgαγ )gαγ gβα The cocycle condition implies that dgβγ = (dgβα )gαγ + gβα (dgαγ ). More precisely. we consider a family of nonnegative smooth functions uα : M → R. and gluing cocycle gβα : Uαβ → G. CHARACTERISTIC CLASSES Proposition 8. (b) We need to check that the forms Fα satisfy the gluing rules Fβ = g −1 Fα g. There exists a natural operation ∧ : Ωk (Uα ) ⊗ gl(n. 2 ] = 0. (8.5) 1 +d(g −1 Aα g ) + [ . uniquely defined by (ω k ⊗ A) ∧ (η ⊗ B ) = (ω k ∧ η ) ⊗ (A · B ). we deduce gαγ γβ βα −1 −1 uγ gβα (dgαγ )gαγ gβα = γ γ uγ gβα (dgαγ )gγβ . This shows that the collection (Bα ) defines a connection on P . −1 where g = gαβ = gβα .6) . 2 ] (8. Let gl(n.8.1. 2 We will check two things. Using (8. K). K) → Ωk+ (Uα ) ⊗ gl(n. The Maurer-Cartan structural equations. CHERN-WEIL THEORY Hence − γ −1 uγ (dgβγ )gβγ = −(dgβα )gαβ − γ 309 uγ gβα (dgαγ )gγβ . K) denote the associative algebra of K-valued n × n matrices.2) we get Fβ = d 1 + [ . on the overlap Uαβ we have −1 −1 −1 Bβ = −(dgβα )gβα gβα Bα gβα = gαβ dgαβ + gαβ Bα gαβ . We have 1 Fβ = dAβ + [Aβ .1.1.1. K) × Ω (Uα ) ⊗ gl(n. d(g −1 Aα g ) + [ . Hence. g −1 Aα g ]. −1 g −1 = g . 2 Set := g −1 dg . g −1 Aα g ] = g −1 (dAα )g. g −1 Aα g ] + [g −1 Aα g. g −1 dg + g −1 Aα g ]. so that Using the cocycle condition again. Proof of A. d B. Let us first introduce a new operation. A. Aβ ] 2 1 = d(g −1 dg + g −1 Aα g ) + [g −1 dg + g −1 Aα g. 1 + [ . 1 ω ∧ η = [ω.1.2) d(Fα ) = {[dAα .1. so that its Lie algebra g lies inside gl(n. Aα ] − [[Aα .3. dAα ]} = [dAα . Aα ]. K) → Ωk+ (Uα ) ⊗ gl(n.7) = − [ . 2 The proposition is proved.1). η ] + (−1)|ω| [ω. The space gl(n. the Bianchi identity can be rewritten as dA F (A) = 0. Exercise 8. g 1 Aα g ] + g −1 (dAα )g 2 2 = −[ .1.5). the collection ωα defines a global k -form ω ∈ Ωk (Ad(P )). (8. η ∈ Ω1 (Uα ) ⊗ gl(n. K) defined as in (8.5. Aα ]. η ] ∀ω. η ∈ Ω (Uα ) and A. Thus.1. dη ]. Aα ] − [Aα . K) (see also Example 3. K) is naturally a Lie algebra with respect to the commutator of two matrices. where ω. Using the above equality we get 1 (8. (c) First. .1. K) × Ω (Uα ) ⊗ gl(n. Let ωα ∈ Ωk (Uα ) ⊗ g satisfy the gluing rules −1 ωβ = gβα ωα gβα on Uαβ .1. η ] = [dω. η ∈ Ω∗ (Uα ) ⊗ g. ωα ]. Aα ] 2 1 (8. for any A ∈ A(P ). We compute d(g −1 Aα g ) = (dg −1 Aα ) · g + g −1 (dAα )g + g −1 Aα dg = −g −1 · dg · g −1 ∧ Aα · g + g −1 (dAα )g + (g −1 Aα g ) ∧ g −1 dg =− ∧ g −1 Aα g + g −1 Aα g ∧ + g −1 (dAα )g 1 1 (8. A very simple computation yields the following identity. Prove that the collection dωα + [Aα . We can think of the map gαβ as a matrix valued map.3) =[Fα . we let the reader check the following identity d[ω.12). B ∈ gl(n. (8. g −1 Aα g ] + [ . g −1 Aα g ] + g −1 (dAα )g. K).2) Part (b) of the proposition now follows from A.1.1.7) =− ∧ = − [ . CHARACTERISTIC CLASSES where ω k ∈ Ωk (Uα ). Aα ] = [Fα . K). (8. •] : Ωk (Uα ) ⊗ gl(n. K).1. defines a global Ad(P )-valued (k + 1)-form on M which we denote by dA ω .310 CHAPTER 8. B and (8.1. so that we have d =d(g −1 dg ) = (dg −1 ) ∧ dg = − (g −1 · dg · g −1 )dg = −(g −1 dg ) ∧ (g −1 dg ) 1 (8.8) In other words. ]. 2 Proof of B.7) 2 The Lie group lies inside GL(n. This structure induces a bracket [•. for any A ∈ A(P ). g•• ) acts on A(U. and a gluing cocycle gαβ : Uαβ → G.1. h•• ) are isomorphic. . The group G(P ) also acts on the vector spaces Ω• Ad(P ) T ∈ G(P ) we have FT AT −1 = T FA T −1 . then the groups G(U. One can verify that if P is described the (open cover. g•• ) and G(V). In other words. A G-structure on E is defined by the following collection of data. ∀x ∈ U αβ. We denote by G(P ) the isomorphism class of all these groups. A gauge transformation of P is by definition. a collection of smooth maps Tα : Uα → G satisfying the gluing rules Tβ (x) = gβα (x)Tα (x)gβα (x)−1 . §8. where the symmetry group G acts on itself by conjugation G×G (g. A) → T AT −1 := −(dTα )Tα + Tα Aα Tα ∈ A(P ).1. such that the vector bundle E can be defined by the cocycle ρ(gαβ ) : Uαβ → GL(V ).1. C The set of gauge transformations forms a group with respect to the operation (Sα ) · (Tα ) := (Sα Tα ). there exists an open cover (Uα ) of M .7.3. We say that two connections A0 .8. A1 ∈ A(P ) are gauge equivalent if there exists T ∈ G(P ) such that A1 = T A0 T −1 . h•• ). and E → M a vector bundle with standard fiber a vector space V . Observe that a gauge transformation is a section of the G-fiber bundle G → C (P ) M. We denote by G(U. Suppose P → M is a principal G-bundle given by the open cover U = (Uα ) and gluing cocycle gβα : Uαβ → G. (a) A representation ρ : G → GL(V ). g•• ) this group. CHERN-WEIL THEORY 311 Remark 8. gluing cocycle)-pair (V.6. (see Definition 2. and.15) with standard fiber G. h) −→ Cg (h) := ghg 1 ∈ G. (b) A principal G-bundle P over M such that E is associated to P via ρ. The group G(U. g•• ) according to G(P ) × A(P ) −1 −1 (T. Let G be a Lie group.1. symmetry group G.2 G-vector bundles Definition 8. C. which consists of symmetric. ρ). In other words. a Hermitian metric on a rank k complex vector bundle defines an U (k )structure on that bundle. . Example 8. i = 1. Consider S k (V ∗ ) ⊂ (V ∗ )⊗k . The representation ρ is in this case the natural injection O(r) → GL(r. The curvature of such a connection is skew-symmetric which shows the infinitesimal holonomy is an infinitesimal orthogonal/unitary transformation of a given fiber. CHARACTERISTIC CLASSES We denote a G-structure by the pair (P. . 2. then the collection ρ∗ (F (Aα )) coincides with the curvature F (ρ∗ (Aα )) of the connection ρ∗ (Aα ). This follows immediately from the polarization formula ϕ(v1 . Similarly. a connection compatible with some metric on a vector bundle is compatible with the orthogonal/unitary structure of that bundle.312 CHAPTER 8. Let E = (P.1. ρ. Let E → M be a rank r real vector bundle over a smooth manifold M . and the principal G-bundles Pi are isomorphic. ρ∗ : g → End(V ) denotes the derivative of ρ at 1 ∈ G. Assume P is defined by an open cover (Uα ). A metric on E allows us to talk about orthonormal moving frames. k ! ∂t1 · · · ∂tk . ρi ) on E . then the collection ρ∗ (Aα ) defines a connection on the vector bundle E . two different orthonormal local trivializations are related by a transition map valued in the orthogonal group O(r). They are easily produced from arbitrary ones via the Gramm-Schimdt orthonormalization technique. Above. . a metric on a bundle induces an O(r) structure. . . and gluing cocycle gαβ : Uαβ → G. . Note that any ϕ ∈ S k (V ∗ ) is completely determined by Pϕ (v ) = ϕ(v. . V ) be a G-vector bundle.3 Invariant polynomials the symmetric power Let V be a vector space over K = R. §8. so that a metric on a bundle allows one to replace an arbitrary collection of gluing data by an equivalent (cohomologous) one with transitions in O(r).1. If the collection {Aα ∈ Ω1 (Uα )⊗g} defines a connection on the principal bundle P . For example. multilinear maps ϕ : V × · · · × V → K. Note that if F (Aα ) is the curvature of the connection on P . if the representations ρi are isomorphic. . In particular. an O(r) structure on a rank r real vector bundle is tantamount to choosing a metric on that bundle. v ).8. vk ) = ∂k 1 Pϕ (t1 v1 + · · · + tk vk ). Conversely. are said to be isomorphic. Two G-structures (Pi . A connection of E obtained in this manner is said to be compatible with the G-structure. R). Then. For each T ∈ gl(n. we do get a symmetric function. . ck (T ) defines an element of S k (gl(n. · · · ) = −P (a2 X2 . · · · ). When restricted to the commutative subalgebra Ωeven (M ) = k≥0 Ω2k (M ) ⊗ C. c1 (T ) = − 1 tr T. A will be the algebra Ω• (M ) of complex valued differential forms on a smooth manifold M . x) x ∈ A ⊗ V. Starting with ϕ ∈ S k (V ∗ ) we can produce a K-multilinear map ϕ = ϕA : (A ⊗ V ) × · · · × (A ⊗ V ) → A.1. Remark 8.1. Let us emphasize that when A is not commutative. . . C)∗ ). then A ⊗ gl(n. A) of n × n matrices with entries in A. . . then P (a1 X1 . . a1 X1 . A) we have det  − λ T 2π i ∈ A[λ].10. Assume now that A is a K-algebra with 1. . if a1 a2 = −a2 a1 . (i = √ −1). CHERN-WEIL THEORY 313 If dim V = n then. If A is a commutative C-algebra with 1. . For example. cn (T ) = 2π i − 1 2π i n det T. Via polarization. then the above function is not symmetric in its variables.8. we can identify S k (V ∗ ) with the space of degree k homogeneous polynomials in n variables. . C).9. .1. ak ⊗ vk ) = ϕ(v1 . a2 X2 . ck (T ) is a degree k homogeneous polynomial in the entries of T . uniquely determined by ϕ(a1 ⊗ v1 . . vk )a1 a2 · · · ak ∈ A. . For each matrix T ∈ V we denote by ck (T ) the coefficient of λk in the characteristic polynomial cλ (T ) := det  − λ T 2π i = k ≥0 ck (T )λk . then ϕA is uniquely determined by the polynomial Pϕ (x) = ϕA (x. C) can be identified with the space gl(n. If moreover the algebra A is commutative. and ck (T ) continues to be the coefficient of λk in the above polynomial. For applications to geometry. For example. fixing a basis of V . Let V = gl(n. . Example 8. Xi ]. X1 ].314 CHAPTER 8.1. (8. Xk ]) = 0. . We denote by I k (G) the Ad-invariant elements of S k (g∗ ). Proof. . gXk g −1 ) = ϕ(X1 . X2 . . . . we use the “Taylor expansion” exp(X ) = k ≥0 1 k X . [X. . Xk ). [A. If P ∈ I k (g). U is an open subset of Rn . The space I •• (G) can be identified (as vector space) with the space of Ad-invariant formal power series with variables from g∗ . . C)) since tr (gXg −1 )k = tr gX k g −1 = tr X k . It consists of those ϕ ∈ S k (g∗ ) such that ϕ(gX1 g −1 . . C) X → tr exp(X ). Fk ) = (−1)d(d1 +···di−1 ) ωω1 · · · ωk P (X1 . · · · . k! which yields tr exp(X ) = k ≥0 1 tr X k . · · · Xk ). . C).11. . Example 8.12. Set I • (G) := k ≥0 I k (G) and I •• (G) := k≥0 I k (G). . . . Fi−1 . . The proposition follows immediately from the equality d |t=0 ϕ(etX X1 e−tX . . The adjoint action of G on its Lie algebra g induces an action on S k (g∗ ) still denoted by Ad. .9) . C)). k! For each k . . and Fi = ωi ⊗ Xi ∈ Ωdi (U ) ⊗ g. Let G = GL(n. . Xk ∈ g we have ϕ([X. dt Let us point out a useful identity. tr X k ∈ I k (GL(n. Then for any X. The map gl(n. Xk ∈ g. Xk ) + · · · + ϕ(X1 . . . . To see this.1. . . defines an element of I •• (GL(n. . . . [X. . Fi+1 . The elements of I • (G) are usually called invariant polynomials. A = ω ⊗ X ∈ Ωd (U ) ⊗ g then P (F1 . CHARACTERISTIC CLASSES Consider now a matrix Lie group G. etX Xk e−tX ) = 0. .1. . for all X1 . Let ϕ ∈ I k (G). X2 . . C). . Fi ]. . Proposition 8. so that g = gl(n. . . . X1 . . . Fk ) = 0. . Fα ) ∈ Ω2k (Uα ). . Fα ) − · · · − φ(Fα .1. Fi+1 . Fi ]. . . so that the locally defined forms Pφ (Fα ) patch-up to a global 2k -form on M which we denote by φ(FA ). . . . .10) ∀F1 . . 2 Given φ ∈ I k (G). Theorem 8. For 0 ≤ t ≤ 1 we define Aα ∈ Ω (Uα ) ⊗ g by Aα := Aα + tCα . In other words. . . . we deduce that for every i = 1. CHERN-WEIL THEORY 315 In particular. then the forms φ(FA0 ). (b) If A0 . we deduce Pφ (Fα ) = Pφ (Fβ ) on Uαβ . . . Fα ) + · · · + φ(Fα . Fk−1 ∈ Ωeven (U ) ⊗ g. Pick A ∈ A(P ) defined by the collection Aα ∈ Ω1 (Uα ) ⊗ g. . . Fi−1 . and a gluing cocycle gαβ : Uαβ → G. .8. . [X. . Xi ]. Fα . . k we have P (F1 . 1) be defined by the collections 1 Ai α ∈ Ω (Uα ) ⊗ g. . . Fα ) = φ(dFα . . . dFα ) = −φ([Aα . . Fk ) = ωω1 · · · ωk P (X1 . .1. Fα . Fi ]. [A. Fi−1 .1. . . . . . Aα ]. . . . and Fβ = gβα Fα gβα . 0 1 t t 0 Set Cα := A1 α − Aα . Assume P is defined by an open cover (Uα ).1. Fk . we deduce k P (F1 . Fα ]. . Xk ). i=1 (8. . and using the Ad-invariance of the polynomial P .13 (Chern-Weil). §8. . (b) Let Ai ∈ A(P ) (i = 0. . . Fα ]) = 0. −1 Because φ is Ad-invariant. . . Proof. . we can define as in the previous section (with A = Ωeven (Uα ). Fα ]. . . Fi+1 . Fα . The Leibniz’ rule yields dφ(Fα . . [Aα . . . and φ(FA1 ) are cohomologous. . . Summing over i. . .9) . the closed form φ(FA ) defines a cohomology class in H 2k (M ) which is independent of the connection A ∈ A(P ). (a) The form φ(FA ) is closed. Fα . We use the Bianchi identity dFα = −[Aα . . . V = g) Pφ (Fα ) := φ(Fα . . A ∈ Ω• (U ) ⊗ g. . ∀A ∈ A(P ).4 The Chern-Weil Theory Let G be a matrix Lie group with Lie algebra g. . [A. (8. . A1 ∈ A(P ). Its curvature is then defined by the collection 1 Fα = dAα + [Aα . if F1 . . and P → M be a principal G-bundle over the smooth manifold M .1. Fk−1 have even degree. Fα ) + · · · + φ(Fα . Cα ] ) = dφ( Fα . . .1. . . Fα . dFα . for all σ ∈ Sk and any ω1 . and t → A ∈ A(P ) is an (affine) path connecting A0 to A1 . . Fα . Fα . . . We claim that t t t t φ( Fα . . .1. Fα . Fα ) + · · · + φ(Fα . . · · · . . Cα ]. . Cα ] + t[Cα . 1 0 φ(Fα ) − φ(Fα )= 1 1 0 t t t t t ˙t ˙α φ(F . [Aα . . dCα ) + φ(dFα . Cα ]. Fα . [At α . Fα ) dt = 0 1 t t t t φ(dCα . . . . Consequently. . Fα . . . . Cα ] ) − φ( Fα . . Cα ) t t = φ( Fα . Fα . . . . Cα ] ) = φ( dCα + [Aα . Fα . . Note that ˙ t. Cα ]. . . (8. 1 0 Hence 1 0 φ(Fα ) − φ(Fα )=d t t ˙t kφ(A α . Fα ). Fα . Fα . · · · . . . . Fα ] ) t t t t t = φ( Fα . . . . . . . Fα . . . . . . Hence 1 0 φ(Fα ) − φ(Fα )=k 1 0 t t φ(Fα . . ωσ(k) ) = φ(ω1 . . . . dCα + [At α Cα ])dt. Fα . dCα ) t t t t t t t −φ( Cα . . Cα ] ) t t t t t t t −φ( [At α . Fα . . . . . . C = (Cα ) = A t ) the curvature of At .11) Hence. . Cα ] = dCα + [Aα . . . . . A simple computation yields We denote by F t = (Fα t 0 Fα = Fα + t(dCα + [A0 α . Fα ]. Fα ]. Fα . we deduce φ(ωσ(1) . . . Cα ) + · · · + φ(Fα . Fα . . . . Cα ) − · · · − φ( Fα . CHARACTERISTIC CLASSES t t The collection (At α ) defines a connection A ∈ A(P ). ωk ). Fα . 2 (8. . Fα ) − · · · − φ( Cα . . . . Using the Bianchi identity we get t t dφ( Fα . Fα . . . Fα . . . Fα . . .316 CHAPTER 8. Fα . Cα ]) dt. . . . . . ωk ∈ Ωeven (Uα ) ⊗ g. . . [Aα .12) . [Aα . . . . . Cα ). dCα + [At α . Fα . . Fα )dt. t t ˙α F = dCα + [A0 α . dCα ) dt + 0 t t t t t φ([At α . Fα . . . Cα ]) + t2 [Cα . · · · . Cα ) (8. .10) = t t t t t φ( Fα . Cα ]. [Aα .1. . dCα + [At α . dCα + [At α . Fα ) + · · · + φ(Fα . . . . Cα ) t t t t t t = φ(Fα . . Fα . Because the algebra Ωeven (Uα ) is commutative. . Fα . Fα ]. The Chern-Weil theorem is proved. . [ . We say that the Chern-Weil construction is gauge invariant. [ . [ . 2k−1 (2k )! 2k = (−1)k−1 2k · φ( . The transgression formula implies that the form 1 τφ = Tφ (d + .1. Observe that for every Ad-invariant polynomial φ ∈ I k (g). Denote by d the trivial connection on P0 . · · · . . ]. We have thus established the transgression formula φ(FA1 ) − φ(FA0 ) ) = d Tφ (A1 . and Fβ = gβα Fα gβα on Uαβ . (8. Clearly d is a flat connection. [ . Thus. Note that for every left invariant vector field X ∈ g we have (X ) = X.15. ]) ]) k 22k−1 k !(k − 1)! φ( . Fα . Aα ) := 0 317 1 t t ˙t kφ(A α . Consider a matrix Lie group G with Lie algebra g. the Maurer-Cartan equation implies that1 d + is also a flat connection. P0 = G × G. Fd+t )dt ∈ Ω2k−1 (G) is closed. ]. A simple computation using the Maurer-Cartan equations shows that τφ = k 2k−1 0 1 (t2 − 1)k−1 dt · φ( . ]. .14. and denote by P0 the trivial principal G-bundle over G. Fd+t . . Fα )dt. and any gauge transformation T ∈ G(P ) we have φ FA = φ FT AT −1 . [ . = (−1)k−1 k 1 The connections d and d + are in fact gauge equivalent. . for any k φ ∈ I (G) φ(Fd ) = φ(F (d+ ) = 0. . any principal G-bundle P . .1.1. Example 8. and we name it the φ-transgression from A0 to A1 .13) Remark 8. d) = k 0 φ( . −1 −1 Since Cβ = gβα Cα gβα . A0 ). . [ . · · · . Aα ) defines a global (2k − 1)-form on M which we denote by Tφ (A1 . we conclude from the Ad-invariance of 1 0 φ that the collection Tφ (Aα . A0 ). . · · · . Moreover.1.8. any connection A ∈ A(P ). ]). Denote by the tautological 1-form = g −1 dg ∈ Ω1 (G) ⊗ g. CHERN-WEIL THEORY We set 0 Tφ (A1 α . then φ(P ) = 0 ∈ H • (M ). When G is compact and connected.1. The Killing form κ is a degree 2 Ad-invariant polynomial on su(2). One can check easily the map cwP is a morphism of R-algebras.and gluing cocycle gαβ : Uαβ → G. Exercise 8.318 CHAPTER 8. (b) A connection A ∈ A(P ) defined by the collection Aα ∈ Ω1 (Uα ) ⊗ g. and gluing cocycle F −1 (Uαβ ) → Uαβ → G. Chevalley. We refer to [21] for a beautiful survey of this subject. Proposition 8. The pullback of a connection on P is defined similarly. then the pullback of P by F is the principal bundle F ∗ (P ) over M defined by the open cover F −1 (Uα ). Thus.16. Koszul states that the cohomology of G is generated as an R-algebra by the transgressive elements. Let M and N be two smooth manifolds. Hopf. Let G = SU (2). defined by an open cover (Uα ) . The following result should be obvious. Cartan. CHARACTERISTIC CLASSES We thus have a natural map τ : I • (G) → H odd (G) called transgression.1. (c) φ ∈ I k (G).18.3. Let us now analyze the essentials of the Chern-Weil construction. Output: A closed form φ(F (A)) ∈ Ω2k (M ). H. A.1. then a nontrivial result due to the combined efforts of H. C. If P is a principal G-bundle over N defined by an open cover (Uα ). Input: (a) A principal G-bundle P over a smooth manifold M . the principal bundle P defines a map. and F : M → N be a smooth map. The elements in the range of τ are called transgressive. for every principal G-bundle over N . and gluing cocycle gαβ : Uαβ → G. (a) If P is a trivial G-bundle over the smooth manifold M . Then. Definition 8. and any φ ∈ I • (G) we have φ(F ∗ (P )) = F ∗ (φ(P )). We denote this cohomology class by φ(P ). G Compare this result with the similar computations in Subsection 7.17.4. Describe τκ ∈ Ω3 (G) and then compute τκ . called the Chern-Weil correspondence cwP : I • (G) → H • (M ) φ → φ(P ). for any φ ∈ I • (G). F F gαβ . whose cohomology class is independent of the connection A. (b) Let M → N be a smooth map between the smooth manifolds M and N . Weil and L. satisfying the transition rules −1 −1 Aβ = gαβ dgαβ + gαβ Aα gαβ on Uαβ . . Remark 8. For each P ∈ PG we denote by BP the base of P ..8.20.19. and using topological techniques. and any principal Gbundle P → N . In the process we will describe the invariants of some commonly encountered Lie groups. one can produce H • (−.1. For details we refer to [75]. E.2 Important examples P → c(P ) ∈ F(BP ). 8. A). (b) There exist characteristic classes valued in contravariant functors other then the ˇ DeRham cohomology. the Cech cohomology with coefficients in the constant sheaf A defines a contravariant functor H • (−. this means the diagram below is commutative. or the classical [93]. A)-valued characteristic classes. (a) c(P ) = 0 if P is trivial. (b) F(F )(c(P )) = c(F ∗ (P )). (b) A very legitimate question arises. (a) We see that each characteristic class provides a way of measuring the nontriviality of a principal G-bundle. Definition 8. but the proof requires an elaborate topological technology which is beyond the reach of this course.2. Do there exist characteristic classes (in the DeRham cohomology) not obtainable via the Chern-Weil construction? The answer is negative.1. We devote this section to the description of some of the most important examples of characteristic classes. Finally. for any smooth map F : M → N . Hence. An F-valued G-characteristic class is a correspondence PG such that the following hold. for each Abelian group A.g. IMPORTANT EXAMPLES Equivalently. we denote by F a contravariant functor from the category of smooth manifolds (and smooth maps) to the category of Abelian groups. The interested reader can find the details in the monograph [75] which is the ultimate reference on the subject of characteristic classes. the Chern-Weil construction is just a method of producing G-characteristic classes valued in the DeRham cohomology. I • (G) 319 cwF • (P ) M H (N )   KF cwP • • H • (M ) Denote by PG the collection of smooth principal G-bundles (over smooth manifolds). ∇X λu. C). A connection on this U (r)-bundle is then equivalent with a linear connection ∇ on E compatible with a Hermitian metric •. xn ]. . and set xj := − 1 dθj . . . we can identify such a bundle with the tautological principal U (r)-bundle of unitary frames. Hence I • (T n ) = S • ((tn )∗ ). 2π i The xj s form a basis of (tn )∗ . To describe these characteristic classes we need to elucidate the structure of the ring of invariants I • (U (r)). such that T XT −1 is diagonal T XT −1 = i diag(λ1 . The characteristic classes of E are by definition the characteristic classes of the tautological principal U (r)-bundle. so that the adjoint action on its Lie algebra tn is trivial. • . . 1 ≤ i ≤ n. ∀λ ∈ C. . The Lie algebra u(r) consists of r × r complex skew-hermitian matrices. invariant with respect to the adjoint action u(r) X → T XT −1 ∈ u(r).2 Chern classes Let E be a rank r complex vector bundle over the smooth manifold M . Prove that different Hermitian metrics on E define isomorphic U (r)structures. .2. ∇X v }. and now we can identify I • (T n ) ∼ = R[x1 . We have seen that a Hermitian metric on E induces an U (r)-structure (P. X ). v = λ{ ∇X u.1. for any X ∈ u(r). CHARACTERISTIC CLASSES §8. Classical results of linear algebra show that. In practice one uses a more explicit description obtained as follows. Pick angular coordinates 0 ≤ θi ≤ 2π . The ring I • (U (r)) consists of symmetric. . . Exercise 8. . where ρ is the tautological representation ρ : U (r) → GL(r.2. . §8. Thus. r-linear maps φ : u(r) × · · · × u(r) → R.320 CHAPTER 8.e. . . It is convenient to identify such a map with its polynomial form Pφ (X ) = φ(X. there exists T ∈ U (r). i. v + u. .1 The invariants of the torus T n The n-dimensional torus T n = U (1) × · · · × U (1) is an Abelian Lie group. X ∈ Vect (M ). T ∈ U (r). ρ).2.. λr ). v ∈ C ∞ (E ). u. .4. The Weyl group Sr permutes these variables. T XT −1 = X. so we can form the quotient WU (r) := NU (r) /SU (r) called the Weyl group of U (r). . It is a (maximal) Abelian Lie subalgebra of u(r). xr ) is a symmetric polynomial in its variables. . . . . . . 2π i T ∈ U (r) . .8. . so that Pφ is Ad-invariant if and only if Pφ (x1 . . . Consider the stabilizer SU (r) := and the normalizer NU (r) := T ∈ U (r). we obtain the Chern classes of E ck (E ) := ck (F (∇)) ∈ H 2k (M ). . As in the previous subsection we introduce the variables xj := − 1 dθj . and its coefficients are called the universal. . The restriction of Pφ to Cu(r) is a polynomial in the variables x1 . . . The stabilizer SU (r) is a normal subgroup of NU (r) . .4. cr = Thus I • (U (r)) = R[c1 .2. rank r Chern classes. . As in Subsection 7. cr ]. ∀X ∈ Cu(r) . including multiplicities. The Cartan algebra is the Lie algebra of the (maximal) torus T n consisting of diagonal unitary matrices. According to the fundamental theorem of symmetric polynomials. We see that Pφ is Ad-invariant if and only if its restriction to the Cartan algebra is invariant under the action of the Weyl group. . and we will denote it by Cu(r) . . The above polynomial is known as the universal rank r Chern polynomial. T Cu(r) T −1 ⊂ Cu(r) . . Returning to our rank r vector bundle E . . we see that the Weyl group is isomorphic with the symmetric group Sr because two diagonal skew-Hermitian matrices are unitarily equivalent if and only if they have the same eigenvalues. the ring of these polynomials is generated (as an R-algebra) by the elementary ones c1 = j xj c2 = i<j xi xj . c2 . xr . IMPORTANT EXAMPLES 321 The set of diagonal matrices in u(r) is called the Cartan algebra of u(r). . In terms of matrices X ∈ u(r) we have ck (X )tk = det  − k x1 · · · xr t X 2π i ∈ I • (U (r))[t]. ˇ In other words. This terminology is motivated by the following result of Kodaira and Spencer. The class δ (L).2.2. Upon choosing a good open cover (Uα ) of M we can describe L by a collection of smooth maps zαβ : Uαβ → U (1) ∼ = S 1 satisfying the cocycle condition zαβ zβγ zγα =  ∀α. It is not difficult to see that. The long exact sequence in cohomology then gives 0 → Pic∞ (M ) → H 2 (M. S 1 ) the sheaf of multiplicative groups of smooth S 1 -valued functions. To give the reader a feeling of what the Chern-Weil construction is unable to capture we will sketch a different definition of the first Chern class of a complex line bundle. R)) we deduce from (8. R) exp(2π i·) −→ C ∞ (·. • on E . Let L → M be a smooth complex Hermitian line bundle over the smooth manifold M . ∀Uαβγδ = ∅. The Chern classes produced via the Chern-Weil method capture only a part of what topologists usually refer to characteristic classes of complex bundles. L ∈ Pic∞ (M ) is called the topological first Chern class and is denoted by ctop 1 (L). The group multiplication is precisely the tensor product of two line bundles. while F (∇) denotes its curvature. The cocycle ( nαβγ ) represents precisely the class δ (L). The middle sheaf is a fine sheaf so its cohomology vanishes in positive dimensions. Above. If we write zαβ = exp(2π iθαβ ) (θβα = −θαβ ∈ C ∞ (Uαβ . The following facts are essentially due to Kodaira and Spencer [61]. S 1 ) → 0.2. This group is called the smooth Picard group of M . ˇ we see that the family of complex line bundles on M can be identified with the Cech 1 ∞ 1 group H (M. C (·.2. ∇ denotes a connection compatible with a Hermitian metric •. On a more formal level.322 and the Chern polynomial of E CHAPTER 8. we can capture the above cocycle starting from the exact sequence of sheaves 0 → Z → C ∞ (·. We will denote it by Pic∞ (M ). Z) → 0. we have nβγδ − nαγδ + nαβδ − nαβγ = 0. γ. [61]: δ . (8. the collection nαβγ defines a Cech 2-cocycle of the constant sheaf Z.1) If we denote by C ∞ (·. see also [40] for a nice presentation. S )).1) that ∀Uαβγ = ∅ θαβ + θβγ + θγα = nαβγ ∈ Z. CHARACTERISTIC CLASSES ct (E ) := det  − t F (∇) 2π i ∈ H • (M )[t]. Remark 8. β. r = 2m + 1 0 −1 1 0 . λj ∈ R} .2. R) ∼ (M ) = HDR 323 coincides with the first Chern class obtained via the Chern-Weil procedure. As usual. An Euclidean metric on E induces an O(r) structure (P. 2π ] we denoted by Rθ the 2 × 2 rotation Rθ := cos θ − sin θ sin θ cos θ . Prove that two metrics on E induce isomorphic O(r)-structures. Ad-invariant polynomials on the Lie algebra o(r) consisting of skew-symmetric r × r real matrices. A connection on the principal bundle can be viewed as a metric compatible connection in the associated vector bundle. r = 2m . For example. if a line bundle admits a flat connection then its first Chern class is trivial. Z). §8. IMPORTANT EXAMPLES The image of ctop 1 (L) in the DeRham cohomology via the natural morphism ∗ H ∗ (M. As in Subsection 8. The Cartan algebra Co(r) is the Lie algebra of the (maximal) torus Tm = Rθ1 ⊕ · · · ⊕ Rθm ∈ O(r) . Fix P ∈ I k (O(r)). Exercise 8. exactly as in the complex case. ρ). Set m = [r/2]. To describe the various characteristic classes we need to understand the ring of invariants I • (O(r)). R). 2π . The Chern-Weil construction misses precisely the torsion elements in H 2 (M. Hence.1 we introduce the variables xj = − 1 dθj .2. r = 2m {λ1 J ⊕ · · · ⊕ λm J ⊕ 0 ∈ o(r) .3. because line bundle may not be topologically trivial.3 Pontryagin classes Let E be a rank r real vector bundle over the smooth manifold M .2.8. Z) → H ∗ (M. and denote by J the 2 × 2 matrix J := Consider the Cartan algebra Co(r) = {λ1 J ⊕ · · · ⊕ λm J ∈ o(r) . λj ∈ R} . Rθ1 ⊕ · · · ⊕ Rθm ⊕ R ∈ O(r) . The representation ρ is the tautological one ρ : O(r) → GL(r.2. we will identify the elements of I k (O(r)) with the degree k . r = 2m + 1 where for each θ ∈ [0. This may not be the case with the topological one. we can naturally identify the rank r-real vector bundles equipped with metric with principal O(r)-bundles. . pj (X )t2j = det  − t X 2π ∈ I • (O(r))[t]. separately even in each variable. . . . . . we deduce that. xm invariant under the action of the Weyl group. . . . while its coefficients pj (X ) are called the universal rank r Pontryagin classes. . . . Consequently. . R) generated by the involutions σij : (x1 . Invoking once again the fundamental theorem of symmetric polynomials we conclude that P must be a polynomial in the elementary symmetric ones p1 p2 . . T Co(r) T −1 ⊂ Co(r) . . Hence I • (O(r)) = R[p1 . . . Exercise 8. there exists T ∈ O(r) such that T XT −1 ∈ Co(r) . Using the above exercise we deduce that P must be a symmetric polynomial P = P (x1 .324 CHAPTER 8. xi . . The Pontryagin classes p1 (E ). The restriction of P ∈ I k (O(r)) to Co(r) is a degree k homogeneous polynomial in the variables x1 . xm ) → (x1 . . . xm ). Prove that WO(r) is the subgroup of GL(m.2. . . . where ∇ denotes a connection compatible with some (real) metric on E . xm ) εj : (x1 . Note that pj (E ) ∈ H 4j (M ). we consider SO(r) = T ∈ O(r). T XT −1 = X. . NO(r) = The stabilizer SO(r) is a normal subgroup in N(O(r)). . T ∈ O(r). . . . . . . . any Ad-invariant polynomial on o(r) is uniquely defined by its restriction to the Cartan algebra. . p In terms of X ∈ o(r) we have pt ( X ) = j r/2 = = . . . . . . −xj . . . . . xj . . . . . ∀X ∈ Co(r) . pm (E ) of our real vector bundle E are then defined by the equality pt (E ) = j pj (E )t2j = det  − t F (∇) 2π ∈ H • (M )[t]. xj . . while F (∇) denotes its curvature. . . . . j 2 x2 1 · · · xm pm = ]. The polynomial pt (E ) ∈ H • (M )[t] is called the Pontryagin polynomial of E . . . Following the approach in the complex case. xj . . x2 j 2 2 i<j xi xj . The above polynomial is called the universal rank r Pontryagin polynomial. . . CHARACTERISTIC CLASSES Using standard results concerning the normal Jordan form of a skew-symmetric matrix. xi . . . . . . so we can form the Weyl group WO(r) := NO(r) /SO(r) . . .4. xm ). xm ) → (x1 . for every X ∈ o(r). . . . . λm ) → (ε1 λ1 . where ε1 . and ε1 · · · εm = 1. The inclusion ı : SO(r) → O(r). IMPORTANT EXAMPLES 325 §8. ∆(x1 . λj . . When r is odd then ı∗ : I • (O(r)) → I • (SO(r)) is an isomorphism. A metric on E induces an O(r)-structure. . Exercise 8. . The Pontryagin O(2m)-invariants pj (x1 . . . . Lemma 8. εm λm ). where P f denotes the pfaffian of the skewsymmetric matrix X viewed as a linear map R2m → R2m . namely an SO(r)-symmetry. (Check this!) Set as usual xi := −λi /2π . . . The groups O(r) and SO(r) share the same Lie algebra so(r) = o(r). . . . . λm ) → (λ1 . . . . In terms of X = λ1 J ⊕ · · · ⊕ λm J ∈ CSO(2m) . . . . . . . . . Prove the above lemma. I • (SO(2m)). . . . . defined as usual as the quotient WSO(2m) = NSO(2m) /SSO(2m) . . . The situation is different when r is even. λi . λi . but the existence of an orientation implies the existence of a finer structure. induces a morphism of R-algebras ı∗ : I • (O(r)) → I • (SO(r)). and ε : (λ1 .2.6. is isomorphic to the subgroup of GL(Co(2m) ) generated by the involutions σij : (λ1 . . . . .2. . . λj . . εm = ±1. real oriented vector bundle. . Because so(r) = o(r) one deduces immediately that ı∗ is injective. 1≤i1 <···<ij ≤m continue to be WSO(2m) invariants. Note that pm = ∆2 . . λm ). .8. To describe the ring of invariants we need to study in greater detail the Cartan algebra Co(2m) = λ1 J ⊕ · · · ⊕ λm J ∈ o(2m) and the corresponding Weyl group action. . xm ) = (xi1 · · · xij )2 . The Weyl group WSO(2m) .2. . when R2m is endowed with the canonical orientation.5. . .2. There is however a new invariant. we can write ∆(X ) = −1 2π m P f (X ). xm ) := j xj .4 The Euler class Let E be a rank r. r = 2m. . . xm ). e} acts on I • (SO(2m)) by (eF )(x1 . For each F ∈ I • (SO(2m)) we define F + := ( + e)F . . . . . Step 1.e. Thus WSO(2m) is a normal subgroup. Zm . .2). −xm ). Step 1 follows from 1 1 F = (F + + F − ) = (P (p1 . Indeed. . . . . Proof. . . Z2 . Note that WSO(2m) has index 2 as a subgroup in WO(2m) . . and moreover. pm )). and e∆ = −∆ we deduce eG = G. . . Y ] → I • (SO(2m)) defined by Zj → pj . xm ) = F (−x1 . . . . . On the other hand. Since eF − = −F − . The group G = {. CHAPTER 8. . . . . Y → ∆ is the ideal (Y 2 − Zm ). . F − = ∆ · G. Weyl in describing the invariants of the alternate group ([100]. Consequently. . . pm ). x2 .326 Proposition 8. The space I • (SO(2m)) is generated as an R-algebra by the polynomials p1 .. . xm ) = eF − (x1 . . Y = ∆) where (Y 2 − Zm ) denotes the ideal generated by the polynomial Y 2 − Zm . . . x2 . . . . . . . . Proof of Step 1. . xm ) = −( − e)F (x1 . The kernel of the morphism R[Z1 . . . . xm ) = · · · = F (x1 . . . the polynomial F − := ( − e)F is separately odd in each of its variables. Step 2. . x2 . . The isomorphism will be established in two steps. . · · · . We follow the approach used by H. xm ) = −F − (x1 . Sec. pm ) + ∆ · Q(p1 . and G = WO(2m) /WSO(2m) ∼ = Z2 . i. I • (O(2m)) = ker( − e).2. . . . . G can be written as G = Q(p1 . . .7. pm ). II. . Zm . ∆. pm ). pm . Hence F + = P (p1 . F − vanishes when any of its variables vanishes so that F − is divisible by their product ∆ = x1 · · · xm . and we observe that F + ∈ ker(1 − e). . x2 . . . . 2 2 ψ . CHARACTERISTIC CLASSES I • (SO(2m)) ∼ = R[Z1 . . Y ]/(Y 2 − Zm ) (Zj = pj . . G ∈ I • (O(2m)). xm ) = e( − e)F (x1 . Hence. . so that F − = ∆ · Q(p1 . F − (−x1 . . .2. Divide P by the quadratic polynomial (in Y ) Y 2 − Zm . . . From the equality det X = P f (X )2 ∀X ∈ so(2m). . pm )∆ + B (p1 . IMPORTANT EXAMPLES Proof of Step 2. . . As in the previous subsection we deduce that we can use a metric to naturally identify E with a principal SO(2m)-bundle and in fact. is the cohomology class e(E ) ∈ H 2m (M ) represented by the Euler form 1 P f (−F (∇)) ∈ Ω2m (M ). we deduce R ∈ ker ψ . . Thus A(p1 . . . 327 so that we only need to establish the opposite inclusion. Let P = P (Z1 . . . . . . . Zm ]. . choose a connection ∇ compatible with some metric on E .2. e(∇) = (2π )m (According to the Chern-Weil theorem this cohomology class is independent of the metric and the connection. P ∈ ker ψ . Consider P as a polynomial in Y with coefficients in R[Z1 .8.9. Zm ). Y ) ∈ ker ψ . . . . Definition 8. this principal bundle is independent of the metric.) Example 8. . (2π )m (b) The Euler class of E . . . g ) be a compact. . Riemann surface and denote by ∇g the Levi-Civita connection. . we deduce (Y 2 − Zm ) ⊂ ker ψ. Z2 . . pm ) = 0. Zm . oriented.2. Hence A ≡ B ≡ 0 so that P is divisible by Y 2 − Zm . Applying the morphism e we get −A(p1 .8. real. . . pm ) = 0. Zm )Y + B (Z1 . Let E be a rank 2m. . . Let (Σ. . oriented vector bundle over the smooth manifold M . The remainder is linear R = A(Z1 . Since Y 2 − Zm . . . The the Euler form ε(g ) = 1 s(g )dvg 4π coincides with the Euler form e(∇g ) obtained via the Chern-Weil construction. . Finally. pm )∆ + B (p1 . (a) The universal Euler class is defined by e = e(X ) = 1 P f (−X ) ∈ I m (SO(2m)). (a) Let R[[x]] denote the set of formal power series F ∈ R[[x]] such that F (0) = 1. We begin by first describing a special example of inverse limit. Let E be a rank 2m. . .5 Universal classes In each of the situations discussed so far we defined characteristic classes for vector bundles with a given rank. . oriented manifold M .2.]]. A projective sequence of rings is a a sequence of rings {Rn }n≥0 .10. We now have two apparently conflicting notions of Euler classes. we can from the sequence of sums (0. φn ) is the subring lim Rn ⊂ Rn ← n≥0 φn consisting of the sequences (x1 . . Similarly. In this subsection we show how one can coherently present these facts all at once. .) such that φn (xn+1 ) = xn . oriented vector bundle over the smooth. X2 . Xn ]] be the ring of formal power series in n variables with coefficients in the commutative ring with unit R. When G1 = G2 = · · · = Fn = · · · = F we denote the corresponding element (G)∞ . G1 (X1 ) + G2 (X2 ).2. Let Rn = R[[X1 . . . we can form the sequence of products (1. . . . . which will be established later in this chapter. When F1 = F2 = · · · Fn = · · · = F the corresponding elements is denoted by (F )∞ . X2 .).) Denote by φn : Rn+1 → Rn the natural morphism defined by setting Xn+1 = 0. (b) Prove that ∀F. Prove that (R[[x]] . The inverse limit of a projective system (Rn . . . compact. . . . . real. which defines an element in R[[X1 . G1 (X1 ). together with a sequence of ring morphisms Rn ←− Rn+1 . which defines an element in R[[X1 . Exercise 8. . . G ∈ R[[x]] (F · G)∞ = (F )∞ · (G)∞ .328 CHAPTER 8. given Gn ∈ R[[x]]. . Given a sequence Fn ∈ R[[X ]] (n ≥ 1) such that Fn (0) = 1. (R0 = R. . X2 . . will show that these two notions coincide! §8. ·) is an abelian group.2. .]] denoted by G1 (X1 ) + G2 (X2 ) + · · · . ∀n ≥ 0. CHARACTERISTIC CLASSES Remark 8.11. (n ≥ 1) such that Fn (0) = 0. The algebraic machinery which will achieve this end is called inverse limit. . A topological Euler class etop (E ) ∈ H 2m (M ) defined as the pullback of the Thom class via an arbitrary section of E . . . . G1 (X1 ) + · · · ⊕ Gn (Xn ). F1 (X1 ). . Example 8. A geometric Euler class egeom (E ) ∈ H 2m (M ) defined via the Chern-Weil construction.).2. irrespective of rank. . x2 . . F1 (X1 ) · · · Fn (Xn ). F1 (X1 )F2 (X2 ).12. . The inverse limit of this projective system is denoted by R[[X1 . The most general version of the Gauss-Bonnet theorem.]] denoted by F1 (X1 )F2 (X2 ) · · · . . The sequence in (8. . . (F )∞ ∈ S∞ [[xj ]] [[t]] and moreover (F )∞ (x1 . . we can write ck = xi1 xi2 · · · xik . Then (F )∞ defines an element in R[[X1 . . An element φ = (φ1 . . IMPORTANT EXAMPLES 329 In dealing with characteristic classes of vector bundles one naturally encounters the increasing sequences U (1) → U (2) → · · · (8. where R = R[[t]] is the ring of formal power series in the variable t. ck . . x2 . ck (k ) (k+1) . . φ2 . . for any u ∈ Ω∗ (M ). X2 . Example 8.) defines an element in S∞ [[xj ]] denoted by ck . 1≤i1 <···<ik ≤n (n) Then the sequence (0. (8. 1≤i1 <···<ik <∞ We can present the above arguments in a more concise form as follows. . . the characteristic class φ(E ) is well defined since udim M +1 = 0. More precisely.2. .4) (8. which we call the universal k -th Chern class.) ∈ I •• (U (∞)) := lim I •• (U (n)) = S∞ [[xj ]] ← is called universal characteristic class . . where r = rank E . to define φr (E ) we need to pick a connection ∇ compatible with some Hermitian metric on E . and φ ∈ I •• (U (n)).3) We know that I •• (U (n)) = Sn [[xj ]]= the ring of symmetric formal power series in n variables with coefficients in R. We denote by ck (n ≥ k ) the elementary symmetric polynomial in n variables (n) ck = xi1 · · · xik . . 0.]]. .2. Formally.) = (1 + tx1 )(1 + tx2 ) · · · = 1 + c1 t + c2 t2 + · · · .2. Consider the function F (x) = (1 + tx) ∈ R[[x]]. If E is a complex vector bundle. and then set φ(E ) = φr (F (∇)). We will discuss these two situations separately.8.2) (in the complex case) and (in the real case) O(1) → O(2) → · · · . The complex case. . ← Given a rank n vector bundle E over a smooth manifold M . One sees immediately that in fact.2) induces a projective sequence of rings R ← I •• (U (1)) ← I •• (U (2)) ← · · · .2. we set φ(E )” = φr (E ).13.2.2. Thus we can work with the ring I •• (U (n)) rather than I • (U (n)) as we have done so far. Set S∞ [[xj ]] := lim Sn . . We have two examples in mind. . . x2 . We denote it by Td. where Tdn ∈ S∞ [[xj ]] is an universal symmetric. P2 .] = lim R[x1 . . . CHARACTERISTIC CLASSES (1 + txj ) j ≥1 is the universal Chern polynomial. By an universal“polynomial” we understand element in the inverse limit R[x1 . x2 . ← An universal “polynomial” P is said to be homogeneous of degree d. (2k )! The coefficients Bk are known as the Bernoulli numbers. c2 . . . . Td2 = (c2 1 + c2 ). ·) F → (F )∞ ∈ (S∞ [[xj ]]. . . .. hence expressible as a combination of the elementary symmetric “polynomials” c1 . Td3 = 2 12 24 Analogously. We get the symmetric function k sk = (xk )∞ = (G)∞ = xk 1 + x2 + · · · . We can perform a similar operation with any F ∈ R[[x]] such that F (0) = 1. any function G ∈ R[[x]] such that G(0) = 0 defines an element (G)∞ = G(x1 ) + G(x2 ) + · · · ∈ S∞ [[xj ]]. and Pm+1 (x1 . xm ). 0) = Pm (x1 . . where Pm is a homogeneous polynomial of degree d in m variables. consider G(x) = xk . homogeneous “polynomial” of degree n. .]. 1 1 1 c1 c2 etc.). Using the fundamental theorem of symmetric polynomials we can write Td = 1 + Td1 + Td2 + · · · .12. The product (F )∞ = x1 1 − e−x1 · x2 1 − e−x2 ··· . . . First. .330 We see that ct := CHAPTER 8. For example. if it can be represented as a sequence P = (P1 . defines an element in S∞ [[xj ]] called the universal Todd class. . We get a semigroup morphism (R[[x]] . x2 . ·). (see Exercise 8.2. . Td1 = c1 . xm . .) One very important example is F (x) = x 1 1 1 = 1 + x + x2 + · · · = 1 + x + − x 1−e 2 12 2 ∞ (−1)k−1 k=1 Bk 2k x ∈ R[[x]]. . . k! where ∧ is the bilinear map Ωi (End (E )) × Ωj (End (E )) → Ωi+j (End (E )) defined in Example 3. consider xm G(x) = ex − 1 = .12. from which we deduce the first equality. Hence exp(F (∇1 ⊕ ∇2 )) = exp(F (∇1 )) ⊕ exp(F (∇2 )). If ∇ is a connection on E compatible with some Hermitian metric. 2. Proposition 8. The cohomology class ch(E ) is called the Chern character of the bundle E . consider the connection ∇ on E1 ⊗ E2 uniquely defined by the product rule ∇(s1 ⊗ s2 ) = (∇1 s1 ) ⊗ s2 + s1 ⊗ (∇2 s2 ).8. where the operation ⊗ : Ωk (E1 ) × Ω (E2 ) → Ωk+ (E1 ⊗ E2 ) . i = 1.14. then we can express the Chern character of E as ∞ ch(E ) = tr eF (∇) = rank (E ) + k=1 1 tr (F (∇)∧k ). Then ch(E1 ⊕ E2 ) = ch(E1 ) + ch(E2 ). Proof. and moreover F (∇1 ⊕ ∇2 ) = F (∇1 ) ⊕ F (∇2 ). and ch(E1 ⊗ E2 ) = ch(E1 ) ∧ ch(E2 ) ∈ H • (M ). We will denote these by sk . Then ∇1 ⊕∇2 is a connection on E1 ⊕ E2 compatible with the metric h1 ⊕ h2 . m! m≥1 331 We have (ex − 1)∞ = m≥1 1 m (x )∞ = m! m=1 1 sm ∈ S∞ [[xj ]] m! Given a complex vector bundle E we define ch(E ) = rank E + (ex − 1)∞ (E ).3. si ∈ C ∞ (Ei ). Next. As for the second equality. IMPORTANT EXAMPLES called the universal k-th power sum. E2 over the same manifold M . Consider two complex vector bundles E1 .2. Consider a connection ∇i on Ei compatible with some Hermitian metric hi .2. xn ]. . We compute the curvature of ∇ using the equality F (∇) = (d∇ )2 . µm )). r . Then tr (exp(A ⊗ Cm + Cn ⊗ B )) = tr (exp(A)) · tr (exp(B )). . λn ) and B = diag(µ1 . .16. Pick an orthonormal basis (ei ) of Cn and an orthonormal basis (fj ) of Cm ) such that. . .) Consider the symmetric polynomials ck = 1≤i1 ≤···ik ≤n eλi eµj = tr (eA ) · tr (eB ). and (r ≥ 0) sr = j xr j ∈ R[x1 . The second equality in the proposition is a consequence the following technical lemma.2. ∀ω k ∈ Ωk (M ). then F (∇)(s1 ⊗ s2 ) = d∇ {(∇1 s1 ) ⊗ s2 + s1 ⊗ (∇2 s2 )} = (F (∇1 )s1 ) ⊗ s2 − (∇1 s1 ) ⊗ (∇2 s2 ) + (∇1 s1 ) ⊗ (∇2 s2 ) + s1 ⊗ (F (∇2 )) = F (∇1 ) ⊗ E2 + E1 ⊗ F (∇2 ). Lemma 8. Proof. . Then. n Set f (t) := (a) Show that (1 − xj t).2. . with respect to these bases A = diag(λ1 . . and ∀η ∈ Ω (M ). Exercise 8. so that. xn ]. j =1 f (t) =− f (t) s r tr − 1 . n × n (respectively m × m) complex matrix. . Let A (respectively B ) be a skew-adjoint. with respect to the basis (ei ⊗ fj ) of Cn ⊗ Cm . tr (A ⊗ Cm + Cn ⊗ B ) = The proposition is proved.2.15. Hence exp(A ⊗ Cm + Cn ⊗ B ) = diag(eλi eµj ). . . .5) ∀si ∈ C ∞ (Ei ). .332 is defined by CHAPTER 8. xi1 · · · xik ∈ R[x1 . CHARACTERISTIC CLASSES (ω k ⊗ s1 ) ⊗ (η s2 ) = (ω k ∧ η ) ⊗ (s1 ⊗ s2 ) (8. we have (A ⊗ Cm + Cn ⊗ B ) = diag( λi + µj ). . . If si ∈ C ∞ (Ei ). (Newton’s formulæ. . As before.3) induces a projective system I •• (O(1)) ← I •• (O(2)) ← · · · . s3 = c1 − 3c1 c2 + 3c3 . In fact. The sequence (8. A2 = 7 2 (−4p2 + 7p2 1 ) etc. j =1 (c) Deduce from the above formulae the following identities between universal symmetric polynomials. (F (x))∞ The first couple of terms are A1 = − B. even.8. using metric compatible connections. any element of this ring is called a universal characteristic class. =1+ F (x) = 2 (2k )! sinh( x/2) k≥1 The universal characteristic class is denoted by A. L2 = (7p2 − p2 1 ) etc. Consider 1 p1 . symmetric power series in n/2 variables. If F ∈ R[[x]] then (F (x2 ))∞ defines an element of S∞ [[x2 j ]]. IMPORTANT EXAMPLES (b) Prove the Newton formulæ r 333 (−1)j sr−j cj = 0 ∀1 ≤ r ≤ n. 24 2 ·3 ·5 (−1)k−1 k≥1 √ x 1 1 √ = 1 + x + x2 + · · · = 1 + F (x) = 3 45 tanh x 22k Bk xk . the most commonly encountered situations are the following. s2 = c2 1 − 2c2 . even. symmetric. The inverse limit of this system is m 2 I • • (O(∞) = lim I •• (O(r)) = S∞ [[x2 j ]] := lim S [[xj ]]. 3 s1 = c1 . The first few terms are 1 1 L1 = p1 . and as such they can be described using universal Pontryagin classes pm = 1≤j1 <···jm 2 x2 j1 · · · xjm . where Ak are universal. ← ← As in the complex case. √ x/2 22k−1 − 1 √ (−1)k 2k−1 Bk xk . where the Lj ’s are universal. homogeneous “polynomials”. symmetric. for any real vector bundle E and any φ ∈ S∞ [[x2 j ]] there is a well defined characteristic class φ(E ) which can be expressed exactly as in the complex case. 3 45 . and it is called the A-genus. They can be expressed in terms of the universal Pontryagin classes. homogeneous “polynomials”. we can write L = 1 + L1 + L2 + · · · . In topology.2.2. A. The real case. and it is called the L-genus. We have proved that I •• (O(n)) = S n/2 [[x2 j ]] = the ring of even. (2k )! The universal class (F (x2 ))∞ is denoted by L. We can write A = 1 + A1 + A2 + · · · . We will discuss in some detail a few invariant theoretic methods and we will present one topological result more precisely the Gauss-Bonnet-Chern theorem. In particular.2. In terms of the Whitney splitting E = E1 ⊕ E2 above. Then for every η ∈ ker ϕ∗ we have η (P ) = 0 in H ∗ (M ). For example. Let ϕ : H → G be a smooth morphism of (matrix) Lie groups. and using the G structure. then E . P is a principal G bundle. ρ). More generally.3. Let P be a principal G-bundle which can be reduced to a principal H -bundle Q.1. . One natural question is whether there is any relationship between them. (b) A principal G-bundle Q over M is said to be ϕ-reducible.334 8. can be given a finer structure of U (r1 ) × U (r2 ) vector bundle. In this section we will describe some methods of doing this. if a rank r complex vector bundle E splits as a Whitney sum E = E1 ⊕ E2 with rank Ei = ri .3 CHAPTER 8. Proposition 8. (a) If P is a principal H -bundle over the smooth manifold M defined by the open cover (Uα ). and ρ : G → U (r) is a representation of G. The following result is immediate. where G is a Lie group. Most concrete applications require the aplication of a combination of techniques from topology. differential and algebraic geometry and Lie group theory that go beyond the scope of this book. we obtain two collections of characteristic classes associated to E . which has a natural U (r).3. Then we can perform two types of Chern-Weil constructions: using the U (r) structure. §8. The morphism ϕ : H → G in the above definition induces a morphism of R-algebras ϕ∗ : I • (G) → I • (H ). assume that a given rank r complex vector bundle E admits a Gstructure (P. Definition 8. then the principal G-bundle defined by the gluing cocycle gαβ = ϕ ◦ hαβ : Uαβ → G is said to be the ϕ-associate of P . and gluing cocycle hαβ : Uαβ → H.1 Reductions In applications. CHARACTERISTIC CLASSES Computing characteristic classes The theory of characteristic classes is as useful as one’s ability to compute them.3. if there exists a principal H -bundle P → M such that Q = ϕ(P ). the problem takes a more concrete form: compute the Chern classes of E in terms of the Chern classes of E1 and E2 . Our next definition formalizes the above situations.symmetry. and it is denoted by ϕ(P ). The elements of ker ϕ∗ ⊂ I • (G) are called universal identities. the symmetries of a vector bundle are implicitly described through topological or geometric properties. Equivalently. rather than a rigid result. Example 8. COMPUTING CHARACTERISTIC CLASSES 335 Proof. We have t (r+s) ı∗ (ct )(Xr ⊕ Xs ) = det r+s − Xr ⊕ Xs 2π i = det r − t Xr 2π i · det s − t Xs 2π i = ct · ct (s). g ⊕ h is a Hermitian metric on E ⊕ F . E ⊕ F has an U (r + s) structure reducible to an U (r) × U (s) structure. and by ϕ∗ the differential of ϕ at 1 ∈ H .3. We conclude this subsection with some simple. and denote by (Fα ) its curvature.1) now follows using the argument in the proof of Proposition 8. Let E and F be two complex vector bundles over the same smooth manifold M of ranks r and respectively s.3. Now η (ϕ∗ (Fα )) = (ϕ∗ η )(Fα ) = 0. but very important applications of the above principle. Pick a connection (Aα ) on Q. Denote by LG (respectively LH ) the Lie algebra of G (respectively H ). . Then the collection ϕ∗ (Aα ) defines a connection on P with curvature ϕ∗ (Fα ).3. (r) (s) The equality (8.1) ci (E ) · cj (F ).3. Hence.8. F and E ⊕ F are related by the identity ct (E ⊕ F ) = ct (E ) · ct (F ). The above result should be seen as a guiding principle in proving identities between characteristic classes. this means ck (E ⊕ F ) = i+j =k (8. The Lie algebra of U (r) × U (s) is the direct sum u(r) ⊕ u(s). (ν ) Let ı denote the natural inclusion u(r) ⊕ u(s) → u(r + s) and denote by ct ∈ I ∗ (U (ν ))[t] the Chern polynomial. where the “·” denotes the ∧-multiplication in H even (M ). Then the Chern polynomials of E .2.3. What is important about this result is the simple argument used to prove it. ϕ∗ : LH → LG . pick a Hermitian metric g on E and a Hermitian metric h on F . skew-hermitian matrix. where Xr (respectively Xs ) is an r × r (respectively s × s) complex. To check this.3. Any element X in this algebra has a block decomposition XE X= Xr 0 0 Xs = Xr ⊕ Xs . Let E and F be two complex vector bundles over the same manifold M .3.n = =0 j vt.n and by v the -th Chern class of Qk. Denote by uj the j -th Chern class of Uk. and we claim that i∗ (c2k+1 ) = 0. R) ∼ = R[ui . The universal complex vector bundle Uk.n the orthogonal complement of Uk. Exercise 8. deg ui = 2i.6.8. Exercise 8. This is mirrored at the Lie algebra level by an inclusion o(n) → u(n). vj . Show that L(E ⊕ F ) = L(E ) · L(F ) A(E ⊕ F ) = A(E ) · A(F ).n ct Qk. More formally H • Grk (Cn ). Chapter v3.2) where pt denotes the Pontryagin polynomials. From the above example we deduce that ct Uk. Then k n−k ct Uk. The trivial bundle Cn is equipped with a canonical Hermitian metric. Section 6.336 CHAPTER 8. vj . 0 ≤ i ≤ k.3. CHARACTERISTIC CLASSES Remark 8.n .n = j =0 uj t .n ⊕ Qk. and ı∗ (c2k ) = (−1)k pk .n → Grk (Cn ) is a subbundle of the trivial bundle Cn → Grk (Cn ). deg vj = 2j . The natural inclusion Rn → Cn induces an embedding ı : O(n) → U (n).3. One can then prove that the cohomology ring H • Grk (Cn ).n in Cn so that we have an isomorphism Cn ∼ = Uk. Example 8. ct Qk.4. Let E and F be two real vector bundles over the same smooth manifold M . (8. (An orthogonal map T : Rn → Rn extends by complexification to an unitary map TC : Cn → Cn ).n . R) is generated by the classes ui . The proof requires a more sophisticated topological machinery.5. For details we refer to [76]. Show that Td(E ⊕ F ) = Td(E ) · Td(F ).3.n = ct (Cn ) = 1. 0 ≤ j ≤ n − k ]/ i ui )( j vj ) = 1 . Let E and F be two real vector bundles over the same manifold M . We obtain a morphism ı∗ : I • (U (n)) → I • (O(n)).7. Exercise 8.3. . We denote by Qk. which are subject to the single relation above. Prove that pt (E ⊕ F ) = pt (E ) · pt (F ). Consider the Grassmannian Grk (Cn ) of complex k -dimensional subspaces in Cn .3. . 2.3. ct (E ∗ ) = c−t (E ).9. for X ∈ o(n) we have ı∗ (c2k+1 )(X ) = − 1 2π i 2k+1 337 λi1 (X ) · · · λi2k+1 (X ). and cr (E ) = e(ER ).11. Consequently. (a) The natural morphisms I ∗∗ (U (r)) → I ∗∗ (O(r)) described above induce a morphism Φ∞ : I •• (U (∞)) → I •• (O(∞)). Since X is. Let E → M be a complex vector bundle of rank r. (8. .3) In a more concentrated form. If E → M is a real vector bundle and E ⊗ C is its complexification then pk (E ) = (−1)k c2k (E ⊗ C)..3. . a real skew-symmetric matrix we have λj (X ) = λj (X ) = −λj (X ). (b) One can also regard E as a real. Proposition 8. Exercise 8. 1≤i1 <···<i2k+1 ≤n where λj (X ) are the eigenvalues of X over C. k = 1. The equality ı∗ (c2k ) = (−1)k pk is proved similarly.3.3.3. in effect. oriented vector bundle ER . (a) Show that ck (E ∗ ) = (−1)k ck (E ). ı∗ (c2k+1 )(X ) = ı∗ (c2k+1 (X )) = 2k+1 1≤i1 <···<i2k+1 ≤n − 1 2π i 2k+1 λi1 (X ) · · · λi2k+1 (X ) 1≤i1 <···<i2k+1 ≤n = (−1)2k+1 − 1 2π i λi1 (X ) · · · λi2k+1 (X ) = −ı∗ (c2k+1 )(X ). .8. COMPUTING CHARACTERISTIC CLASSES Indeed.10. this means pt (E ) = p−t (E ) = cit (E ⊗ C). e. Prove that pit (ER ) = k (−1)k t2k pk (ER ) = ct (E ) · c−t (E ). i. Exercise 8. . From the above example we deduce immediately the following consequence. and if moreover F is even. CHARACTERISTIC CLASSES As we already know. Denote by e(ν ) the Euler class in I • (SO(2ν )). The connection ∇ induces a Hermitian connection ∇c on the complexification Ec := E ⊗R C. (b) Let E be a real vector bundle. Show that for every F ∈ R[[x]] Φ∞ ((F )∞ ) = (F · F − )∞ ∈ I •• (O(∞)). 2 We are not worried about convergence issues because the matrices for which we intend to apply the formula have nilpotent entries. . We want to prove that ı∗ (e(k+ ) ) = e(k) ⊗ e( ) . Example 8.3.338 CHAPTER 8. Note that I • (SO(2k ) × SO(2 )) ∼ = I • (SO(2k )) ⊗R I • (SO(2 ). then (F )∞ can be regarded as an element of I ∗∗ (O(∞)). equipped with a metric and a compatible connection ∇. Exercise 8.12. Modulo a conjugation by (S. k! and det1/2 ( + X ) := det( + X )1/2 . the product (F )∞ is an element of I •• (U (∞)). k r k := r(r − 1) · · · (r − k + 1) .3. This induces a ring morphism ı∗ : I • (SO(2k + 2 ) → I • (SO(2k ) × SO(2 )). Let X = Xk ⊕ X ∈ so(2k ) ⊕ so(2 ). For every square matrix X we set2 ∞ ( + X )1/2 = k=0 (−1)k 1/2 Xk. A(∇) = det1/2 i c 4π F (∇ ) sinh( 4iπ F (∇c ) . Suppose that E → M is a rank r real vector bundle. where F − (x) = F (−x). Deduce from part (a) that Td(E ⊗ C) = A(E )2 . Prove that L(∇) = det1/2 i c 2π F (∇ ) tanh 2iπ F (∇c ) . Consider the inclusion ı : SO(2k ) × SO(2 )) → SO(2k + 2 ).13. for any F ∈ R[[x]] . T ) ∈ SO(2k ) × SO(2 ) we may assume that Xk = λ1 J ⊕ · · · ⊕ λk J and X = µ1 J ⊕ · · · ⊕ µ J. where · denotes the ∧-multiplication in H even (M ).3. Since ı∗ (e(k) ) = 0.3. the SO(2k ) structure of E can be reduced to an SO(1) × SO(2k − 1) ∼ = ∗ SO(2k − 1)-structure.2 that e(E ) = 0.14. oriented vector bundle over the smooth manifold M . oriented vector bundle of even ranks over the same manifold M . Denote by L the real line subbundle of E generated by the section ξ .3. λ1 · · · λk · µ1 · · · µ = e(k) (Xk ) · e( ) (X ) = e(k) ⊗ e( ) (Xk ⊕ X ). and the orientation on L defined by ξ induce an orientation on L⊥ . we deduce from the Proposition 8.3.3.8. The orientation on E . so that L⊥ has an SO(2k − 1)-structure. then any section of E must vanish somewhere on M ! . In other words.15. Let E be a rank 2k real. fix an Euclidean metric on E so that E is now endowed with an SO(2k )structure. COMPUTING CHARACTERISTIC CLASSES where J denotes the 2 × 2 matrix J= We have ı∗ (e(k+ ) )(X ) = − 1 2π k+ 339 0 −1 1 0 . Clearly. If e(E ) = 0. We claim that if E admits a nowhere vanishing section ξ then e(E ) = 0. The result proved in the above example can be reformulated more suggestively as follows. Proposition 8. Let E and F be two real. Example 8. Then e(E ⊕ F ) = e(E ) · e(F ). Let E be a real oriented vector bundle of even rank over the smooth manifold M . L is a trivial line bundle.16. The above example has an interesting consequence. and E splits as an orthogonal sum E = L ⊕ L⊥ . Denote by ı the inclusion induced morphism I ∗ (SO(2k )) → I ∗ (SO(2k − 1)). Corollary 8. To see this. oriented vector bundle over the compact oriented manifold M . Then etop (E ) = egeom (E ). Theorem 8.3. oriented manifold M then there are two apparently conflicting notions of Euler class naturally associated to E .3. • The topological Euler class ∗ etop (E ) = ζ0 τE . The strategy of proof is very k simple.4) On the other hand. We will use a variation of the original argument of Chern. notice that ∗ ∗ ζ0 i = ζ0 . Since π∗ is an isomorphism (Thom isomorphism theorem). will show that these two notions of Euler class coincide. 0 if rank (E ) is odd where ∇ is a connection on E compatible with some metric and 2r = rank (E ). We will distinguish two cases. Since the fibers of E are odd dimensional.3. Hence ∗ ∗ etop (E ) = −ζ0 i τE . The equality etop = egeom now follows from (8. which generalizes the Gauss-Bonnet theorem. where π∗ denotes the integration along fibers.4).340 CHAPTER 8. A. where τE is the Thom class of E and ζ0 : M → E is the zero section. we deduce i∗ τE = −τE . • The geometric Euler class egeom (E ) = 1 (2π )r P f (−F (∇)) if rank (E ) is even . rank (E ) is odd. this implies π∗ i∗ τE = −π∗ τE = π∗ (−τE ). In particular. The next result. CHARACTERISTIC CLASSES §8. rank (E ) = 2k . Let E → M be a real. B. we deduce that i reverses the orientation in the fibers. (8. ∗ ∗ ζ0 i = (iζ0 )∗ = (−ζ0 )∗ = (ζ )∗ (ζ0 = −ζ0 ). Indeed.2 The Gauss-Bonnet-Chern theorem If E → M is a real oriented vector bundle over a smooth. π Proof. Let ∇ denote a connection on E compatible with a metric g . [22].17 (Gauss-Bonnet-Chern).3. We will explicitly construct a closed form ω ∈ Ω2 cpt (E ) such that . Consider the automorphism of E i : E → E u → −u ∀u ∈ E. compact. 5) and Ψ(∇) = −1 ∈ Ω0 (M ).6) The form Ψ(∇) is sometimes referred to as the global angular form of the pair (E.18 which will be proved later on. and moreover. There exists Ψ = Ψ(∇) ∈ Ω2k−1 (S (E )) such that ∗ dΨ(∇) = egeom (π0 (E )) (8. The decisive step in the proof of Gauss-Bonnet-Chern theorem is contained in the following lemma.8.3. For the clarity of the exposition we will conclude the proof of the Gauss-Bonnet-Chern theorem assuming Lemma 8. Lemma 8. S (E )/M (8. and dim S (E ) = dim M + 2k − 1. Denote by r : E → R+ the norm function E e → |e|g .3. S (E ) = {u ∈ E .3.15 we must have ∗ egeom (π0 (E )) = 0 ∈ H 2k (S (E )).3. The condition (ii) simply states the sought for equality e class in Hcpt top = egeom . it admits a tautological.3.3. |u|g = 1}. COMPUTING CHARACTERISTIC CLASSES (i) π∗ ω = 1 ∈ Ω0 (M ). Then S (E ) is a compact manifold.18. ∗ E ) denotes the differential form where egeom (π0 ∗ egeom (π0 ∇) = 1 ∗ P f (−F (π0 ∇)). Denote by S (E ) the unit sphere bundle of E . ∇). according to Example 8. and by π0 ∗ of E to S (E ) via the map π0 . (ii) ζ0 341 The Thom isomorphism theorem coupled with (i) implies that ω represents the Thom 2k (E ). nowhere vanishing section Υ : S (Ex ) ∗ e → e ∈ Ex ≡ (π0 (E )x )e (x ∈ M ). (2π )k Hence there must exist ψ ∈ Ω2k−1 (S (E )) such that ∗ dψ = egeom (π0 E ). The vector bundle π0 (E ) has an SO(2k )-structure. ∗ ω = e(∇) = (2π )−k P f (−F (∇)). ∗ (E ) → S (E ) the pullback Denote by π0 the natural projection S (E ) → M . Thus. ∗ E . where L is the real line bundle spanned by Υ. To complete the program outlined at the beginning of the proof we need to show that ω is closed.3. then we can identify 1 E0 ∼ = (0. dω = ρ (r)dr ∧ dΨ(∇) − ρ (r) ∧ π ∗ e(∇) (8. The differential form ω is well defined since ρ (r) ≡ 0 near the zero section.342 CHAPTER 8. we can produce a new metric compatible ˆ on π ∗ E by connection ∇ 0 ˆ = (trivial connection on L) ⊕ P ∇P. and satisfies the condition (ii) since ∗ ∗ ∗ ζ0 ω = −ρ(0)ζ0 π e(∇) = e(∇). Thus ω The above form is identically zero since π0 is closed and the theorem is proved. ∞) × S (E ) e → (|e|. ∗ e(∇) = π ∗ e(∇) on the support of ρ . Obviously ω has compact support on E .6) Ψ(∇) = −(ρ(1) − ρ(0)) S (E )/M Ψ(∇) = 1.18 We denote by ∇ the pullback of ∇ to π0 ∗ section Υ : S (E ) → π0 E can be used to produce an orthogonal splitting ∗ π0 E = L ⊕ L⊥ . and ρ(r) = 0 for r ≥ 3/4. e). define ω = ω (∇) = −ρ (r)dr ∧ Ψ(∇) − ρ(r)π ∗ (e(∇)). Using P .3. CHARACTERISTIC CLASSES If we set E 0 = E \ { zero section }.3. The tautological Proof of Lemma 8. such that ρ(r) = −1 for r ∈ [0. Denote by π0 ∗ ∗ P : π0 E → π0 E the orthogonal projection onto L⊥ .5) ∗ = ρ (r)dr ∧ {π0 e(∇) − π ∗ e(∇)}. From the equality ρ(r)π ∗ e(∇) = 0. 1/4]. while L⊥ is its orthogonal complement in ∗ E with respect to the pullback metric g . E/M we deduce ∞ ω=− E/M E/M ρ (r)dr ∧ Ψ(∇) = − 0 ρ (r)dr · S (E )/M (8. ∞) → R. Finally. |e| Consider the smooth cutoff function ρ = ρ(r) : [0. ∇ . Moreover. . (2π )k ˆ splits as a direct sum Since the curvature of ∇ ˆ ) = 0 ⊕ F (∇ ˆ ). so all that we need to prove is Ψ(∇) = −1 ∈ Ω0 (M ). . S (E )/M It suffices to show that for each fiber Ex of E we have Ψ(∇) = −1. COMPUTING CHARACTERISTIC CLASSES We have an equality of differential forms ∗ π0 e(∇) = e(∇) = 343 1 P f (−F (∇)). we deduce where F (∇ ˆ )) = 0. P f (F (∇ ˆ + t(∇ − ∇ ˆ ). and the Levi-Civita connection on T S 2k−1 . . Ex ∗ E is naturally isomorphic with a trivial bundle Along this fiber π0 ∗ π0 E |Ex ∼ = (Ex × Ex → Ex ). ˆ is then the direct where ν denotes the normal bundle of S 2k−1 → R2k . and ∇1 = ∇. F t . . ∗ dΨ(∇) = π0 e(∇).3. .1. F t )dt .8. . By construction. so that ∇0 = ∇ ˆ .3. If F t is We denote by ∇t the connection ∇ t the curvature of ∇ . F (∇ ˆ ) denotes the curvature of ∇ ˆ |L⊥ . the connection ∇ restricts as the trivial connection. By choosing an orthonormal 2k ∗E | basis of Ex we can identify π0 over R2k . we deduce from the transgression formula (8. F t . . F t )dt P f (∇ − ∇ satisfies all the conditions in Lemma 8. .18. . The connection ∇ sum between the trivial connection on ν .12) that ∗ ˆ) = d π0 e(∇) = e(∇) − e(∇ −1 2π k 1 k 0 ˆ . The unit sphere Ex with the trivial bundle R 2 k − 1 2 k S (Ex ) is identified with the unit sphere S ⊂ R . The splitting L ⊕ L⊥ over S (E ) restricts over S (Ex ) as the splitting R2k = ν ⊕ T S 2k−1 . P f (∇ − ∇ We claim that the form Ψ(∇) = −1 2π k 1 k 0 ˆ . then we can rephrase the above equality as ∇∂j = −(θ1 + · · · + θ2k−1 ) ⊗ ∂0 . . .   ∂ ∂0 |p = f i = ∂i |p . at p. Hence ∇i ∂j = (∇i ∂j . . Since (x1 . ∇i ∂0 = ∂i . On the other hand. . 0=∇ we deduce ∇i ∂j = (∇i ∂j )ν at p. For α = 0. we deduce ∇i ∂0 |p = Consequently. . ∂ fi ˆ at p. Set ∂i := ∂x for i = 1. . . ∀α so that ∇  0 −θ1 · · · −θ2k−1  θ1 0 ··· 0   θ2 0 · · · 0 ˆ A = (∇ − ∇) |p =   . .  . ∇∂0 = θ1 ⊗ ∂1 + · · · + θ2k−1 ⊗ ∂2k−1 . Since at p ˆ i ∂j = (∇i ∂j )τ . . . .2. Riemann curvature of ∇ The computations in Example 4. . ∇i ∂j = −δji ∂0 . Then F 0 = 0 ⊕ R. . . while the superscript τ indicates the tangential component. and Greek letters indices running from 0 to 2k − 1. . 2k − 1. . · · · . where R denotes the Denote by F 0 the curvature of ∇0 = ∇ ˆ at p. We will use Latin letters to denote indices running from 1 to 2k − 1. ∇i ∂α = (∇i ∂α )ν + (∇i ∂α )τ . i. we deduce that ∇ ˆ ∂α = 0. If we denote by θi the local frame of T ∗ S 2k−1 dual to (∂i ). x2k−1 ) are normal coordinates with respect to the ˆ . such that the basis ( ∂x |p ) is positively oriented.. ∇i ∂0 )∂0 . .344 CHAPTER 8. . ∂0 )∂0 = −(∂j . 2 k − 1 θ 0 ··· 0 Levi-Civita connection     .19 show that the second fundamental form of the embedding S 2k−1 → R2k . The vectors f α form a positively oriented orthonormal basis of R2k . 1. . . and denote by (x1 . Recalling that ∇ is the trivial connection in R2k . 2k − 1 set f α = ∂α |p . CHARACTERISTIC CLASSES Fix a point p ∈ S 2k−1 . x2k−1 ) a collection of normal coordinates ∂ ∂ near p. . . where the superscript ν indicates the normal component. i i Denote the unit outer normal vector field by ∂0 .e. . 1.2. for σ ∈ S1 . ˆ + tA can be computed using the where Ωij = θi ∧ θj . . · · · . . (8. such that (σ ) = (φ).65 in Subsection 2. σ2 . ∂ In matrix format we have F 0 = 0 ⊕ (Ωij ). Using the polarization formula and Exercise 2. · · · . F ) = 2(−1)k+1 φ∈S2k−1 (φ)θφ1 ∧ · · · · · · θφ2k−1 . We deduce 2k k ! P f (A. COMPUTING CHARACTERISTIC CLASSES 345 coincides with the induced Riemann metric (which is the first fundamental form). such that (σ ) = (φ).4. The curvature F t at p of ∇t = ∇ equation (8. σ2k−1 ) ∈ S2k−1 . · · · .2. (1 − t2 )F 0 . F ) = (−1)k σ ∈S0 (−1)k 2k k ! (σ )Aσ0 σ1 Fσ2 σ3 · · · Fσ2k−2 σ2k−1 . . σ2k−1 ) ∈ S2k−1 . F. . and we get F t = F 0 + t2 A ∧ A = 0 ⊕ (1 − t2 )F 0 . k 0 1 P f (A. we get a permutation φ = (σ0 .3. · · · .11) of subsection 8.7) We need to evaluate the pfaffian in the right-hand-side of the above formula. We can now proceed to evaluate Ψ(∇). we get P f (A. define Si = {σ ∈ S2k . 1. Hence 2k k ! P f (A. F.8. Similarly. · · · .4. F 0 . σi = 0}. Ψ(∇) |p = = −1 2π −1 2π k k 1 = δi δjk − δik δj . F ) = For i = 0. Set for simplicity F := F 0 . F 0 ). σ ∈S2k (σ )(−θσ1 ) ∧ θσ2 ∧ θσ3 ∧ · · · ∧ θσ2k−2 ∧ θσ2k−1 +(−1)k σ ∈S1 (σ )θσ0 ∧ θσ2 ∧ θσ3 ∧ · · · ∧ θσ2k−2 ∧ θσ2k−1 . F. σ2 .1.3. Using Teorema Egregium we get R(∂i . ∂j )∂k . · · · . F. For each σ ∈ S0 we get a permutation φ : (σ1 . (1 − t2 )F 0 )dt k 0 (1 − t2 )k−1 dt P f (A. F 0 . 346 CHAPTER 8.3. Let (M. Using the Exercise 4. S 2 k −1 The above integral can be evaluated inductively using the substitution t = cos ϕ. g ) be a compact.19 (Chern).Using the last equality in (8. and let τE be a compactly supported form representing the Thom class. oriented Riemann Manifold of dimension 2n. (k − 1)! (1 − t2 )k−1 dt (1 − t2 )k−1 dt . 2k − 2 Lemma 8. . We have 1 Ik = (1 − t2 )k−1 dt π/2 π/2 = 0 π/2 (cos ϕ)2k−1 dϕ 0 = (cos ϕ)2k−2 sin ϕ|0 + (2k − 2) 0 (cos ϕ)2k−3 (sin ϕ)2 dϕ = (2k − 1)Ik−1 − (2k − 2)Ik so that Ik = One now sees immediately that Ψ(∇) = −1. CHARACTERISTIC CLASSES = 2(−1)k+1 (2k − 1)!θ1 ∧ · · · ∧ θ2k−1 = 2(−1)k+1 dVS 2k−1 . Corollary 8.60 we get that ω 2k−1 = and consequently Ψ(∇) = −ω 2k−1 (2k )! (4π )k k ! (2k )! = − 2k−1 2 k !(k − 1)! 1 0 1 0 S 2k−1 dVS 2k−1 = 2π k . Set r = rank E . S 2k−1 2k − 3 Ik−1 .18 is proved.3.7) we get Ψ(∇) |p = −1 2π k 1 k 0 (1 − t2 )k−1 dt · 2(−1)k+1 (2k − 1)!dVS 2k−1 1 0 =− (2k )! (4π )k k ! (1 − t2 )k−1 dt dVS 2k−1 . If R denotes the Riemann curvature then χ(M ) = 1 (2π )n P f (−R). where dVS 2k−1 denotes the Riemannian volume form on the unit sphere S 2k−1 .1. M The next exercises provide another description of the Euler class of a real oriented vector bundle E → M over the compact oriented manifold M in terms of the homological Poincar´ e duality.3. In the next exercise we will also assume M is endowed with a Riemann structure. COMPUTING CHARACTERISTIC CLASSES 347 Exercise 8. In many instances one can explicitly describe a nondegenerate section and its zero set. Show that the adjunction map a∇ : NZ → E |Z .20.10 c1 (U) = e(U). i. (a) (Adjunction formula. The local index of each zero measures the difference between two orientations of the normal bundle of this finite collection of points: one is the orientation obtained if one tautologically identifies this normal bundle with the restriction of T M to this finite collection of points while.3. defined by X → ∇X s X ∈ C ∞ (NZ ). Show that Φ∗ τE := (Φ# )∗ τE ∈ Ωr (Φ∗ E ) is compactly supported. where N is compact and oriented.3.44. and represents the Thom class of Φ∗ E . and (ii) there exists a connection ∇0 on E such that ∇0 X s = 0 along Z for any vector field X normal to Z (along Z). Let U denote the tautological line bundle over the complex projective space CPn . the section was a nondegenerate vector field. Example 8. According to Exercise 8. (Compare with Lemma 7.3. We denote by Φ# the bundle map Φ∗ E → E induced by the pullback operation. (b) Show that there exists an open neighborhood N of Z → NZ → T M such that exp : N → exp(N) is a diffeomorphism.3. and the other one is obtained via the adjunction map.3. and its zero set was a finite collection of points. The part (c) of the above exercise generalizes the Poincar´ e-Hopf theorem (see Corollary 7.) Deduce that the Poincar´ e dual of Z in M can be identified (via the above diffeomorphism) with the Thom class of NZ . and denote by NZ the normal bundle of the embedding Z → M . and thus one gets a description of the Euler class which is satisfactory for most topological applications.48).3.8.) Let ∇ be an arbitrary connection on E . Conclude that Z is endowed with a natural orientation.21. .3.22.. In that case. Let Φ : N → M be a smooth map. (c) Prove that the Euler class of E coincides with the Poincar´ e dual of Z . Consider a nondegenerate smooth section s of E .e. Exercise 8. (i) the set Z = s−1 (0) is a codimension r smooth submanifold of M . is a bundle isomorphism. . . and conclude that c1 (U∗ ) is indeed the Poincar´ e dual of [H ]. Let P be a degree 1 homogeneous polynomial P ∈ C[z0 . We claim that the (homological) Poincar´ e dual of c1 (τn ) is −[H ]. where deg g denotes the degree of the smooth map g |S 1 ⊂C∗ → S 1 . . . We thus have a well defined map CPn L → P |L ∈ L∗ = U∗ |L . For each map g : C∗ → U (1) ∼ = S 1 denote by Lg the line bundle over S 2 defined by the gluing map gN S : UN ∩ US → U (1). . and the reader can check easily that this is a smooth section of U∗ . The ∗ overlap UN ∩ US is diffeomorphic with the punctured plane C = C \ {0}. Show that S2 c1 (Lg ) = deg g. . . [z0 . The zero set of the section [P0 ] is precisely the image of [H ]. real vector bundle.3. CHARACTERISTIC CLASSES when we view U as a rank 2 oriented. . which we denote by P |L . Let US = S 2 \ {north pole}. zn ].21. zn−1 ] → [z0 . gN S (z ) = g (z ). the polynomial P defines a complex linear map L → C. We let the reader keep track of all the orientation conventions in Exercise 8. zn−1 .23. For each complex line L → Cn+1 . .348 CHAPTER 8. . . 0] ∈ CPn . which we denote by [P ]. . Consider the special case P0 = zn . Denote by [H ] the (2n − 2)-cycle defined by the natural inclusion ı : CPn−1 → CPn . and UN = S 2 \ {south pole} ∼ = C. We will achieve this by showing that c1 (U∗ ) = −c1 (U) is the Poincar´ e dual of [H ]. Exercise 8.3. and hence an element of L∗ . . Chapter 9 Classical Integral Geometry Ultimately. is a mixed breed subject. we are going against the natural course of things.1. we will present some very beautiful applications of the techniques developed so far. for various reasons. namely the change of order of summation. since the problems we will solve in this chapter were some of the catalysts for the discoveries of many of the geometric results discussed in the previous chapters. We define its “average” over X as the “integral” f X := f (x). Suppose we are given a function f : X → R. mathematics is about solving problems. the main trick of classical integral geometry is a very elementary one. One should think of α and β as defining two different fibrations with the same total space X .1 The integral geometry of real Grassmannians §9. In a sense. and make him/her appreciate the power of the technology developed in the previous chapters. x∈X We can compute the average f X in two different ways. Integral geometry.1 Co-area formulæ As Gelfand and his school pointed out. Let us explain the bare bones version of this trick. either summing first over the 349 . 9. The question it addresses have purely geometric formulations. In this chapter. Consider a “roof” X α β A α β B where X → A and X → B are maps between finite sets. unencumbered by various technical assumptions. also known as geometric probability. but the solutions borrow ideas from many mathematical areas. and we can trace the roots of most remarkable achievements in mathematics to attempts of solving concrete problems which. such as representation theory and probability. were deemed very interesting by the mathematical community. This chapter is intended to wet the reader’s appetite for unusual questions. . α(x)=a and β∗ (f ) : B → C. i. Note that any basis (v1 .1.1. A. All the integral geometric results that we will prove in this chapter will be of this form.e. and the function f is not really a function. . The geometric situation we will investigate is a bit more complicated than the above baby case. vn ) of V defines linear isomorphisms |Λ|s V → R. (9. a density. To produce such results we need to gain a deeper understanding of the notion of density introduced in Subsection §3. ∞). B are smooth manifolds. Given a real number s we define an s-density on V to be a map λ : det V → R such that λ(tΩ) = |t|s λ(Ω). and we denote the corresponding space by |Λ|(V ). The reason such formulæ have captured the imagination of geometers comes from the fact that the two sides α∗ (f ) A and β∗ (f ) B have rather unrelated interpretations. We denote by |Λ|+ s (V ) the cone of positive densities.1) We can reformulate the above equality in a more conceptual way by introducing the functions α∗ (f ) : A → C. . . λ → λ(v1 ∧ · · · ∧ vn ). It turns out that the push forward operations do make sense for densities resulting in formulas of the type f X = α∗ (f ) A .4. because in the geometric case the sets X. f (x) a∈A α(x)=a = f X = b∈B β (x)=b f (x) . We denote by det V the top exterior power of V . The equality (9.1) implies that α∗ (f ) A = β∗ (f ) B . i. Ω ∈ det V. . ∀t ∈ R∗ . Suppose V is a finite dimensional real vector space. α∗ (f )(a) = f (x).350 CHAPTER 9. We will refer to 1-densities simply as densities. We denote by |Λ|s (V ) the one dimensional space of s-densities. We say that an s-density λ : det V → R is positive if λ(det V \ 0) ⊂ (0. the maps α and β define smooth fibrations.e. β∗ (f )(b) = β (x)=b f (x). but an object which can be integrated over X . Note that we have a canonical identification |Λ|0 (E ) = R.1. CLASSICAL INTEGRAL GEOMETRY fibers of α. or summing first over the fibers of β . These formulæ generalize the classical Fubini theorem and they are often referred to as coarea formulæ for reasons which will become apparent a bit later. . then g ∗ λ = | det g |s λ. Now choose lifts vj ∈ V of wj . v1 . . dim W = p we can associate maps \ : |Λ|+ s (U ) × |Λ|s (V ) → |Λ|s (W ). (λ. such that β (vj ) = wj . α(um ). λ) → g∗ λ = (g −1 )∗ λ = | det g |−s λ. . If V0 and V1 are vector spaces of the same dimension n. and g : V0 → V1 is a linear isomorphism then we get a linear map g ∗ : |Λ|s (V1 ) → |Λ|s (V0 ). . dim V = m. is a basis of V . . · · · . It is easily seen that the above definition is independent of the choices of v ’s and u’s. . THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS In particular. . . . For every g. . µ ∧i ui α β λ → g ∗ λ. vp . and a basis (ui )1≤i≤m of U such that α(u1 ). . • Let µ ∈ |Λ|+ s (U ). . we have a canonical identification |Λ|s (Rn ) ∼ = R. and thus we have a left action of Aut(V ) on |Λ|s (V ) Aut(V ) × |Λ|s (V ) → |Λ|s (V ). . Aut(V ) × |Λ|s (V ) We have bilinear maps |Λ|s (V ) ⊗ |Λ|t (V ) → |Λ|s+t (V ). If V0 = V1 = V so that g ∈ Aut(V ). ∀v1 . / : |Λ|s (V ) × |Λ|+ s (W ) → |Λ|s (U ). We set (µ\λ) ∧j wj := λ (∧i α(ui ) ) ∧ (∧j vj ) .1. |Λ|s (V1 ) where (g ∗ λ) (∧i vi ) = λ ∧i (gvi ) . en ) is the canonical basis of Rn . To any short exact sequence of vector spaces 0 → U → V → W → 0. λ → λ(e1 ∧ · · · ∧ en ). dim U = m. µ) → λ · µ. . h ∈ Aut(V ) we gave (gh)∗ = h∗ g ∗ . 351 where (e1 . λ ∈ |Λ|s (V ). and × : |Λ|s (U ) × |Λ|s (W ) → |Λ|s (V ) as follows. (g.9. vn ∈ V0 . and suppose (wj )1≤j ≤p is a basis of W . of V and now define (λ/ν ) ∧i ui := λ (∧i α(ui )) ∧ (∧j vj ) . λ ×αt . e2 ) the canonical basis of V and by f the canonical basis of W . β (x.1. CLASSICAL INTEGRAL GEOMETRY • Let λ ∈ |Λ|s (V ) and ν ∈ |Λ|+ s (W ). Then we set (µ × ν ) (∧i α(ui )) ∧ (∧j wj ) = µ( ∧i ui )ν (∧j vj ).βτ ν = (t/τ )−s µ ×α. . λ/αt . and × associated to a short exact sequence of vector spaces 0 → U −→ V −→ W → 0. Exercise 9. λ ∈ |Λ|s (V ). τ > 0.1. . Choose lifts (vj ) of wj to V . Remark 9. The next example illustrates this. then µ\αt . Consider the short exact sequence 0 → U = R −→ V = R2 −→ W = R → 0 given by α(s) = (4s. y ) = 5x − 2y. · · · . 10s).β ν. ν ∧j β (vj ) Again it is easily verified that the above definition is independent of the various choices.β ν. /.βτ λ = (t/τ )s µ\α.352 CHAPTER 9.β λ. We obtain canonical densities λU on U . vp .βτ ν = (t/τ )s λ/α. λV on V and λW on W given by λU (e) = λV (e1 ∧ e2 ) = λW (f ) = 1. Fix a basis (ui )1≤i≤m of U and a basis (wj )1≤j ≤p of W . α β α β . Given a basis (ui )1≤i≤m of U . Prove that the above constructions are indeed independent of the various choices of bases. Again one can check that this is independent of the various bases (ui ) and (wj ). To define µ × ν : det V → R it suffices to indicate its value on a single nonzero vector of the line det V . do depend on the maps α and β ! For example if we replace α by αt = tα and β by βτ = τ β .1. . and if µ ∈ |Λ|s (U ). by (e1 . t. Denote by e the canonical basis of U . extend the linearly independent set α(ui ) ⊂ V to a basis α(u1 ). . α(um ). v1 . Example 9. • Let µ ∈ |Λ|s (U ) and ν ∈ |Λ|s (W ).2.3. The constructions \. ν ∈ |Λ|s (W ).1. and E → M is a smooth real vector bundle. Then λV /β ∗ λW (e) = λV (f 1 ∧ f 2 )/λW (f ) = det Hence λV /β ∗ λW = 2λU . then we have a canonical identification |Λ|s (Rn ) → R. Let us point out that if V = Rn . Assume that it is given by the open cover (Uα ) and gluing cocycle gβα : Uαβ → Aut(V ). we obtain maps ∞ ∞ \ : C ∞ (|Λ|+ s E0 ) × C (|Λ|s E1 ) → C (|Λ|s E2 ). We choose f 2 ∈ V such that β (f 2 ) = f . We denote by C ∞ (|Λ|s E ) the space of smooth sections of |Λ|s E . An s-density λ ∈ C ∞ (|Λ|s E ) is called positive if for every x ∈ M we have λ(x) ∈ + |Λ|s (Ex ). We have canonical isomorphisms |Λ|s π ∗ E ∼ = π ∗ |Λ|s E. where V is a fixed real vector space of dimension n. . then we obtain the pullback bundle π ∗ E → N . 353 Suppose now that E → M is a real vector bundle of rank n over the smooth manifold M .9. and in this case a density can be regarded as a collection of smooth functions λα : Uα → R satisfying the above gluing conditions. β. f 2 = (1. and a natural pullback map φ∗ : C ∞ (|Λ|s E ) → C ∞ (π ∗ |Λ|s E ) ∼ = C ∞ (|Λ|s π ∗ E ). Such a section is given by a collection of smooth functions λα : Uα → |Λ|s (V ) satisfying the gluing conditions −1 ∗ λβ (x) = (gβα ) λα (x) = | det gβα |−s λα (x). for example. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS We would like to describe the density λV /β ∗ λW on V . If φ : N → M is a smooth map. Given a short exact sequence of vector bundles 0 → E0 → E1 → E2 → 0 over M .1. Set f 1 = α(e) = (4. 10). 4 1 10 2 = 2. ∀α. ∞ / : C ∞ (|Λ|s E1 ) × C ∞ (|Λ|+ s E2 ) → C (|Λ|s E0 ). 2). x ∈ Uαβ . Then the bundle of s-densities associated to E is the real line bundle |Λ|s E given by the open cover (Uα ) and gluing cocycle | det gβα |−s : Uαβ → Aut( |Λ|s (V ) ) ∼ = R∗ . When s = 1. a choice of orientation produces a linear map Ωn (M ) → C ∞ (|Λ|(M ) ). (xi α ) .1. 1 The functions λα satisfy the gluing conditions λβ = d− βα λα . Because of this fact. (c) Any Riemann metric g on M defines a canonical density on M denoted by |dVg | and called the volume density. then the collection of functions λα defines a density on M . this map is a bijection. ∞) we have (f µ)\λ = (f −1 (µ\λ). As explained in Subsection §3. As explained in Subsection §3. λ/(f ν ) = (f −1 )(µ/λ). and we conclude that the collection of functions |λα |s defines an s-density on M which we will denote by |ω |s . dxα := dx1 α ∧ · · · ∧ dxα .2) We deduce that the smooth 0-densities on M are precisely the smooth functions. where dβα = det ∂xj β ∂xi α 1≤i.j ≤n . the s-densities are traditionally described as collections n λα |dxα |s . Observe that for every positive smooth function f : M → (0. where |gα | denotes the determinant of the symmetric matrix representing the metric g in the coordinates (xi α ). and we will refer to its sections as (smooth) s-densities on M . Example 9. If ω is a top dimensional form on M described locally by forms ωα = λα dxα . (b) Suppose M is an orientable manifold. (a) Suppose ω ∈ Ωn (M ) is a top degree differential form on M . It is locally described by the collection |gα ||dxα |. . Thus. By fixing an orientation we choose an atlas (Uα .1.1. we will use the simpler notation |Λ|M to denote |Λ|1 (M ). (xi α ) ) so that all the determinants dβα are positive. (9.4. n = dim M. In the sequel we will almost exclusively use a special case of the above construction. Then in a coordinate atlas (Uα .2. CLASSICAL INTEGRAL GEOMETRY × : C ∞ (|Λ|s E0 ) × C ∞ (|Λ|s E2 ) → C ∞ (|Λ|s E1 ). and smooth functions λα : Uα → R satisfying the conditions λβ = |dβα | −s λα . We will denote by |Λ|s (M ) the line bundle |Λ|s (T M ).354 and CHAPTER 9. an s-density on M is described by a coordinate atlas Uα .4. when E is the tangent bundle of the smooth manifold M . (xi α ) ) this form is described by a collection of forms n ωα = λα dx1 α ∧ · · · ∧ dxα .4. ρα |dyα |) is a density on N . |dµ| → |dµ|. Suppose φ : M → N is a diffeomorphism between two smooth mdimensional manifolds. (f. and |dµ| is a density. ρα |dyα |). then f |dµ| is a continuous compactly supported density. Then φ∗ |ω | = |φ∗ ω |. yα = xi α ◦ φ. ∀g ∈ G.1. and |ω | is the associated density. and any density |dρ|. compactly supported function on M . The density |dρ| is called G-invariant if g ∗ |dρ| = |dρ|.5. For every x ∈ M we have U |dρ f (x) = lim . THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 355 The densities on a manifold serve one major purpose: they can be integrated. The classical change in variables formula now takes the form |dρ| = N M φ∗ |dρ|. ∀g ∈ G. M Let us observe that if |dρ| and |dτ | are two positive densities.1. The existence of this function follows from the Radon-Nicodym theorem. ω ∈ Ωm (M ). Consider the integration map defined in Subsection §3. Then for every g ∈ G.1 M : C0 (|Λ|(M ) ) → R. If φ : M → N is a diffeomorphism and |dρ| = (Uα . We denote by C0 (|Λ|(M ) ) the space of continuous densities with compact support. |dτ | and we will refer to f as the jacobian of |dρ| relative to |dτ |. M We obtain in this fashion a natural pairing C0 (M ) × C (|Λ|M ). .4. |dµ|) → f |dµ|. Suppose a Lie group G acts smoothly on M . we get a new density g ∗ |dρ|. We will use the notation f := |dρ| .9. U →{x} U |dτ | where the above limit is taken over open sets shrinking to x. then there exists a positive function f such that |dρ| = f |dτ |. Note that a density is G-invariant if and only if the associated Borel measure is G-invariant. ∗ |dρ| A positive density is invariant if the jacobian g|dρ | is identically equal to 1. M Note that if f is a continuous. then we define the pullback of |dρ| by φ to be the density φ∗ |dρ| on M defined by i φ∗ |dρ| = φ−1 (Uα ). and thus there is a well defined integral f |dµ|. Example 9. Fix a positive density |dν | on B . then we obtain a density λ × Φ∗ |dν | on M . The kernels of the differentials of Φ form a vector subbundle T V M → T M consisting of the planes tangent to the fibers of Φ. positive function on G. Then dρ| the jacobian J = ||dτ | is a G-invariant smooth. There exists a smooth density Φ∗ |dµ| on B uniquely characterized by the following condition. CLASSICAL INTEGRAL GEOMETRY Proposition 9. up to a positive multiplicative constant there exists at most one invariant positive density.6. Observe that any (positive) density |dν | on B defines by pullback a (positive) density Φ∗ |dν | associated to the bundle Φ∗ T B → M . If G acts smoothly and transitively on the smooth manifold M then. Since Φ is a submersion. Φ−1 (U ) DΦ U Proposition 9. Suppose |dµ| is a density on M such that Φ is proper on the support of |dµ|. Intuitively. Φ∗ |dµ| is the unique density on B such that for any open subset U ⊂ B we have Φ∗ |dµ| = |dµ|.356 CHAPTER 9. Then for every open neighborhood U of x we have |dρ| = U g (U ) |dρ|. and then letting U → {x} we deduce J (x) = J (gx). Set k := dim B .1. We would like to describe a density Φ∗ |dµ| on B called the pushforward of |dµ| by Φ.8 (The pushforward of a density). Let x ∈ M and g ∈ G. If λ is a density associated to the vertical tangent bundle T V M . Φ−1 (b) Proof. DΦ . Suppose |dρ| and |dτ | are two G-invariant positive densities. 0 → T V M → T M −→ Φ∗ T B → 0. Along every fiber Mb = Φ−1 (b) we have a density |dµ|b /Φ∗ |dν | ∈ C ∞ |Λ|(Mb ) corresponding to the short exact sequence 0 → T Mb → (T M )|Mb −→ (Φ∗ T B )|Mb → 0. where Vν ∈ C ∞ (B ) is given by Vν (b) := |dµ|/Φ∗ |dν |. U |dτ | = g (U ) |dτ | =⇒ U |dτ | = U |dρ| g (U ) |dτ | g (U ) |dρ| . Corollary 9.1. we have a short exact sequence of bundles over M . r := dim M − dim B .7. Suppose Φ : M → B is a submersion. We will refer to it as the vertical tangent bundle. g ∈ G. ∀x ∈ M.1. For every density |dν | on B we have Φ∗ |dµ| = Vν |dν |. Proof. a nowhere vanishing form Ω ∈ Ωk+r (MU ). We can associate to Vν the density Vν |dν | which a priori depends on ν . then there exists a positive smooth function w : B → R such that |dν ˆ| = w|dν |.1.1.3) Remark 9. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 357 To understand this density fix x ∈ Mb . and a form η ∈ Ωr (MU ) such that Ω = η ∧ π ∗ ω. where for every u ∈ U ⊂ B we have f (u) = Mu |dµ|/f ∗ |ω | = Mu ρ|Ω|/|f ∗ ω | = Mu ρ|η |.1. For every point on the base b ∈ B there exist an open neighborhood U of b in B . ˆ |dν In other words. Vν ˆ = w Vν so that Vν ˆ| = Vν |dν |. . and we conclude that |dµ|/f ∗ |ω | = ρ|η |. Then −1 |dµ|b /Φ∗ |dν ˆ| = w−1 |dµ|b /Φ∗ |dν |. (MU := Φ−1 (U )). (9. ρB (y ) |dµ|b /Φ∗ |dν | The function Vν : B → R is smooth function. y ) |dx|. y ) → y . Then we can write |dµ| = ρ|Ω|. Then we can find local coordinates (yj )1≤j ≤k near b ∈ B . for any open subset U ⊂ B . and smooth functions (xi )1≤i≤r defined in a neighborhood V of x in M such that the collection of functions (xi .9. a nowhere vanishing form ω ∈ Ωk (U ). and in these coordinates the map Φ is given by the projection (x. y )|dx ∧ dy |. for some ρ ∈ C ∞ (MU ). The form ω defines a density |ω | on U .9. In the coordinates y on B we can write |dν | = ρB (y )|dy | and |dµ| = ρM (x. the density Vν |dν | on B is independent on ν . Very often the submersion Φ : M → B satisfies the following condition. Φ∗ |dµ| = f |ω |. Observe that if |dν ˆ| is another density. It depends only on |dµ|. Then along the fibers y = const we have |dµ|b /Φ∗ |dν | = We set Vν (b) := Mb ρM (x. Using partitions of unity and the classical Fubini theorem we obtain the Fubini formula for densities |dµ| = Φ−1 (U ) U Φ∗ |dµ|. y j ) defines local coordinates near x on M . v (t) = Mt 1 |dVt0 | |U ∩Mt0 . so that we have the equality |dVt0 | |U ∩Mt0 = ρ n (df ∧ dx1 ∧ · · · ∧ dxm ) |U ∩Mt0 = ρ|∇f | · (dx1 ∧ · · · ∧ dxm ) |U ∩Mt0 Now observe that along U we have ρ(df ∧ dx1 ∧ · · · ∧ dxm ) = ρ(f ∗ dt ∧ dx1 ∧ · · · ∧ dxm ). For simplicity. and smooth function x1 . .1.1.1. . . xm ) are local coordinates on U . On M we have a volume density |dVg |. The vector field n is a unit normal vector field along Mt0 ∩ U . x1 .4) 1 |dVt |. For t ∈ R. and f : M → R is a smooth function without critical points.10. |∇f | and we obtain in this fashion the classical coarea formula |dVg | = 1 |dVt | |dt|. xm such that (f. g ) is a Riemann manifold of dimension m + 1. Fix t0 ∈ R. (The classical coarea formula). so that |dVt0 | |U ∩Mt0 = |∇f | · |dVg |/f ∗ |dt| and |dVg |/f ∗ |dt| = Hence f∗ |dVg | = v (t)|dt|. We denote by |dVt | the volume density on Mt defined by the induced metric gt := g |Mt .5) M R Mt . Then along U we can write |dVg | = ρ|df ∧ dx1 ∧ · · · ∧ dxm |. . Hence |dVg |/f ∗ |dt| = ρ|(dx1 ∧ · · · ∧ dxm )|. . |∇f | (9. We denote by 1 ∇f the g -gradient of f . . .358 CHAPTER 9. we also assume that the level sets Mt = f −1 (t) are compact. . For every point p ∈ Mt0 we have df (p) = 0. and we set n := |∇ f | ∇f . and v is a smooth function. We would like to compute the pushforward density f∗ |dVg | on R. Example 9. We seek f∗ |dVg | of the form f∗ |dVg | = v (t)|dt|. where ρ is a smooth function. where |dt| is the Euclidean volume density on R. and from the implicit function theorem we deduce that we can find an open neighborhood U . Suppose (M. using (9.3) we obtain the generalized coarea formula |dµ| = MU MU ρ|η ∧ π ∗ ω | = U f (u)|ω | = U Mu ρ|η | |ω |. the fiber Mt is a compact codimension 1 submanifold of M . |∇f | (9. CLASSICAL INTEGRAL GEOMETRY In particular.1. We denote by θ the angle between ∂t and Tp S n . p θ 1 t θ 1 t 0 p' Figure 9. where B denotes the Beta function 1 B (p. We let P± ∈ S n denote the poles given t = ±1. xn ) → t. (9. Hence |∇π (p)| = (1 − t2 )1/2 . x1 .1. p] and the segment [p. set p = π (p). Then θ is equal to the angle at p between the radius [0. . p ] = (1 − t2 )1/2 . x1 . q > 0. p. π (t. M = S n \ {P± }.6) . and consequently. We denote the coordinates in Rn+1 by (t. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 359 To see how this works in practice. 1).9. The last formula implies 1 σ n = 2σn (1 − t2 ) n−2 2 1 0 |dt| = σ n−1 (1 − s) 0 n−2 2 s−1/2 |ds| = σ n−1 B 1 n . .1: Slicing a sphere by hyperplanes Observe that π −1 (t) is the (n − 1)-dimensional sphere of radius (1 − t2 )1/2 . because ∂t is the gradient with respect to the Euclidean metric on Rn+1 of the linear function π : Rn+1 → R. p ]. xn ). and by t the coordinate of p (see Figure 9. . xi ) = t. . The map π is a submersion on the complement of the poles. π −1 (t) 1 |dVt | = (1 − t2 )−1/2 |∇π | π −1 (t) |dVt | = σn−1 (1 − t2 ) n−2 2 . q ) := 0 sp−1 (1 − s)q−1 ds. consider the unit sphere S n ⊂ Rn+1 . . To find the gradient ∇π observe that for every p ∈ S n the tangent vector ∇π (p) is the projection of the vector ∂t on the tangent space Tp S n . where σ m denotes the m-dimensional area of the unit m-dimensional sphere S m . 2 2 .1. and π (M ) = (−1. . We want to compute π∗ |dVn |. We denote by |dVn | the volume density on S n .1). . and by π : S n → R the natural projection given by (t. . We deduce cos θ = length [p. and we deduce that the action of O(V ) on Grk (V ) preserves the metric h.1.2. the Grassmannian of k dimensional subspaces of V . We would like to investigate a certain natural density on Grk (V ). q ) = 0 xp dx Γ(p)Γ(q ) · = . Denote by O(V ) the group of orthogonal transformations of V . In Example 1. B ∈ End(V ). for every t ∈ R.2. we can regard Grk (V ) as a submanifold of End+ (V ). More precisely. via the (orthogonal) projection map U → PU . Then.k on Grk (V ). Ut := ΓtS ∈ Grk (V.2. (1 + x)p+q x Γ(p + q ) (9. L⊥ ). . then If U 2 ˙ 0. defined by A.20. The action of O(V ) on End+ (V ) by conjugation preserves the inner product on End+ (V ).2) on End(V ). Grk (V ) is orientable if and only if dim V is even. we denote by Ut the graph of the map tS : L → L⊥ .1. We denote by • the inner product on V . CLASSICAL INTEGRAL GEOMETRY It is known (see [101]. the vector space of symmetric endomorphisms of V . and S ∈ Hom(L. ˙ 0 denotes the tangent to the path t → Ut at t = 0.7) where Γ(x) denotes Euler’s Gamma function ∞ Γ(x) = 0 e−t tx−1 dt. 2 induces a metric h = hn. We are forced into considering densities rather than forms because very often the Grassmannians Grk (V ) are not orientable. We would like to express this metric in the graph coordinates introduced in Example 1. The Euclidean metric (1. The group O(V ) acts smoothly and transitively on Grk (V ) O(V ) × Grk (V ) Note that PgL = gPL g −1 . For every subspace U ⊂ V we denote by PU the orthogonal projection onto U . B = 1 tr(AB † ). L). Consider L ∈ Grk (V ). P tS 2 dt (g.360 CHAPTER 9. Section 12. We deduce n+1 2Γ( 1 2) σn = n+1 . U ˙ 0 ) = 1 tr(P ˙0 ˙0 := d |t=0 PΓ . Γ( 2 ) (9. L) → g (L) ∈ Grk (V ).8) §9.41) that ∞ B (p.20 we have shown that.1. h(U ). ∀A.2 Invariant measures on linear Grassmannians Suppose that V is an Euclidean real vector space of dimension n. We would like to compute h Observe that we have a smooth path t → gt ∈ O(V ). With respect to these bases. . • We will use lower case Greek letters α. to denote indices in the range {k + 1. the map S : L → L⊥ is described by a matrix (sαi )1≤i≤k<α≤n . γ. ˆ (L ˙ 0. . . |t| This defines a smooth path t → Lt = span (ei (t) ) ∈ Grk (V ).1. . n}. · · · . . . k }. U ˙ 0) = h(U i. C.α |sαi |2 . j.. In the sequel we make the following notational conventions. B..α |sαi |2 . L) ⊂ Grk (V ) consisting of k -planes intersecting L⊥ transversally. • We will use upper case Latin letters A. . to denote indices in the range {1.9. .9) ˙ 0... and an orthonormal basis (eα )k<α≤n of L⊥ .10) In integral geometric computations we will find convenient to relate the above coordinates to the classical language of moving frames. to denote indices in the range {1. . defined by gt eA (0) = eA (t).1. β.. h(U 2 We can be even more concrete.. and then tr(SS ∗ ) = tr(S ∗ S ) = i. Let us choose an orthonormal basis (ei )1≤i≤k of L. . L ˙ 0 ). U ˙ 0 ) = 1 tr(SS ∗ ) + tr(S ∗ S ) = tr(SS ∗ ). 361 (9.2... Hence ˙ 0. Suppose we have a smooth 1-parameter family of orthonormal frames (eA ) = (eA (t)).1.4) that Pt = Hence ˙t=0 = P so that (L + t2 S ∗ S )−1 t(L + t2 S ∗ S )−1 S ∗ tS (L + t2 S ∗ S )−1 t2 S (L + t2 S ∗ S )−1 S ∗ 0 S∗ S 0 = S ∗ PL⊥ + SPL . • We will use lower case Latin letters i.. 1. (9. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS If we write Pt := PΓtS we deduce from (1. We can think of the collection (sαi ) as defining local coordinates on the open subset Grk (V. n}. this means that if X = (u ˙ a ) ∈ Tp0 M .11) We want to interpret this in the language of moving frames. such that L(u) = span ( ei (u) ). Then −1 Pt = gt P0 gt . the orthogonal transformation gt is given by a matrix ( sAB (t) ). (9.L 0 0 0 .e. xAB = s ˙ AB = eA • e which has the block form X= XL0 . P0 ] = Hence 0 XL⊥ . so that. Suppose M is a smooth m-dimensional manifold and L : M → Grk (V ) is a smooth map.1. α. P With respect to the decomposition V = L0 + L⊥ 0 the projector P0 has the block decomposition L0 0 P0 = . Observe that g0 = V . L ˙ 0 ) = 1 tr(P ˙0 .L0 0 ∗ XL ⊥ .362 CHAPTER 9.12) .L0 0 ∗ −XL ⊥ . P0 ]. P ˙0 ) = tr(X ⊥ X ∗ ⊥ ) = h(L L0 . varying smoothly with (ui ).L 0 0 0 0 XL⊥ . θαi := eα • dei . CLASSICAL INTEGRAL GEOMETRY With respect to the fixed frame ( eA (0) ). then ˙i = Dp0 L(X ) = (xαi ) ∈ TL(0) Grk (V ). and we let “ ˙ ” denote the differentiation along the flow lines of X . xαi = eα • e a eα • dei (X ). if we set X = d dt |t=0 gt . where XL⊥ . The above computations show that the differential of L at p0 is described by the (n − k ) × k matrix of 1-forms on M Dp0 L = (θαi ). Fix a point p0 ∈ M and local coordinates (ui )1≤i≤m near p0 such that ui (p0 ) = 0.L0 L0 . We deduce 0 [X. More precisely..i (9. ˙ 0. i. The map Lt can be described near p0 via a moving frame. where sAB = eA (0) • gt eB (0) .L0 XL⊥ .L0 2 |s ˙ αi |2 .1. we have ˙0 = [X. an orthonormal frame (eA ) of V .L0 denotes a map L0 → L⊥ 0 etc.L⊥ . 0 0 The operator X is represented by a skew-symmetric matrix with entries ˙ B. Let Pt denote the projection onto Lt . 11) show that h= α. To understand the above construction it is helpful to consider the special case of the Grassmannian Gr1 (R2 ) of lines through the origin in R2 .i θαi . with the following properties. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 363 If M happens to be an open subset of Grk (V ).10) and (9. Example 9. π ] that it forms with the x-axis. where n = dim V . then we can use the forms θαi to describe the metric h. (9. We would like to give a local description of |dγn.i θαi ⊗ θαi .k |. A = 1. If O is a sufficiently small open subset of Grk (V ) then we can find smooth maps eA : O → V.11. and |dγ2. The metric h = hn.9. and the associated volume density is described by |dγn. Set n := dim V .1. More precisely.1.13) In other words. Since the action of the group O(V ) is transitive. A line L in R2 is uniquely determined by the angle θ ∈ [0. e2 (θ) = (− sin θ. and we will refer to it as the kinematic density on Grk (V ). we denote by Lθ the corresponding line.k is O(V )-invariant so that the associated Riemannian volume defines an invariant density.k | = α. the equalities (9. where the first vector e1 (θ) gives the direction of the line. Then θ21 = e2 • de1 = dθ. θαi = eα • dei = −deα • ei .1. which in turn. is an orthonormal frame of L. we deduce that any other invariant density is equivalent to a constant multiple of this metric density. and every k + 1 ≤ α ≤ 1. This space is diffeomorphic to the real projective line RP1 . · · · . we have a 1-form θαi ∈ Ω1 (O). sin θ). the metric h is described along O by the symmetric tensor h= α.i θαi := α. Lθ = span ( e1 (θ) ). • For every L ∈ O the collection eA (L) • For every L ∈ O the collection ei (L) 1≤A≤n 1≤i≤k is an orthonormal frame of V . cos θ). For such an angle θ. it is diffeomorphic to a circle. We will denote this invariant density by |dγn. n.1.k |.1 | = |dθ|. As explained above. The line Lθ is also represented by the the orthonormal frame e1 (θ) = (cos θ. For every 1 ≤ i ≤ k .1. . the collection (θαi ) is an orthonormal frame of the cotangent bundle T ∗ Grk (V ).i θαi ⊗ θαi . k |. CLASSICAL INTEGRAL GEOMETRY We would like to compute the volumes of the Grassmannians Grk (V ). we would like to compute Cn. We show that nω n 1 . dim V = n with respect to the kinematic density |dγn. Ck Cn−k |dγn |. To compute the volume of the Grassmannians we need to use yet another description of the Grassmannians. O(Rn ) . n ωn 0 1 1 2 2 π 3 4π 3 4 π2 2 .k := |dγn.364 CHAPTER 9. Grk (V ) Denote by ω n the volume of the unit ball B n ⊂ Rn . and ω n = = Γ(1 + n/2)   22k+1 π k k !    (2k + 1)! where Γ(x) is the Gamma function. We list below the values of ω n for small n. i.. Note first that the group O(V ) acts transitively on Grk (V ). as homogeneous spaces. Cn . Step 1. Fix L0 ∈ Grk (V ). Cn. We show that Cn.1 = σ n−1 = 2 2 and then compute Cn inductively using the recurrence relation from Step 2 Cn+1 = (C1 Cn. The computation of Cn.e. Then the stabilizer of L0 with respect to the action of O(V ) on Grk (V ) is the subgroup O(L0 ) × O(L⊥ 0 ) and thus we can identify Grk (V ) with the homogeneous space of left cosets O(V )/O(L0 ) × O(L⊥ 0 ). n = 2k + 1 σ n−1 Γ(1/2)n = nω n .k = Step 3.1 )Cn . and by σ n−1 the (n − 1)-dimensional “surface area” of the unit sphere S n−1 .k is carried out in three steps. Set Cn := Step 2. so that  k π      k! n = 2k .k |. and the initial condition C2 = vol ( O(2) ) = 2σ 1 . We equip the orthogonal groups O(Rn ) with a canonical invariant density |dγn | called the kinematic density on O(n). xAB = eA • f B A B More generally. Two skew-symmetric endomorphisms X. Y ˙ Y (0) ∈ TS0 O(V ) ⊂ End(V ). ˙ = S0 X . if we define f A : O(V ) → V. f A (S ) := S eA . This defines a diffeomorphism from a neighborhood of 0 in End− (V ) to a neighborhood of S0 in O(V ). originating at S0 . For any S0 ∈ O(V ) we have a map expS0 : End− (V ) → O(V ). 2 This metric induces an invariant metric h on O(V ). We set ˙ := γ ˙ := γ X ˙ X (0) ∈ TS0 O(V ) ⊂ End(V ). . Y ˙ = S0 Y . Denote by End− (V ) the subspace of End(V ) consisting of skew-symmetric operators. Y ∈ End− (V ) define paths γX .1. B = 1 tr(AB ∗ ). If we set f A (t) := exp(tX )eA . A>B The associated volume density is |dγn | = A>B θAB . γY : R → O(V ). γX (t) = S0 exp(tX ). then we deduce ˙ Y ˙)= h(X. 1 1 1 ∗ ∗ tr (S0 X )(S0 Y )∗ = tr S0 XY ∗ S0 = tr S0 S0 XY ∗ 2 2 2 = 1 tr XY ∗ 2 If we choose an orthonormal basis (eA ) of V so that X and Y are given by the skew symmetric matrices (xAB ). End− (V ) −→ S0 · exp(X ). then we deduce ˙ (0) = f (0) • f ˙ (0). and Then X ˙ Y ˙)= h(X. We equip End(V ) with the inner product A. then we obtain the angular forms θAB := f A • df B . The group O(V ) is a submanifold of End(V ) consisting of endomorphisms S satisfying SS ∗ = S ∗ S = V . We would like to give a more concrete description of this metric. A>B xAB yAB . (yAB ). The above metric metric has the description h= θAB ⊗ θAB .9. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 365 Step 1. γY (t) = S0 exp(tY ). defined as follows. such that L = span (φi (L). p(T ) = T (L0 ). . We can identify V with Rn .k . Fix an orthonormal frame (eA ) of V such that L0 = span (ei . An orthogonal n × n matrix T is uniquely determined by the orthonormal frame (T eA ) via the equalities TAB = eA • T eB . It is a given by a matrix with entries φ(L)AB = eA • φB . and L0 with the subspace Rk ⊕ 0n−k ⊂ Rn . For every sufficiently small open subset U ⊂ Grk (V ) we can find a smooth section φ : U → O(n) of the map p : O(n) → Grk (Rn ). Define p : O(n) → Grk (Rn ). Then we have a smooth map Ψ : O(k ) × O(n − k ) × U → p−1 (U ). Once we have this we deduce from the Fubini formula (9. The section can be identified with a smooth family of orthonormal frames (φA (L). Let us prove the above facts.3) that Cn = Ck Cn−k Cn. O(V ) with O(n). L ∈ U )1≤A≤n of Rn . 1 ≤ i ≤ k ). we have p(T ) = span (T ei )1≤i≤k . To such a frame we associate the orthogonal matrix φ(L) ∈ O(n) which maps the fixed frame eA to the frame φB .1. More explicitly.k |. We will prove that we have a principal fibration O(k ) × O(n − k ) O M O(n) Kpn .366 CHAPTER 9. Grk (R ) and that p∗ |dγn | = Ck Cn−k |dγn. CLASSICAL INTEGRAL GEOMETRY Step 2. 1 ≤ i ≤ k ). . The map Ψ is a homeomorphism with inverse O(n) T → Ψ−1 (T ) = (s. t. 1 ≤ A ≤ n. We have a commutative diagram O(k ) × O(n − k ) × U Ψ DD D DD π U ⊂ Grk (Rn ) M p−1(U )  p In particular. express s as a k × k matrix s = (si j ). • Define the orthonormal frame of Rn ( f A ) := (φB ) ∗ (s. ∀A.15). sβ = φα (LT ) • f β . More precisely. This constant is given by the integral of the density |dγn |/p∗ |dγn. Observe that. α while the matrices (si j ) and (tβ ) are obtained via (9. t1 ∈ O(n − k ).k | along the fiber p−1 (L0 ). We set f A = f A (T ) := T eA . (9.e. t0 ) ∗ (s1 . and t α as a (n − k ) × (n − k ) matrix (tβ ). t. t. L) ∈ O(k ) × O(n − k ) × U .1. L) ∈ O(k ) × O(n − k ) × L defined as follows. i. Observe now that p∗ |dγn | is an invariant density on Grk (Rn ). f A = ΨeA . t1 ) = (φB ) ∗ (s0 s1 . t0 . This means that Ψ is equivariant with respect to the right actions of O(k ) × O(n − k ) on O(k ) × O(n − k ) × U and O(n). this shows that p defines a principal O(k ) × O(n − k )-bundle. ⊥ tβ α φβ (L) ∈ L . L) = β • Now define Ψ = Ψ(s. t0 t1 ). k + 1 ≤ α ≤ n.1. L) to be the orthogonal transformation of Rn which maps the frame (eA ) to the frame (f B ). we have (φB ) ∗ (s0 . THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 367 • Given (s.1.14) (9.1. ∀s0 . we have α si j = φi (LT ) • f j .15) f α = f α (t. t). and thus there exists a constant c such that p∗ |dγn | = c|dγn. Then L = LT = span (f i )1≤i≤k .k |. s1 ∈ O(k ). 1 ≤ i ≤ k.1.14) and (9. L) = j sj i φj (L) ∈ L. via the equalities f i = f i (s.9. and θAB := f A • df B . Now observe that ψ is in fact an isometry.368 Hence Cn = O(n) CHAPTER 9. Ck Cn−k n−1 Step 3.1 = area (S n−1 ) = 2 2 2 Hence Cn+1 = Cn C1 Cn+1. Fix an orthonormal basis {eA } of V .1. This map is a diffeomorphism. + Note that Gr1 (V ) ∼ = RPn−1 is the Grassmannian of lines in V . Cn. whose associated density is |dγn. and denote by S + the open hemisphere −1 Sn = v ∈ V . and thus we deduce σ n−1 nω n 1 = .1 = σ n Cn .1 |. We now find ourselves in the situation described in Remark 9.i is the pullback of a nowhere vanishing form defined in a neighborhood of L0 in Grk (Rn ). We denote by Gr1 (V )∗ the open subset consisting of lines −1 intersecting S n + . n−1 → ∩ S+ .i θα.k | = cCn.1 | = |dγn. The set of lines that do n−1 not intersect S + is a smooth hypersurface of Gr1 (V ) diffeomorphic to RPn−2 and thus has kinematic measure zero. |v | = 1.9. CLASSICAL INTEGRAL GEOMETRY |dγn | = Grk (Rn ) p∗ |dγn | = c Grk (Rn ) |dγn.k . and we have Cn.1 | = −1 Sn + Gr1 (V ) Gr∗ 1 (V ) (ψ −1 )∗ |dγn.i θα.i .1 = |dγn. Recall that if we define f A : O(n) → Rn by f A (T ) := T eA . v • e1 > 0 . then |dγn | = | A>B θAB |. . = O(k) |dγk | O(n−k) Hence Cn. We write this as θij ∧ i>j α>β θαβ ∧ α.k |.k = Cn . We deduce c= p−1 (L0 ) i>j θij ∧ α>β θαβ |dγn−k | = Ck Cn−k . The form α. We thus have a map n−1 ψ : Gr∗ 1 (V ) → S + . 17) Grk (V ) We have |dνn. Example 9. k (9. [n]! := 2 ω n−1 2ω n−1 n [k ] = k=1 ω n n! .1 = 0 π |dθ| = 2 ω2 2.1. Using the computation in Example 9. 0 ≤ θ < π. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS which implies inductively that n−1 n− 1 369 Cn = σ n−1 · · · σ 2 C2 = 2 k=1 σk = j =0 σj . Proposition 9. For every 1 ≤ k < n we have Cn.1 .k = = |dγn.k |dγnk |.1. we deduce the following result. In particular.13.9.1.k | = n j =1 ω j k i=1 ω i n−1 j =0 σ j k −1 n−k−1 σj i=0 σ i · j =0 n−k j =1 Grk (Rn ) n k · ωj .12.k | = n . dim V = n such that |dνn. 1 ω1 so that |dν2.12. which implies that C2. .1 | = |dγ2.1 |. Following [59].16) n [n]! = := ([k ]!)([n − k ]!) k n ωn . 2n (9.k | the unique invariant density on Grk (V ).11 we deduce |dγ2.1. ω1 as predicted by Proposition 9.k | = n k Cn.1.1. We have 2 ω2 = 2 2 = C2. k ω k ω n−k Denote by |dνn. we set [n] := 1 σ n− 1 nω n = .1.1 | = |dθ|. s1 . .k |. We set c(S ) := S ∩ [S ]⊥ . where L ∈ Grk (V ).k |.k | on Graff k (V ) given by |dγn. and we denote by U⊥ → Grk (V ) the orthogonal complement of U in V . The points of U⊥ are pairs (c. It is naturally a subbundle of the trivial vector bundle V = V × Grk (V ) → Grk (V ) whose fiber over L ∈ Grk (V ) is the vector subspace L. We have the tautological vector bundle U = Un. We have a natural map A : U⊥ → Graff k (V ) given by (c. and we say that c(S ) is the center of the affine plane S . The fiber of U⊥ over L ∈ Grk (V ) is canonically identified with the orthogonal complement L⊥ of L in V . 0 → ker Dπ −→ T Graff k (V ) −→ π ∗ T Grk (V ) → 0. Using the short exact sequence of vector bundles over Graff k (V ).1. where PL denotes the orthogonal projection onto L. and thus we obtain a density π ∗ |dγn.370 CHAPTER 9. This map is a bijection with inverse Graff k (V ) S → (S ∩ [S ]⊥ . We would like to describe a natural structure of smooth manifold on Graff k (V ).3 Affine Grassmannians We denote by Graff k (V ) the set of k -dimensional affine subspaces of V . we identify Graff k (V ) with a vector subbundle of the trivial bundle V × Grk (V ) described by U⊥ = (c. s2 ∈ S . L) −→ c + L. The projection π : U⊥ → Grk (V ) is a submersion. Thus.k | = |dVL⊥ | × π ∗ |dγn. [S ]). The base Grk (V ) of the submersion π : Graff k (V ) → Grk (V ) is equipped with a density |dγn. We obtain in this fashion a density |dVL⊥ | associated to the vertical subbundle ker Dπ ⊂ T U⊥ . The fiber of this submersion over L ∈ Grk (V ) is canonically identified with the vector subspace L⊥ ⊂ V .k | associated to the bundle π ∗ T Grk (V ) → Graff k (V ). We equip Graff k (V ) with the structure of smooth manifold which makes A a diffeomorphism. it is equipped with a volume density |dVL⊥ |. where [S ] ∈ Grk (V ) denotes the affine subspace through the origin parallel to S . L). and c is a vector in L⊥ . CLASSICAL INTEGRAL GEOMETRY §9. L) ∈ V × Grk (V ). The trivial vector bundle V is equipped with a natural metric. As such. we obtain a density |γn.k → Grk (V ). PL c = 0 . [S ] = S − S = s1 − s2 . and denote by O its preimage in Graff k (V ) via the projection π .9. Observe that the center of this affine plane is the projection of r onto [S ]⊥ c(S ) = α ( eα • r )eα . • For every S ∈ O. The Fubini formula for densities (9. Theorem 9. Suppose f : Graff k (V ) → R is a compactly supported |dγ ˆn.1. Following the tradition we introduce the (locally defined) 1-forms θα := eα • dr.14. r : O → V with the following properties. Then we can find smooth maps eA : O → V.k | = α θα ∧ α.k (L)|.k |-integrable function. • For every S ∈ O we have S = r(S ) + [S ].k (S )| = f (p + L)|dVL⊥ (p)| |dγn. θαi := eα • dei . (eA (S ) ) is an orthonormal frame of V .i θαi .1. THE INTEGRAL GEOMETRY OF REAL GRASSMANNIANS 371 Let us provide a local description for this density. ⊥ For fixed L ∈ Grk (V ) the density on the fiber U⊥ L = L is given by |dVL⊥ | = θα . The volume density on Graff k (V ) is described along O by |dγn. Graff k (V ) Grk (V ) L⊥ where |dVL⊥ | denotes the Euclidean volume density on L⊥ . ei ). . and [L] = span ei ([L]) .1. We rewrite the last equality as S = S (r. Fix a small open subset O ⊂ Grk (V ). Then f (S )|dγn.3) implies the following result. as p varies along M . this tangent plane changes its location in the Grassmannian Grm (Rn ). and in fact. for obvious reasons. and a simple computation shows that the associated volume density |dγn.k |.372 CHAPTER 9. is named the Gauss map of the submanifold M .k | is Iso(V ) invariant. |dνn.k | on Grk (V ) we use the density |dνn.k |. π ) we denote by Lθ. i. ∀v ∈ V. 9.k | = n k Cn. then the tangent plane Tp M does not vary as p moves around M . Any affine isometry T : V → V is described by a unique pair (t. Such a line L is determined by two quantities: the angle θ ∈ [0.2. Note that the tangent plane is uniquely determined by the unit normal. where the variation of the unit normal along the surface detects the Riemann curvature of the surface. the Grassmannians of affine lines in R2 . If instead of the density |dγn. The goal of this section is to analyze in great detail the map M p → Tp M ∈ Grm (Rn ).e. Observe that if M is flat.1.14 can now be rewritten ∞ Graff 1 (R2 ) ∞ (Graff (R2 ) ). The group Iso(V ) acts in an obvious fashion on Graff k (V ). CLASSICAL INTEGRAL GEOMETRY Denote by Iso(V ) the group of affine isometries of V .ρ )|dθ| |dρ|. m = dim M .. To understand the shape of a submanifold M → Rn .e. and the signed distance ρ ∈ (−∞. π ) is makes with the x-axis. it is an open subset of an affine subspace. which. ∀f ∈ Ccpt 1 π 0 f (L) |dγ2.18) Example 9. i. S ) ∈ V × O(V ) so that T (v ) = Sv + t.2 Gauss-Bonnet again?!? The Grassmannians enter into the study of the geometry of a submanifold via a fundamental observation going back to Gauss. ∞) from the origin. The Fubini formula in Theorem 9.1 |(L) = −∞ f (Lθ. In Example 4. Let us unravel the above definition in the special case Graff 1 (R2 ). for every ρ ∈ R and θ ∈ [0. . we obtain a density |dνn. “the more curved is M . the faster will the tangent plane move inside this Grassmannian”.. the subgroup of the group of affine transformations generated by translations. the Grassmannian Graff 1 (R2 ) is diffeomorphic to the interior of the M¨ obius band.15. However.1. if M is curved.ρ the line is given in Euclidean coordinates by the equation x sin θ − y cos θ = ρ As a manifold.1. (9. it is very productive to understand how a tangent plane Tp M varies. More precisely.20 we have seen this heuristic principle at work in the case of a hypersurface in R3 .k | on Graff k (V ) which is a constant multiple of |dγn.k |dγnk |. and by the rotations about a fixed point. Denote by D the Levi-Civita connection of the Euclidean metric on Rn .. • We will use Einstein’s summation convention.e. . . Xn Then  dX 1  . . θ A (eB ) = eA • eB = δAB . and a normal part X ν . . and a local orthonormal frame (eA ) of T Rn along U such that the collection (ei |M ) is a local orthonormal frame of T M along U ∩ M . C ≤ n. To minimize the notational burden we will use the following conventions. dX n  The restriction to M of the tangent bundle T Rn admits an orthogonal decomposition (T Rn )|M = T M ⊕ (T M )⊥ . ≤ m = dim M. . . • We will use lower case Latin letters i. . C to denote indices in the range 1 ≤ A. X ν = θ α (X )eα . ≤ n. Correspondingly.  DX =  . B. GAUSS-BONNET AGAIN?!? 373 §9. β. j. • We will use the capital Latin letters A. If X is a section of T Rn M then X τ = θ i (X )ei . a section X of (T Rn )|M decomposes into a tangential part X τ . . .1 The shape operator and the second fundamental form of a submanifold in Rn Suppose M is a smooth submanifold M of Rn . The Euclidean metric of Rn induces a metric g on M . to denote indices in the range 1 ≤ i.  = dX. . i. Let us recall this construction using notations more appropriate for the applications in this chapter.2. B.2. . We denote by (θ A ) the dual coframe of (eA ). . β. • We will use lower case Greek letters α. Fix a a point p0 ∈ M .2. .4 we explained how to determine the Levi-Civita connection ∇M of g using Cartan’s moving frame method. In Subsection §4. Every vector field on Rn can be regarded as an n-uple of functions  1  X  . j. to denote indices in the range m < α.  X= . and set m := dim M . .9. . . an open neighborhood U of p0 in Rn . the “moving plane” x → Tx M can be represented by the orthonormal frame (eA ) which has the property that the first m vectors e1 (x). that is the differential of the Gauss map.i = eα • D e j . Proposition 9. then. D ej = ΦA j eA .374 CHAPTER 9. and thus.1. D eB = ΘB eA . j. The differential of the Gauss map at x ∈ U ∩ M is a linear map DG : Tx M → TG(x) Grm (Rm ) = Hom(Tx M. We denote the shape operator by S M . Intuitively. (Tx M )⊥ ) is given by S M (ei )ej = (D ei ej )ν = Φα ij eα . the shape operator measures how fast the tanget plane changes its location as a point in a Grassmannian. .i α On the other hand. Define ΦA ij := ei so that 0 ΦA j ∈ Ω (M ∩ U ). as explained Subsection §4. we deduce that (Φi j ) are the 1-forms associated to M the Levi-Civita connection ∇ by the local orthonormal frame (ei |M ). The shape operator of the submanifold M → Rn is. . Y ∈ Vect (M ). ΦA B := ΘB |M .1. As explained in Subsection §9. this differential described by the (n − m) × m matrix of 1-forms eα • dej .2. Consider the Gauss map G = GM : M → Grm (Rn ). . A i ΦA j = Φij ∧ φ .4. in the neighborhood U of p0 .2. and we would like to relate it to the structural coefficients ΦA B. α. (Tx M )⊥ ). This implies that A ∇M ej = Φi j ei = the tangential component of Φj eA = D ej . em (x) span Tx M . As explained in (9. We have thus obtained the following result.12). . α. τ ∇M X Y = (D X Y ) . x → Tx M. More precisely the operator S M (ei ) ∈ Hom(Tx M.2. . The shape operator of M . If we write A φA := θ A |M . is locally described by the matrix of 1-forms (Φα i )1≤i≤m<α<n . by definition. so that eα • D ej = Φj . the differential of the Gauss map. ΘB = −ΘA . CLASSICAL INTEGRAL GEOMETRY We denote by ΘA B the 1-forms associated to D by the frame (eA ). ∀i.1. ∀X. They satisfy Cartan’s equations B A A B dθ A = −ΘA B ∧ θ . x → n(x). X2 ) − SM (X3 . (X. ej ) = φα (D ei ej )eα = Φα ij eα = S (ei )ej . .2. i. . α. em of T M . If we fix an orientation on M . It follows from the Alexander duality theorem with Z/2 coefficients that a compact hypersurface of a vector space is the boundary of a compact domain. an orientable submanifold of codimension 1. (9. . then we obtain a normal vector field n : M → Rm+1 . ∀i. . M x → n(x) ⊥ := the vector subspace orthogonal to n(x).2. Suppose M is a submanifold of Rn .2. We recall this result below. On the other hand. In this case we can identify the second fundamental form with a genuine symmetric bilinear form SM ∈ C ∞ T ∗ M ⊗2 . Theorem 9. Then for any X1 .2.2. . Note that M SM (ei . R(X3 . X4 ∈ Vect (M ) we have g (X1 . em } is an oriented. Suppose M is a compact. where • denotes the inner product in Rn . . . X2 ) • SM (X4 . X1 ). (9. orientable 1 hypersurface of Rm+1 .9. local orthonormal frame of Rm+1 . GAUSS-BONNET AGAIN?!? 375 In Subsection §4. e1 (x). The Gauss map GM : M → Grm (Rm+1 ) of the embedding M → Rm+1 can be given the description. |n(x)| = 1. . Y ) = n • (D X Y ). §9. ej ).10) can be rewritten as ej • (D ei eα ) = −(D ei ej ) • eα = −eα • SM (ei .2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space. We denote by g the induced metric on M and by SM the second fundamental form of the embedding M → Rn .e. If we choose an oriented. X1 ) • SM (X4 . Y ) → (D X Y )ν . Denote by R the Riemann curvature of M with the induced metric. then the ordered set {n(x). . 1 .. SM (X.2) Theorema Egregium states that the Riemann curvature tensor of M is completely determined by the second fundamental form. The results in the previous subsection have very surprising consequences. . ∀x ∈ M.2.4 we have defined the second fundamental form of the submanifold M to be the C ∞ (M )-bilinear and symmetric map SM : Vect (M ) × Vect (M ) → C ∞ (T M )⊥ . and in particular. j.2. local orthonormal frame e1 .1) The equality (4. The orientability assumption is superfluous. n(x) ⊥ Tx M. X4 )X2 ) = SM (X3 . . we have an oriented Gauss map GM : M → S m .2. it is orientable. . .  M Sm   Kπ G G Grm (Rm+1 ) We fix an oriented orthonormal frame (f0 .2. and we orient the unit sphere S m ⊂ Rm+1 so that the orientation of Tf0 S m is given by the ordered frame (f1 . Rijk := g ei . Fix a positively oriented orthonormal frame e = (e1 . and denote by θ = (θ 1 . . . . S m so that the diagram below is commutative M u→ u ⊥ . 2 (2h)! (9. . and by R the curvature of g . where dVM denotes the metric volume form on M and P (RM ) is a universal polynomial of degree m 2 in the curvature R of M . . . . . f1 .3) 1 dAm = 1. σm so that 1 σm M G∗ M dAm = deg GM .2. Theorem 9. . . Denote by dAm ∈ Ωm (S m ) the “area” form on the unit m-dimensional sphere S m . fm ) of Rm+1 . fm ). Observe that dVM = θ 1 ∧ · · · ∧ θ m . em ) of T M defined on some open set U ⊂ M . .376 and a double cover CHAPTER 9. Recall that σm denotes the “area” of S m . . . .2. In the remainder of this subsection we will assume that m is even. m = 2h. ej ) • n. Denote by g the induced metric on M . We set Sij := SM (ei . e )ej .2 implies that Rijk = Sik Sj − Si Sjk = Sik Si Sjk Sj . (9.4) . We would like to prove that the integrand G∗ M dAm has the form G∗ M dAm = P (RM )dVM . σ 2h = (2h + 1)ω 2r+1 =⇒ Hence Sm 22h π h h! σ 2h = . θ m ) the dual coframe. . R(ek . CLASSICAL INTEGRAL GEOMETRY π : S m → Grm (Rm+1 ). 2s−1. regard S as a linear map S : Tx M → T ∗ M.6) We would like to give an alternate description of det S using the concept of pfaffian. ∀1 ≤ i1 < · · · < ip ≤ m. . . . 2. First.5) We want to prove that det S can be described in terms of Λ2 S .4) in a more convenient form. m = 2h} such that ϕ(1) < ϕ(2).9. (9. .4) can now be rephrased as R = Λ2 S.2. To see this observe that (Λ2h S )(e1 ∧ e2 ∧ · · · e2h−1 ∧ e2h ) = (Λ2 S )(e1 ∧ e2 ) ∧ · · · ∧ (Λ2 S )(e2h−1 ∧ e2h ) h h = s=1 i<j Rij. The equality (9. . GAUSS-BONNET AGAIN?!? Observe that Rijk = 0 =⇒ i = j. We can rephrase the equality (9.j ≤m lieing at the intersection of the i. Then S induces linear maps Λp S : Λk Tx M → Λp T ∗ M. S ej = Sij θ i . j -rows with the k. we regard the curvature Rijk at a point x ∈ M as a linear map ∗ Λ2 Tx M → Λ2 Tx M. defined by S (ei1 ∧ · · · ∧ eip ) = (S ei1 ) ∧ · · · ∧ (S eip ). Observe that #Sm = 2h 2h − 2 2 · ··· 2 2 2 = (2h)! .2. . Along U we have the equality m 1 m (G∗ M dAM ) |U = Λ S (e1 ∧ · · · ∧ em ) = (det S )θ ∧ · · · ∧ θ = (det S )dVM .2.2s θ ∧ θ i j = ϕ∈Sm (ϕ) s=1 Rϕ(2s−1)ϕ(2s). ϕ(2h − 1) < ϕ(2h).2s−1. and (ϕ) = ±1 denotes the signature of a permutation.2. -columns. and in this case the matrix Sik Si Sjk Sj 377 is the 2 × 2 submatrix of S = (Sij )1≤i. .2s dVM . . where Sm denotes the set of permutations ϕ of {1.2. . 2h (9. R(ek ∧ e ) = i<j Rijk θ i ∧ θ j . Next. k = . . Observe that P f (Θ) = (−1)h h! (ϕ)Θϕ(1)ϕ(2) ∧ · · · ∧ Θϕ(2h−1)ϕ(2h) . . .4 that the pfaffian of Θ is the differential form P f (Θ) := (−1)h 2h h! (ϕ)Θϕ(1)ϕ(2) ∧ · · · ∧ Θϕ(2h−1)ϕ(2h) ∈ Ω2h (U ). #Sm . can be regarded as a 2-form with coefficients in the bundle of skew-symmetric endomorphisms of T M . It is locally described by a m × m skew-symmetric matrix Θ = Θg := Θij whose entries are 2-forms on U . Θij = k< 1≤i. h! 2 h! (ϕ)Θϕ(1)ϕ(2) ∧ · · · ∧ Θϕ(2h−1)ϕ(2h) . m}. = 1 = (#Sm ) 1 #Sm (ϕ)Θϕ(1)ϕ(2) ∧ · · · ∧ Θϕ(2h−1)ϕ(2h) ϕ∈Sm h (σϕ) (σ. 2. Recall from Subsection §8. CLASSICAL INTEGRAL GEOMETRY The Riemann curvature. .ϕ)∈Sm ×Sm j =1 Rϕ(2j −1)ϕ(2j )σ(2j −1)σ(2j ) dVM = h! P f (−Θ). like the curvature of any metric connection. ϕ∈Sm We can simplify this some more if we introduce the set Sm consisting of permutations ϕ ∈ Sm such that ϕ(1) < ϕ(3) < · · · < ϕ(2h − 1). We have P f (−Θ) = On the other hand.2. k. Observe that #Sm = (#Sm )h! =⇒ #Sm = Then P f (Θ) = (−1)h ϕ∈Sm SM (2h)! = h = 1 · 3 · · · (2h − 1) =: γ (2h). (Λ2h S )(e1 ∧ e2 ∧ · · · ∧ e2h−1 ∧ e2h ) = 1 #Sm (ϕ)Λm S (eϕ(1) ∧ eϕ(2) ∧ · · · ∧ eϕ(2h−1) ∧ eϕ(2h) ) ϕ∈Sm 1 h! h (σϕ) (σ. ϕ∈Sm where Sm denotes the group of permutations of {1.ϕ)∈Sm ×Sm j =1 Rϕ(2j −1)ϕ(2j )σ(2j −1)σ(2j ) dVM .378 CHAPTER 9.j ≤m Rijk θ k ∧ θ = 1 2 Rijk θ k ∧ θ = Λ2 S (ei ∧ ej ) ∈ Ω2 (U ). .2. . 2. 3. . σm M (2π )h and 2 deg GM = 1 (2π )h h e(∇M ) = χ(M ).2. ∀1 ≤ j ≤ h.2. 3. 1. Theorem 9. = 1 −→ Θ1212 Θ3434 . 2. 1. positively oriented orthonormal frame of T M . . then S4 consists of 3 permutations 1. = −1 −→ −Θ1312 Θ2434 = 1 −→ Θ1412 Θ2334 . 4. Using the Gauss-Bonnet-Chern theorem we obtain the following result.3) h! P f (−Θ) = P f (−Θ) 2 σ2h #Sm #Sm 2 h π h h! 1 (9. then 1 P f (−Θg ) = h! (σϕ) (σ. Example 9. .2. #Sm so that G∗ M 2 dAM σ 2h = 2 h! (2h)! (9. .9. (a) If dim M = 2 then P f (−Θg ) = R1212 dVM = (the Gaussian curvature of M ) × dVM (b) If dim M = 4.7a) (9. Sm denotes the set of permutations ϕ ∈ Sm such that ϕ(1) < ϕ(3) < · · · < ϕ(2h − 1).2. where S2h denotes the set of permutations ϕ of {1.7b) = ϕ∈S2h (ϕ)Θϕ(1)ϕ(2) ∧ · · · ∧ Θϕ(2h−1)ϕ(2h) .4.2. . and g denotes the induced metric.3. 2h} such that ϕ(2j − 1) < ϕ(2j ).2.6) P f (−Θ) = (2π )h The last form is the Euler form e(∇M ) associated to the Levi-Civita connection ∇M . If M 2h ⊂ R2h+1 is a compact. M Moreover. e2h ) is a local. if (e1 . 3. GAUSS-BONNET AGAIN?!? Hence 2h G∗ M dAM = (Λ S )(e1 ∧ e2 ∧ · · · ∧ e2h−1 ∧ e2h ) = 379 h! P f (−Θ). then 2 ∗ 1 G dAm = e(∇M ) = P f (−Θg ). 2. 4. 4. oriented hypersurface.ϕ)∈S2h ×Sm j =1 Rϕ(2j −1)ϕ(2j )σ(2j −1)σ(2j ) dVM (9. and Θij = k< Rijk θ k ∧ θ . . . σ m ∂D D . . ∂D If m is even then the Gauss-Bonnet theorem for the hypersurface ∂D implies 1 σm 1 G∗ D dAm = χ(∂D ). because θi ∧ θj = 0. (c) We can choose the positively oriented local orthonormal frame (e1 .380 We deduce CHAPTER 9. so that 1 G∗ dAm = χ(D). where ρ(x) = (2h − 1)!! k=1 m ij = −Rij k . . Then the eigenvalues of SM at x are called the principal curvatures at x and are denoted by κ1 (x). Thus the total number of terms in the above expression is 36. κm (x). . However. It defines an oriented Gauss map GD : ∂D → S m . m ∈ 2Z. CLASSICAL INTEGRAL GEOMETRY P f (−Θg ) = Θ12 ∧ Θ34 − Θ13 ∧ Θ24 + Θ14 ∧ Θ23 1 1 = (R12ij θ i ∧ θ j ) ∧ (R34k θ k ∧ θ ) − (R13ij θ i ∧ θ j ) ∧ (R24k θ k ∧ θ ) 4 4 1 + (R14ij θ i ∧ θ j ) ∧ (R23k θ k ∧ θ ) 4 Each of the three exterior products above in involves only six nontrivial terms. κi (x). etc. Then P f (−Θ) = ρdVM .3 Gauss-Bonnet theorem for domains in an Euclidean space Suppose D is a relatively compact open subset of an Euclidean space Rm+1 with smooth boundary ∂D. when i = j . ∀x ∈ M. We denote by dAm the area form on the unit sphere S m so that deg GD = 1 σm G∗ D dAm . em ) so that it diagonalizes SM at a given point x ∈ M . ρ ∈ C ∞ (M ). 2 ∂D Using the Poincar´ e duality for the oriented manifold with boundary D we deduce χ(∂D) = 2χ(D). there are many repetitions due to the symmetries of the Riemann tensor Rijk = Rk We have (R12ij θ i ∧ θ j ) ∧ (R34k θ k ∧ θ ) = R1212 R3434 + R1213 R3442 + R1214 R3423 +R1223 R3414 + R1242 R3413 + R1234 R3412 dVM . . . We denote by n the outer normal vector field along the boundary. . .2. §9. .e. Tp Rm+1 is invertible. 1 ∗ 1 GD dAm = det(−SD )dV∂D . σm σm where dV∂D denotes the volume form on ∂D. define Let us observe that the map X Yt : ∂D → S m . A smooth vector field on D ¯ → Rm+1 X:D is called admissible if along the boundary it points towards the exterior of D. Suppose X is a nondegenerate admissible vector field. X (pi ) = 0. ∀X. |(1 − t)n + tX Hence ¯ deg GD = deg X. . for any admissible vector field X . |X (p)| We deduce ¯ is homotopic to the map GD .p . . 1]. Indeed.9. Let SD denote the second fundamental form of the hypersurface SD (X. For any ε > 0 sufficiently small the closed balls of radius ε centered at the points in ZX are disjoint. ∀p ∈ ∂D. (p) = sign det AX.2. ZX = {p1 . i. For an admissible vector field X we define ¯ : ∂D → S m . p∈ZX . Define X v → (D v X )(p) : ZX → {±1}. for t ∈ [0. on ∂D. X ¯ (p) := X 1 X (p). This means that X has a finite number of stationary points. Therefore in the remainder of this section we assume m is odd. X • n > 0. ¯. Y ∈ Vect (∂D). and all of them are nondegenerate. Yt (p) = 1 ¯ ¯ | ((1 − t)n + tX |(1 − t)n + tX Observe that this map is well defined since ¯ |2 = t2 + (1 − t)2 + 2t(1 − t)(n • X ¯ ) > 0. We first describe the integrand G∗ D dAm . . Y ) = n • (D X Y ) = −X • (D Y n). Set Dε = D \ Bε (p). for any p ∈ ZX . . pν }. the linear map AX.p : Tp Rm+1 → Tp Rm+1 . GAUSS-BONNET AGAIN?!? 381 We want to prove that the above equality holds even when m is odd. 2. if ∇ is a connection on T D ˆ . The manifold without boundary D is equipped with an orientation reversing involution ˆ →D ˆ whose fixed point set is ∂D. X ¯ = 1 X. where the nondegeneracy of X is defined in terms of ˆ . If X is nondegenerate. an arbitrary connection on T D X then q is nondegenerate if the map AX. The normal vector field n defines a basis of L. and q ∈ Z ˆ . If X is a vector field on D which is equal to n along ˆ by setting ∂D. More precisely. then we obtain a vector field X on D X := X on D −ϕ∗ (X ) on −D.8) X (p). where L is a real line bundle along which the differential of ϕ acts as −L . σm 1 ¯∗ X d(dAm ) = 0 on Dε . D = D ∪∂D (−D). CLASSICAL INTEGRAL GEOMETRY The vector field X does not vanish on Dε . ˆ (q ) X . This is the smooth manifold D obtained by gluing D along ∂D to a copy of itself equipped with the opposite orientation. and we denote by the sign of its determinant. consider the double of D. If we let ε → 0 we deduce deg GD = (9. along ∂D ⊂ D ˆ we have a ϕ-invariant ϕ:D decomposition T D|∂D = T ∂D ⊕ L. This map is independent of the connection ∇. In particular. X |X | Set Ω := Observe that dΩ = Stokes’ theorem then implies that Ω= ∂Dε Dε 1 ¯∗ X dAm . σm dΩ = 0 =⇒ deg GD = Ω= ∂D p∈ZX ∂Bε (p) Ω. To give an interpretation of the right-hand side of the above equality.q : Tq D → Tq D. v → ∇v X is an isomorphism. where the spheres ∂Bε (p) are oriented as boundaries of the balls Bε (p). Moreover ZX = ZX ∪ ϕ(ZX ). then so is X . p∈ZX for any nondegenerate admissible vector field X .382 CHAPTER 9. and we obtain a map ¯ :D ¯ ε → S m−1 . 3. . via the beautiful tube formula of Weyl. Then 1 σm det(−SD )dV∂D = deg GD = χ(D). ∂D p → n(p) = unit outer normal.5 (Gauss-Bonnet for domains). Suppose D is a relatively compact open subset of Rm+1 with smooth boundary ∂D.3. We will introduce these quantities trough the back door. the map X 383 : ZX → {±1} = X satisfies X (p) (ϕ(p)). which describes the volume of a tube of small radius r around a compact submanifold of an Euclidean space as a polynomial in r whose coefficients are these geometric measures. we chose to describe these facts in the slightly more restrictive context of tame geometry. and by SD the second fundamental form of ∂D. On the other hand.9. the general Poincar´ e-Hopf theorem (see Corollary 7. Surprisingly. and averaging the Euler characteristics of such slices.3 Curvature measures In this section we introduce the main characters of classical integral geometry. these coefficients depend only on the Riemann curvature of the induced metric on the submanifold. To keep the geometric ideas as transparent as possible. because m is odd. Theorem 9. the so called curvature measures. and we describe several probabilistic interpretations of them known under the common name of Crofton formulæ. We denote by GD the oriented Gauss map GD : ∂D → S m .8) Hence X q ∈ZX (q ) = 2 p∈ZX X (q ) = 2 deg GD . The Crofton Formulæ state that these curvature measures can be computed by slicing the submanifold with affine subspaces of various codimensions.2. We have thus proved the following result. Since ∂D is odd dimensional and oriented we deduce that χ(∂D) = 0.48) implies that ˆ (q ) X q ∈ZX ˆ ˆ ). CURVATURE MEASURES and. Y ∈ Vect (∂D). Y ) = n • (D X Y ). and the analytical machinery to a minimum. (9. ∀X. ∂D 9. and therefore 2χ(D) = χ(D) = 2 p∈ZX X (q ) = 2 deg GD . = χ(D Using the Mayer-Vietoris theorem we deduce χ(D) = 2χ(D) − χ(∂D).2. SD (X. For every set X we will denote by P(X ) the collection of all subsets of X .. . Unavoidably.e. 29] for more systematic presentations.384 CHAPTER 9. i. then T (A) ∈ Sn . but since this is not yet the case. universal. E2 . such as extensions of R by “infinitesimals”. where Pa (x) = a0 + a1 x + · · · an−1 xn−1 + xn . P1 . . The Tarski-Seidenberg theorem implies that the projection of Zn on the subspace with coordinates (ai ) is a semialgebraic set Rn := a = (a0 . and T : Rm → Rn is an affine map. the subsets described by finitely many polynomial equations. collection of systems of polynomial inequalities. P3 .. or nasty functions such as sin(1/t).3. ∩ and complement. If A ∈ Sm . The Tarski-Seidenberg theorem (see [15]) states that Salg is a structure. E1 . other than R.1 Tame geometry The category of tame spaces and tame maps is sufficiently large to include all the compact triangulable spaces. Hawaiian rings. An R-structure 2 is a collection S = {Sn }n≥1 . yet sufficiently restrictive to rule out pathological situations such as Cantor sets. This is a highly condensed and special version of the traditional definition of structure. Denote by Sn alg the collection of semialgebraic n n subsets of R . we will have to omit many interesting details and contributions.3. The model theoretic definition allows for ordered fields. then A × B ∈ Sm+n . a0 + a1 x + · · · an−1 xn−1 + xn = 0 . Pa (x) = 0 . an−1 ) ∈ Rn+1 . Sn contains all the real algebraic subsets of Rn . . where each Sj is described by finitely many polynomial inequalities. CLASSICAL INTEGRAL GEOMETRY §9. the polynomial Pa has a real root if and only if the coefficients a satisfy at least one system of polynomial inequalities from a finite. . an−1 ) ∈ Rn . Sn is closed under boolean operations. Consider the real algebraic set Zn := (x. i. We believe that the subject of tame geometry is one mathematical gem which should be familiar to a larger geometric audience. To appreciate the strength of this theorem we want to discuss one of its many consequences. with the following properties. . . In other words. 28. 2 . ∪. Example 9.1. but we refer to [23. a0 . ∃x ∈ R. we devote this section to a brief introduction to this topic. (Semialgebraic sets). . If A ∈ Sm . the subsets S ⊂ R which are finite unions S = S1 ∪ · · · ∪ Sν . . and B ∈ Sn . P2 .e. Sn contains all the closed affine half-spaces of Rn . Sn ⊂ P(Rn ). This can come in handy even if one is interested only in the field R. To see this. describe the same set. each describing an S-definable set. ∩.2. (b) The image and the preimage of a definable set via a definable function is a definable set. Suppose S is a structure. the two different looking formulas x ∈ R. suppose we are given a formula φ(a. observe that the operators ∧. Observe that the universal quantifier can be replaced with the operator ¬∃¬.9. b). and taking the complement. In particular. If we start with a collection of formulas. A formula 3 is a property defining a certain set. the resolvent set R2 is described by the well known inequality − 4a0 ≥ 0. Given a collection of formulas. We say that a set is S-definable if it belongs to one of the If A. xn )| = i=1 3 n x2 i 1/2 We are deliberately vague on the meaning of formula. (a. ∀. b) ∈ A × B . B ⊂ Rn . (b. . c) ∈ A × C . For example. when n = 2. ∞).3. ¬. For example. f g Note that any polynomial with real coefficients is a definable function. . |(x1 . When n = 3. the Euclidean norm n | • | : R → R. we can obtain new formulas. Then the formula a ∈ A. b) ∈ Γf . x ∈ R. we know that there cannot exist any algebraic formulæ describing the roots of a degree n polynomial. [0. b) ∈ A × B . a2 1 Sn ’s. using the logical operations ∧. and quantifiers ∃. (a) The composition of two definable functions A → B → C is a definable function because Γg◦f = (a. CURVATURE MEASURES 385 For example. . The existential quantifier corresponds to taking a projection.3. . Example 9. then the projection A × B → A is the restriction to A × B of the linear projection Rm × Rn → Rm and P3 implies that the image of C is also definable. yet we can find algebraic inequalities which can decide if such a polynomial has at least one real root. then any formula obtained from them by applying the above logical transformations will describe a definable set. c) ∈ Γg . x ≥ 0}. ∨. B definable. B are S-definable then a function f : A → B is called S-definable if its graph Γf := (a. b = f (a) is S-definable. A. we can obtain a similar conclusion using Cardano’s formulæ. . ∃y ∈ R : x = y 2 }. ∃b ∈ B : (a. For n ≥ 5. describing a definable set C ⊂ A × B . ∪. ∃b ∈ B : φ(a. (c) Observe that Salg is contained in any structure S. ∨. The reason these sets are called definable has to do with mathematical logic. b) describes the image of the subset C ⊂ A × B via the canonical projection A × B → A. If A ⊂ Rm . ¬ correspond to the boolean operations. Then its closure cl(A) is described by the formula x ∈ Rn . the affine Grassmannian Graff k (Rn ) is a semialgebraic set because its is described by Graff k (Rn ) = (x. ¬ ∃ε¬ (ε > 0) ⇒ ∃a(a ∈ A) ∧ (x ∈ Rn ) ∧ (|x − a| < ε) . In the above description. Λk P = 0. (a ∈ A). to the pair (x. Let us examine the correspondence between the operations on formulas and operations on sets on this example. Similarly. (e) Suppose A ⊂ Rn is S-definable. Λm P = 0. (|x − a| < ε).3.386 CHAPTER 9. which is a vector subspace of dimension k .3. where Λj P denotes the endomorphism of Λj Rn induced by the linear map P ∈ End(Rn ). Finally. We rewrite this formula as ∀ε (ε > 0) ⇒ ∃a(a ∈ A) ∧ (x ∈ Rn ) ∧ (|x − a| < ε) . P ) ∈ Rn × Grk (Rn ). we can replace the universal quantifier to rewrite the formula as a transform of atomic formulas via the basic logical operations. which is the smallest structure containing S and the sets in An . P ) we associate the affine plane x + Range (P ). or o-minimal (order minimal) if it satisfies the property . P x = 0 . In the above description. The above formula is made of the “atomic” formulæ. ∀ε > 0. An ⊂ P(Rn ). and we deduce that cl(A) is also S-definable. P 2 = P. ∀m > k . (ε > 0). (x ∈ Rn ). affine simplicial complex K ⊂ Rn is S-definable because it is a finite union of affine images of finite intersections of half-spaces. we associate its range. (d) Any compact. to the orthogonal projection P of rank k . In the above formula we see one free variable x. An R-structure S is called tame. Observe that any Grassmannian Grk (Rn ) is a semialgebraic subset of the Euclidean space End+ (R) of symmetric operators Rn → Rn because it is defined by the system of algebraic (in)equalities Grk (Rn ) = P ∈ End+ (Rn ). and a collection A = (An )n≥1 . Definition 9. The logical connector ⇒ can be replaced by ∨¬. and the set described by this formula consists of those x for which that formula is a true statement. Arguing in a similar fashion we deduce that the interior of an S-definable set is an Sdefinable set. We say that S(A) is obtained from S by adjoining the collection A. Given an R-structure S. we can form a new structure S(A). CLASSICAL INTEGRAL GEOMETRY is S-definable. ∃a ∈ A : |x − a| < ε . which all describe definable sets. i = 1. . b) then the antiderivative F : (a. ∂xi then f is S -definable.) Suppose A is an S-definable set. For example. is also San -definable. b) → R x (9. Example 9. t → et . . and f : U → R is a C 1 function satisfying ∂f = Fi (x. The definable sets and functions of a tame structure have rather remarkable tame behavior which prohibits many pathologies. Marker ) The structure obtained by adjoining to San the graph of the exponential function R → R.3.9. 91]. connected and S -definable. For example. We will list below some of the nice properties of the sets and function definable in a tame structure S. we denote by San the structure obtained from Salg by adjoining the graphs of all the restricted real analytic functions. Any set A ∈ S1 is a finite union of open intervals (a. and f : A → R is a definable function. 28]. . Their proofs can be found in [23. The structure San is tame. . . and x0 ∈ (a. Fn : U × R → R are S definable and C 1 . . p is a positive integer. all the compact. CURVATURE MEASURES 387 T. Then A can be partitioned into 4 Our definition of pfaffian closure is more restrictive than the original one in [58. and we will denote it by San . (c)(Wilkie. and singletons {r}. −∞ ≤ a < b ≤ ∞. f (x)).4.3. It is perhaps instructive to give an example of function which is not definable in any tame structure. (a) The collection of real semialgebraic sets Salg is a tame structure. van den Dries.3. 1] such that f (x) = ˜(x) x ∈ Cn f 0 x ∈ Rn \ Cn . b) → R is C 1 . The smallest structure satisfying the above two properties. is called the pfaffian closure 4 of San . . .1) F (x) = x0 f (t)dt. (b)(Gabrielov-Hironaka-Hardt ) A restricted real analytic function is a function f : Rn → R ˜ defined in an open neighwith the property that there exists a real analytic function f n borhood U of the cube Cn := [−1. but it suffices for many geometric applications. . is a tame structure. Observe that if f : (a. . n. x ∈ (a. (d2 ) If U ⊂ Rn is open. • (Piecewise smoothness of tame functions and sets. real analytic submanifolds of Rn belong to gSan . the function x → sin x is not definable in a tame structure because the intersection of its graph with the horizontal axis is the countable set π Z which violates the o-minimality condition T. (d)(Khovanski-Speissegger ) There exists a tame structure S with the following properties (d1 ) San ⊂ S . b). San -definable. Macintyre. ∀x ∈ R. b). F1 . then there exists an S-definable C k -map γ : (0. then there exists a definable bijection ϕ : A → B if and only if A and B have the same dimensions and the same Euler characteristics. • (Definable selection. and we denote by ΛS the projection of S on Λ. Then there exists a definable function s : ΛS → S such that s(λ) ∈ Sλ .) If A is an S-definable set. the set χ(Sλ ). λ ∈ Λ. Sk . and each of the restrictions f |Si is a C p -function. ∀λ ∈ ΛS . .) If A and B are two compact definable sets. • (Definability of Euler characteristic. CLASSICAL INTEGRAL GEOMETRY finitely many S definable sets S1 . λ) ∈ S .) Suppose A. (a. then dim(cl(A) \ A) < dim A. (The map ϕ need not be continuous. The dimension of A is then defined as max dim Si .388 CHAPTER 9.) We say that a tame map Φ : X → S is definably trivial if there exists a definable set F . . Then f is continuous if and only if its graph is closed in X × Rn . and a definable homeomorphism Φ:K→A such that all the sets Φ−1 (Si ) are unions of relative interiors of faces of K . Sk }. In particular.) • (Definable triviality of tame maps. • (Curve selection.) If A is an S-definable set. ⊂ Z is finite. . and a definable homeomorphism τ : X → F × S such that the diagram below is commutative X Φ τ M S×F πS . • (Triangulability.) If (Sλ )λ∈Λ is a definable family of definable sets. • Any definable set has finitely many connected components. . We set Sλ := a ∈ A. and any finite collection of definable subsets {S1 .) For every compact definable set A. and x ∈ cl(A) \ A. Then a definable family of subsets of A parameterized by Λ is a subset S ⊂ A × Λ. Λ are S-definable. In particular. its range must be a finite subset of Z.) Suppose X is a tame set and f : X → Rn is a tame bounded function. Then the map Λ λ → χ(Sλ ) = the Euler characteristic of Sλ ∈ Z is definable. 1) → A such that x = limt→0 γ (t). S . • (Dimension of the boundary. . and each of them is definable. such that each Si is a C p -manifold. •(Scissor equivalence. . then the function Λ λ → dim Sλ ∈ R is definable. . . k > 0 an integer.) Suppose (Sλ )λ∈Λ is a definable family of compact tame sets. • (Definability of dimension. there exists a compact simplicial complex K . • (Closed graph theorem. i = 0. . where E is a finite dimensional real vector space. (c) Prove that the curve C = (t. Exercise 9. and each of them has finite length. and p is a positive integer. Yk such that each the restrictions Ψ : Ψ−1 (Yk ) → Yk is definably trivial. (b) Prove that the set (t.3. v = λ(v ). Definition 9. 1]. τ (g )v = gv. g → ρ(g ). 1.2 Invariants of the orthogonal group In the proof of the tube formula we will need to use H. ϕ is differentiable at t and m = ϕ (t) is S-definable. The vector space V has a tautological structure of O(V )-module given by τ : O(V ) → Aut(V ).3. m) ∈ [0. λ ∈ V ∗ = Hom(V. (a) Prove that ϕ is differentiable on the complement of a finite subset of [0. Suppose V is a finite dimensional Euclidean space with metric (−. A morphism of O(V )-modules (Ei . 1] × R2 .3. R). A subset A of some Euclidean space Rn is called tame if it is definable within a tame structure S.Weyl’s characterization of polynomials invariant under the orthogonal group. is a linear map A : E0 → E1 such that for every g ∈ O(V ) the diagram below is commutative E0 ρ0 (g ) A E0 K M E1 K ρ (g) M E1 1 A We will denote by HomO(V ) (E0 . v ∈ V. . − : V ∗ × V → R. ρ). − the canonical pairing −. −). Recall that an O(V )-module is a pair (E. We denote by O(V ) the group of orthogonal transformations of the Euclidean space V . λ. t ∈ [0.3. .6. 1] → R2 is an Sdefinable map. E1 ) the spaces of morphisms of O(V )-modules. Suppose S is a tame structure and Suppose ϕ : [0. while ρ is a group morphism ρ : O(V ) → Aut(E ). . §9. ∀v ∈ V.5.9. then there exists a partition of Y into definable C p -manifolds Y1 . ϕ(t) ). . ∀g ∈ O(V ). CURVATURE MEASURES 389 If Ψ : X → Y is a definable map. We denote by −. 1] has finitely many components. ρi ). ρ) → (V ∗ . Observe that (V ∗ )⊗n can be identified with the vector space of multi-linear maps ω : V n = V × · · · × V → R. Let us observe that if we denote by Sr the group of permutations of {1. r}. ρ† ). this beautiful result of Weyl since it will play an important role in the future. ρ† (g ) = ρ(g −1 )† . u). . (ρ† )⊗n : O(V ) → Aut (V ∗ )⊗n . . . without proof. ∀u. Hermann Weyl has produced in his classic monograph [100] an explicit description n of (V ∗ )⊗ O(V ) . Equivalently. g −1 v .390 CHAPTER 9. n ω ∈ (V ∗ )⊗ O(V ) ⇐⇒ ρ† (g ) ⊗n ω = ω. We obtain an action on (V ∗ )⊗n . ∀λ ∈ V ∗ . . n We denote by (V ∗ )⊗ O(V ) the subspace consisting of invariant tensors. v ∈ V. Tϕ (v1 ⊗ · · · ⊗ vr ) = vϕ(1) ⊗ · · · ⊗ vϕ(r) . Observe first that the metric duality defines a natural isomorphism of vector spaces D : V → V ∗. We follow the elegant and more modern presentation in Appendix I of [6] to which we refer for proofs. R) given by ρ† : O(V ) → Aut(V ∗ ). V ⊗s ) In particular. defined by v † . v ∈ V. n n so that (V ∗ )⊗ O(V ) can be identified with the subspace of O (V )-invariant multilinear maps V n → R. where ρ(h)† denotes the transpose of ρ(h). ∀g ∈ O(V ). v → v†. (V ∗ )⊗(r+s) O(V ) ∼ = Hom(V ⊗r . s we have isomorphisms of G-modules (V ∗ )⊗(r+s) ∼ = (V ∗ ⊗r) ⊗ V ⊗s ∼ = Hom(V ⊗r . V ⊗s ) O(V ) = HomO(V ) (V ⊗r . v = λ. then for every ϕ ∈ Sr we obtain a morphism of O(V )-modules Tφ ∈ HomO(V ) (V ⊗r . u = (v . V ⊗r ). V ⊗s ). This isomorphism induces an isomorphism of O(V )-modules D : (V. g → ρ† (g ). We conclude that for ever nonnegative integers r. . CLASSICAL INTEGRAL GEOMETRY It induces a structure of O(V )-module on V ∗ = Hom(V. where ρ† (g )λ. We would like to present here. g → ρ† (g )⊗n . t1d1 . . v 1 . . . ∀i = 1. . . tn1 . j =1 Observe that f (v 1 . v n ) → f (v 1 . . . . . . . tndn ) = f j =1 tij uj 1. . . v 1 . . . V ⊗r ) = R[Sr ] := ϕ∈Sr 391 cϕ Tϕ .3. v n ). .9. . . 2r and (V ∗ )⊗ O(V ) is spanned by the multilinear forms Pϕ : V 2r → R. . . defined by Pϕ (u1 . . v r ) = u1 . v n . . . d1 dn and f is O(V )-invariant if and only if Polf is O(V ) invariant. . n. and that HomO(V ) (V ⊗r . v j ). . (ϕ ∈ Sr ). . . . . . . . This form determines a multilinear form Polf : V d1 × · · · × V dn → R obtained by polarization in each variables separately. . dn tnj uj n . (v 1 . . . . . . . . t12 . V ⊗s ) = 0 ⇐⇒ r = s. . . . v n ) = Polf (v 1 . . . CURVATURE MEASURES Weyl’s First Main Theorem of Invariant Theory states that HomO(V ) (V ⊗r . . . . cϕ ∈ R. v ϕ(r) . . v n ). . The above result has an immediate consequence. . . n . Thus n (V ∗ )⊗ O(V ) = 0 ⇐⇒ n = 2r. Suppose we have a map f : V × · · · × V → R. . v n . . . . . qij (v 1 . n which is a homogeneous polynomial of degree di in the variable v i . . . d1 1 dn Polf (u1 1 . . Note that every function f : V × ··· × V → R n which is polynomial in each of the variables is a linear combination of functions which is polynomial and homogeneous in each of the variables. v n ) := (v i . . . v n ) = the coefficient of the monomial t11 t12 · · · t1d1 · · · tn1 · · · tndn in the polynomial d1 Pf (t11 . v ϕ(1) · · · ur . . . We can translate this result in terms of invariant multi-linear forms. u1 . . . For every 1 ≤ i ≤ j ≤ n we define qij : V × · · · × V → R. . ur . . r ∈ Z≥0 . . . . · · · .7 (Weyl)..j Ti.. hk = 2m. The polynomial P is called a degree n orthogonal invariant of tensors T ∈ V ⊗k if it is invariant as an element of (V ∗ )⊗kn . .j Tij ei ⊗ ej ∈ V ⊗2 is the trace tr(T ) = i.. (a) Consider the space E = V ⊗k . CLASSICAL INTEGRAL GEOMETRY Theorem 9. ed } of V . If f : V × · · · × V → R is a polynomial map then f is O(V )invariant if and only if there exists a polynomial P in the n+1 variables qij such that 2 f (v 1 .3.. Fix an orthonormal basis { e1 . . Observe that a degree n homogeneous polynomial P on E can by identified with an element in the symmetric tensor product Symd (E ∗ ) ⊂ (V ∗ )⊗2kn . such polynomials exist if and only if hk is even. For example. ϕ−1 ).j Tij Tji .ih . . Q(T ) = ij 2 ˜ (T ) = Tij . we deduce that the number of matchings is 1 2m (m!) · 2 m = (2m)! . First of all. hk } is an equivalence relation ∼ with the property that each equivalence class consists of two elements. Q i. A degree k polynomial P : V ⊗h → R is then a polynomial in the dh variables Ti1 . Weyl’s theorem implies that the only degree 1 invariant of a tensor T = i. . A tensor T ∈ V ⊗h decomposes as T = 1≤i1 . Here is briefly how to find a basis of the space of degree k invariant polynomials P : V ⊗h → R. 2(m!) . v n )1≤i≤j ≤n ). . . The space of degree 2 invariants is spanned by the polynomials (tr(T ))2 .8.j ei ⊗ ej = i Tii . We will construct a bijection between a basis of invariant polynomials and certain combinatorial structures called matchings.3.392 CHAPTER 9... A matching on the set Ihk := {1.. Thus to produce a matching we need to choose a subset A ⊂ Ihk of cardinality m = hk 2 . . .ih ≤d Ti1 . and then a bijection ϕ : A → Ihk \ A. Example 9. v n ) = P ( qij (v 1 .. Observing that the matching associated to (A. ϕ) is the same as the matching associated to (Ihk \ A. .ih ei1 ⊗ · · · ⊗ eih . . d}. and we denote by V (M. dist (x. We want to prove that there exists a polynomial PM (r) = pc rc + · · · + pn rn . . .3. F∼ . but we will not assume that it is orientable.µ(h) · · · Tµ(hk−h+1).9.. We set c := codim M = n − m.. .µ(hk) . §9. . by certain integral. For example. r) its volume. • pc = ω c · vol (M ). • all the coefficients of PM are described. intrinsic geometric invariants of M . where ω c rc is the volume of a ball of dimension c and radius r. we have • V (M. d} 393 such that i ∼ j =⇒ µ(i) = µ(j ). . . for all sufficiently small r. The collection P∼ . M ) ≤ r . .. For r > 0 we define the tube of radius r around M to be the closed set Tr (M ) := x ∈ M . Equivalently. the space of degree 1 invariant polynomials in tensors of order 4 has 4! = 6. For any matching ∼ define P∼ (T ) = µ∈F∼ Tµ(1).. can be identified with the set of functions µ : Ihk / ∼→ {1. up to some universal multiplicative constants. ∼ is a matching of Ihk is a basis of the space of degree k invariant polynomials V ⊗h → R.3 The tube formula and curvature measures dimensional submanifold of Rn . such that. r) is V (M. r) ≈ vol (M ) · ω c rc . Each polynomial in a basis constructed as above is a sum of d2 dimension 2(2!) terms. Suppose that M is an m- . 2. Note that a first approximation for V (M. We see that things are getting “hairy” pretty fast. so there are dm such functions. In this section we will assume that M is compact and without boundary. r) = PM (r). . CURVATURE MEASURES Denote by F∼ the set of functions µ : Ihk → {1.3. A. We define c Dr := t = (tα ) = (tm+1 . and we can use and thus we can identify Dr r c x ∈ M and t ∈ Dr as local coordinates on π −1 (U ). x) → (tα eα (x).2) M |dVn |. α |tα |2 ≤ r . e We have thus extended in a special way the local frame (eA ) of (T Rn )|U to a local frame of (T Rn )|Tr (U ) so that ˜A = 0. Observe that because M is compact. p). CLASSICAL INTEGRAL GEOMETRY Let N(M ) denote the orthogonal complement of T M in (T Rn )|M . The manifold with boundary Dr (Rn ) is a bundle of n-dimensional disks over Rn . . We define Dr (Rn ) := {(v. We will continue to use the indexing conventions we have used in Subsection §9. and ˜A : Tr (U ) → Rn by e ˜A (x + tα eα ) = eA (x). ⊂ T Rn .3. Fix a local orthonormal frame (eA ) of (T Rn )|M defined in a neighborhood U ⊂ M of a point p0 ∈ M such that for all 1 ≤ i ≤ m vector field ei is tangent to U . De (9. Tr (M ) Nr ( M ) If π : Nr (M ) → M denotes the canonical projection. |v | ≤ r and we set Nr (M ) := N(M ) ∩ Dr (Rn ). while Nr (M ) is a bundle of c-dimensional disks over M . and we will call it the normal bundle of M → Rn . c × U with the open subset π −1 (U ) ⊂ N (M ). r0 ). x) ∈ Nr (M ). . there exists r0 = r0 (M ) > 0 such that. r) = π∗ E∗ (9.1. .394 CHAPTER 9. (t.3) ˜α e . Define Tr (U ) := EM ( Nr (U ) ) ⊂ Rn . If we denote by |dVn | the Euclidean volume density on Rn we deduce V (M. The exponential map E : T Rn → Rn restricts to an exponential map EM : N(M ) → Rn . tn ) ∈ Rc . ∀α. the exponential map EM induces a diffeomorphism EM : Nr (M ) → Tr (M ). then we deduce from Fubini’s theorem that V (M.2. p ∈ Rn . Note that we have a diffeomorphism c Dr × U −→ Nr (U ) := Nr (M )|U . . for every r ∈ (0. r) = vol Tr (M ) = |dVn | = E∗ M |dVn |.3. M We want to give a more explicit description of the density π∗ E∗ M |dVn |. v ∈ Tp Rn . x) as coordinates and we have EM (tα eα (x). and set 395 associated to the Levi-Civita connection D by ΘA B . The equalities (9. ΦiB := π (ΘiB |U ) ∈ C (Nr (U ) ). EM θ = dt . ij so that j j E∗ Mθ = φ − i (t • Φij )φi .| 1≤i. CURVATURE MEASURES ˜A . ˜C ΘA CB = e so that D eC eB = ΘA CB eA .3. φα = dtα .4) imply A i ΦA B = ΦiB φ . Using (9. On Nr (M )|U we use (t.9.4) Finally set ∗ A 1 A ∗ A ∞ ΦA B = π (ΘB |M ) ∈ Ω (Nr (U )). ∀α = B = ΘiB θ . We find it convenient to set +1 c Φij = (Φm .3) we deduce A i ΘA ⇒ ΘA αB = 0. . .3. Φn ij ) : U → R . x) = t • Φij (x) Note that the volume density on Rn is |dVn | = |θm+1 ∧ · · · ∧ θn ∧ θ1 ∧ · · · ∧ θm . Define the m × m symmetric matrix S := S (t. ei ) with dual coframe (φA ) defined by φi = π ∗ θ i .j ≤m . We denote by (θ A ) the coframe of Tr (U ) dual to e c α Over Dr × U we have a local frame (∂t . ∀i. (9. Hence j j α j i j E∗ M θ = φ + t Φiα φ = φ − α i ∗ β β tα Φ α ij φ . .3. 1 Consider the 1-forms ΘA B ∈ Ω Tr (U ) the frame (˜ eA ) on Tr (U ).3. . We have A E∗ Mθ = i (eA • D ei EM )φi + α (eA • ∂tα EM )dtα = i eA • (ei + tα D ei eα )φi + α i α δAα dtα = δAi φi + tα ΦA iα φ + δAα dt . x) = x + tα eα . 1 ≤ i. x) = m (−1)ν ρν Pν Φij (x) • ω . Then c−1 E∗ |dρ| × |dω | × |dVM |. x) | |dt ∧ dφ| = det( − S (t. ρ and denote by |dω | the area density on the unit sphere in Rc . (9. 1 ≤ i.3. Now set 1 ρ := |t|. We set P ν Φij (x) := S c−1 Pν Φij (x) • ω |dω |. x) |dt ∧ dφ|.396 CHAPTER 9. x) ρ (9. Theorem 9.3. x) |dt| × π |dVM |. . P ν uij is an O(c)-invariant.3. For every t ∈ Rc = span (em+1 . Assume therefore ν = 2h.7 on invariants of the orthogonal group O(c) implies that P ν must be a polynomial in the quantities qi. ω := t. M |dVn | = det( − ρS (ω. We deduce that (−1)ν Pν ( uij • t ) = tr Λν U (uij . := uik • uj . . m π∗ E∗ M |dVn | = ν =0 (−1)ν c+ν r P ν Φij (x) |dVM (x)|.k. ν =0 where Pν denotes a homogeneous polynomial of degree ν in the m2 variables uij ∈ Rc . c+ν (9. . dφ = dφ1 ∧ · · · ∧ dφm .6) Observe that det( − ρS (ω.3. t) : Rm → Rm given by the m × m matrix ( uij • t )1≤i. em+c ) we form the linear operator U (t) = U (uij . we deduce ∗ E∗ M |dVn | = det( − S (t. Because these quantities are homogeneous of degree 2 in the variables uij we deduce P ν = 0 if ν is odd. where |dt| denotes the volume density on Rc . . .j.5) Recalling that | ∧i dθ i |M | is the metric volume density |dVM | on M . h ∈ Z≥0 .7) We would like to determine the invariant polynomials P ν uij . homogeneous polynomial of degree ν in the variables uij ∈ Rc . We conclude. For simplicity we write |dVM | instead of π ∗ |dVM |. t) = the sum of all the ν × ν minors of U (t) symmetric with respect to the diagonal. j ≤ m.j ≤m . Above. CLASSICAL INTEGRAL GEOMETRY E∗ M dVn = | det( − S (t. dt = dtm+1 ∧ · · · ∧ dtn . j ≤ m. ν.3. .k2 . . . . CURVATURE MEASURES 397 These minors are parametrized by the subsets I ⊂ {1.ϕ = j =1 qϕ2j −1 .j. 2h .σ.c = µI (uij ) . Lemma 9.9. .ϕ . There exists a constant ξ = ξm. . Note that µI is an O(c)-invariant polynomial in the variables {uij }i.σ. . . ν and c such that µ ¯I = ξQI .c depending only on m.3. where {k1 . and by µI its average.j ∈I . . . and by Qσ. . For any σ.k. ϕ)QI. m}.ϕ2j . . 2h. k2h } and { 1 . . • µI is separately linear in each of the variables uij . . The skew-symmetry of µI with respect to the permutations of rows and columns now implies that µI must be a multiple of QI . The constant ξ satisfies ξm. ∀uij ∈ Rc QI (uij ) . m} of cardinality #I = ν .σ2j = j =1 uϕ2j −1 σ2j −1 • uϕ2j σ2j . are permutations of I . 2h−1 . Let I = {1 ≤ i1 < i2 < · · · < i2h ≤ m} ⊂ {1. 2 · · · qk2h−1 k2h . . . We regard µI as a function on the vector space of (2h) × (2h) matrices U with entries in Rc U = [uij ]i. .σ ∈SI (σ. We observe that µI satisfies the following determinant like properties. . 2h } 1.ν. ϕ ∈ SI we denote by (σ. For ϕ ∈ SI we set ϕj := ϕ(ij ). • µI changes sign if we switch two rows (or columns).σ2j −1 . µI (uij ) := S c−1 µI ( uij • ω )dω.ϕ the invariant polynomial h h QI. . For every ω ∈ Rc we denote by µI ( uij • ω ) the corresponding minor of U (ω ). Proof. .j ∈I . ∀j = 1. and denote by SI the group of permutations of I .9. We deduce that µI is a linear combination of monomials of the form qk1 . QI := ϕ. • µI is a homogeneous polynomial of degree h in the variables qi. . ϕ) the signature of the permutation σ ◦ ϕ−1 . . if we set    t=  we deduce   1   ∈ Rc . . . where h = h1 + · · · + hc . t) =   t1 0 · · · 0 t1 · · · . .ν. 0 Then. . 2hc we have |ω 1 |2h1 · · · |ω c |2hc |dω | = S c−1 +1 2hc +1 2Γ( 2h1 2 ) · · · Γ( 2 ) h Γ( c+2 2 ) . . S c−1 At this point we invoke the following result whose proof is deferred to the end of this subsection.ϕ = j =1 uϕ2j −1 σ2j −1 • uϕ2j σ2j . . We can assume I = {1. . which is nonzero if and only if σ = ϕ. . . .. Hence ξm.c = 1 (2h)! |ω 1 |2h |dω |. . we have h |ω 1 |2h |dω |.10. . . tc    U (uij . Lemma 9. µI (uij • t) = |t1 |ν . . 2h} and we choose    uij =   1 0 .  t1 t2 .  Hence µI (uij ) = On the other hand. nonnegative integers 2h1 . We conclude that for this particular choice of uij we have QI = (2h)!. . . For any even. . . . 2. .3. CLASSICAL INTEGRAL GEOMETRY so it suffices to compute the numerator and denominator of the above fraction for some special values of uij . S c−1 (9.σ. t1     . .3. ω = t  |t|     ∈ Rc .8) QI. 0 0 ··· 0 0 .398 CHAPTER 9. ϕ)QI. ϕ) j =1 qϕ2j −1 .11) . .σ. τh ) ∈ G = S2 × · · · × S2 h defines a permutation of I by regarding τ1 as a permutation of {i1 . i4 } etc. We observe that every element τ = (τ1 . (9.ϕ2j .ϕ∈SI (σ. CURVATURE MEASURES We deduce ξm.3.τ ∈G (σ.τ ∈G (στ.c #I =2h QI (R). . We deduce that if uij = Sij (x). i2 }.ϕ = ϕ∈SI h ϕ∈SI .2h. . ϕ)QI. τ2 as a permutation of {i3 . ϕ) j =1 h Rϕ2j −1 ϕ2j σ2j −1 σ2j =2 h σ.9.10) We denote by S2 the group of permutations of a linearly ordered set with two elements. (9.σ2j −1 .ϕ∈SI (σ.ϕ = σ.c = and P ν (uij ) = ξm.ϕτ = ϕ∈SI (σ. ϕτ )QI.ϕ2j −1 . . then for every σ ∈ SI we have (σ.σ. ϕ3 < ϕ4 .σ2j − qϕ2j . The space Sk /G of left cosets of this group can be identified with the subset SI ⊂ SI consisting of bijections ϕ : I → I satisfying the conditions ϕ1 < ϕ2 . ϕ) j =1 k Rϕ2j −1 ϕ2j σ2j −1 σ2j we deduce h Using the skew-symmetry Rijk = −Rij (σ. .3.9) QI (uij ).c #I =ν 399 +1 2Γ( 2h2 )Γ(1/2)c−1 h (2h)!Γ( c+2 2 ) . .3. ϕ2h−1 < ϕ2h . ϕ) σ ∈SI ϕ∈SI h j =1 Rϕ2j −1 ϕ2j σ2j −1 σ2j = σ ∈SI .σ2j h = ϕ∈SI (σ.2h. ϕ) j =1 Rϕ2j −1 ϕ2j σ2j −1 σ2j =:QI (R) We conclude that P ν (ψij ) = 2h ξm.2h.σ.3. Thus G is naturally a subgroup of SI . .σ2j −1 . (9. . .2h. [99]. ϕ) j =1 Rϕ(i2j −1 )ϕ(i2j )σ(i2j −1 )σ(i2j ) . where Qh is a polynomial of degree h in the curvature.3. .c 2h = = h (c + 2h)ω c+2h σ c+2h−1 σ c+2h−1 (2h)!Γ( c+2 2 ) = γ (2h) 1 2h Γ(h + 1/2) = h = . we have +1 2Γ( 2h2 )Γ(1/2)c−1 2h ξm. m = 2h. Theorem 9. m} we define h QI (R) = σ. CLASSICAL INTEGRAL GEOMETRY Using (9.ϕ∈SI (σ. . dim M = m. Let us observe that the constant 2h ξm. .3.2h. Example 9.c 2h ξm. µm−2h (M ) = 1 (2π )h h! M Qh (R)|dVM |. Denote by R the Riemann curvature of the induced metric on M . and oriented. c = n − m. . . em ) of T M we can express the polynomial Qh (R) as Qh (R) := #I =2h QI (R).3. (b) Assume now that m is even.2) and (9. (a) If h = 0 then µm (M ) = vol (M ).7) we deduce that m/2 V (M. Suppose M is a closed.400 CHAPTER 9. Then c + 2h = m + c = n and µ0 (M ) = 1 (2π )h 1 Qh (R)dVM . where for every I = {i1 < i2 < · · · < i2h } ⊂ {1.12. It depends only on h. h! M . . orthonormal frame (e1 .11 (Tube formula). Indeed. . Then for all r > 0 sufficiently small we have m/2 V (M.2h. By choosing a local. Γ(1/2)1+2h (2h)! π (2h)! (2π )h h! We have thus obtained the following celebrated result of Hermann Weyl. compact submanifold of Rn .3. r) = vol (Tr (M )) = h=0 ω c+2h rc+2h µm−2h (M ).c (c + 2h)ω c+2h M Qh (R)|dVM |.2h. r) = vol (Tr (M )) = h=0 ω c+2h rc+2h 2h ξm.c (c + 2h)ω c+2h is independent of the codimension c. 13) (d) Using the definition of the scalar curvature we deduce that for any m-dimensional submanifold M m → Rn we have µm−2 (M.9.3. g ) = constm sg |dVg |. g ).2. . . We denote by cν (κ) the elementary symmetric polynomial of degree ν in the variables κi . and we have m E∗ M dVRm+1 = det( − tS )dt ∧ dVM = ν =0 m r −r (−1)ν tν cν (κ)dt ∧ dVM m/2 π∗ E∗ M dVRm+1 = ν =0 tν dt cν (κ)dVM = 2 h=0 r2h+1 c2h (κ)dVM 2h + 1 so that m/2 m/2 ω 1+2h r1+2h µm−2h (M ) = V (M. M . . The eigenvalues κ1 .j ≤m . . σ 2h 2h (9. . . σ 2h = (2h + 1)ω 1+2h . M If M = S m → Rm+1 is the unit sphere then κi = 1 and we deduce that µm−2h (S m ) = 2 σm m . Fix a point x0 ∈ M and assume that at this point the frame (e1 .3. In this case c = 1.3. We conclude that µm−2h (M ) = 2 σ 2h c2h (κ)dVM . M (9. so that µ0 (M. CURVATURE MEASURES 401 Comparing the definition of Qh (R) with (9. em ) diagonalizes the second fundamental form so that Sij = κi δij . where we recall that em+1 is in fact the oriented unit normal vector field along M . g ) = e(M.7a) we deduce that the top dimensional form 1 1 Qh (R)dVM ∈ Ω2h (M ) (2π )h h! is precisely the Euler form associated with the orientation of M and the induced metric. . r) = 2 h=0 h=0 r2h+1 2h + 1 M c2h (κ)dVM . Sij = (D ei ej ) • em+1 . .12) (c) Suppose now that M is a hypersurface. Consider the second fundamental form S = (Sij )1≤i. κm are the principal curvatures at the point x0 . σ =ϕ∈SI = σ ∈SI Rσ1 σ2 σ1 σ2 Rσ3 σ4 σ3 σ4 + The first sum has only three different monomials. 2 Hence constm = 2 1 1 = =⇒ µm−2 (M. M where sround denotes the scalar curvature of the round metric on the unit sphere. CLASSICAL INTEGRAL GEOMETRY where s denotes the scalar curvature of the induced metric g .1) we deduce sround = i. m = dim M . To find constm we compute µm−2 (M ) when M = S m .3. M We will say that µw (M. the shape operator of S m → Rm+1 is the identity operator From Theorema Egregium we deduce that all the sectional curvatures of S n are equal to one. and w is nonnegative integer. oriented. Riemann manifold. ϕ)Rσ1 σ2 ϕ1 ϕ2 Rσ3 σ4 ϕ3 ϕ4 (σ. Observe that the Grassmannian of oriented codimension one subspaces of Rm+1 can be identified with the unit sphere S m . Definition 4.13) we deduce 2σ m m σ 2 2h = constm sround |dVS m |. If (M. ϕ)Rσ1 σ2 ϕ1 ϕ2 Rσ3 σ4 ϕ3 ϕ4 .22. Using (9. g ) = σ2 2π 2π M sg |dVg |. m − w = 2h. If m − w is odd we set µw (M ) = 0. and constm is an universal constant. compact.13. g ) = 1 (2π )h h! Qh (R)|dVM |. We set |dµw | := 1 Qh (R)|dVM |.5. each of them appearing twice is them sum. If m − w is an even.j Rijij = i. g ) is the weight w curvature measure of (M. g ).3.ϕ∈SI (σ. We see that the map g → µm−2 (M. (2π )h h! . (e) The polynomial Q2 still has a “reasonable form” Q2 (R) = #I =4 QI Then #SI = 6 and QI = σ. then we set µw (M. the oriented Gauss map of the unit sphere S m is the identity. g ) is a closed. nonnegative integer. depending only on m.j 1=2 m .402 CHAPTER 9.2. The second sum has 6 2 different monomials (corresponding to subsets of cardinality 2 of SI ) and each of them appears twice.2. In particular. Using the equalities (4. Definition 9. g ) is precisely the Hilbert-Einstein functional we discussed in Subsection §4. Hence.2. hc ) = We have e−|t| = e−|t+1| · · · e−|t c 2 2 c |2 e−|t| |t1 |2h1 · · · |tc |2hc |dt|. σ : I → I is defined by choosing a linear ordering on I . i. . . CURVATURE MEASURES and we will refer to it as the (weight w) curvature density. . ρ = |t|.14. M ) = r} = ∂ Tr (M ). The fastest way to argue this is by invoking Nash embedding theorem which implies that any compact manifold is can be isometrically embedded in an Euclidean space. 403 Remark 9. for any finite set I . we deduce that I (h1 . g ) as a quantity measured in meterw . . . . S c−1 §9. dist (x.3. We do not assume that M is orientable. using spherical coordinates.e. independent of the choice of local frames used in their definition. . Rc 2 so that ∞ −∞ I (h1 . For every sufficiently small positive real number r we set Mr := {x ∈ Rn . Consider the integral I (h1 . we should think of µw (M. |t| On the other hand. . ω = h = h1 + · · · + hc . compact submanifold of Rn . set c = n − m. the quantities |dµw | are indeed well defined.10. hc ) = j =1 e−s s2hj ds = 2c j =1 0 2 c ∞ e−s s2hj ds 2 (u = s2 ) = c j =1 0 ∞ c e −u hj −1/2 t du = j =1 Γ 2hj + 1 . . but it is independent of this choice. As usual.9. (b) The weight of the curvature density has a very intuitive meaning. Namely. . Proof of Lemma 9. hc ) = (u = ρ2 ) = 1 2 |ω 1 |2h1 · · · |ω c |2hc dω S c−1 0 and recalling that |ω 1 |2h1 · · · |ω c |2hc dω S c−1 ∞ 0 ∞ e−ρ ρ2h+c−1 dρ 2 e−u u c+2h −1 2 du 1 c + 2h = Γ 2 2 |ω 1 |2h1 · · · |ω c |2hc dω . . . the proof of the tube formula then implies that these densities are indeed well defined. We can prove this by more elementary means by observing that.3. For submanifolds of Rn . .3. ϕ) of two permutations ϕ. orientable or not. (a) Let us observe that for any Riemann manifold M . and we denote by g the induced metric on M .3. the relative signature (σ. 2 1 t.4 Tube formula =⇒ Gauss-Bonnet formula for arbitrary submanifolds Suppose M m ⊂ Rn is a closed. Mr = Sr n−1 µp (Sr )=2 n ωn p n p r = 2ω p r . g ). If in the formula (9. ε) = V (M.14) In the above equality it is understood that µw (M ) = 0 if m − w is odd. r + ε) − V (M. g ).404 CHAPTER 9.3.3. gr ) = 2µp (M. which implies that ω 1+2h ε1+2h µn−1−2h (Mr .17) . The last equality agrees with our previous computation (9. then we deduce that n−1 .3. ∀0 < r 1.3. and we denote by gr the induced metric. even though the manifold M may not be orientable.14) we let p = 0 we deduce µ0 (Mr . ∀0 < r 2 1. and we conclude that Mr is the (n − 1)-dimensional sphere of radius r. 1 + 2h We make a change in variables.15) If in the formula (9. In particular. (9. The Gauss-Bonnet theorem for oriented hypersurfaces implies χ(Mr ) = µ0 (Mr . We deduce µn−1−2h (Mr . gr ) = 2µ0 (M. g ) = χ(Mr ). gr ) = 2 ω 1+2h k≥0 ω c+2k c + 2k c−1+2k−2h r µn−c−2k (M.3. n − p ω n−p (9. CLASSICAL INTEGRAL GEOMETRY The closed set Mr is a compact hypersurface of Rn . g ).1. so that we can rewrite the above formula as m µp (Mr . V (Mr . gr ) = 2 w=0 n − w ω n−w p−w r µw (M. (9. gr ) h≥0 = k≥0 ω c+2k (r + ε)c+2k − (r − ε)c+2k µn−c−2k (M.14) we assume that the manifold M is a point. Then c + 2k = n − w.13).16) where n p is defined by (9. ∀0 ≤ p ≤ m = dim M. w := n − c − 2k = m − 2k. We set p := n − 1 − 2h. g ).3. g ). c + 2k − 1 − 2h = p − w.3. Observe that the tube Tr (M ) is naturally oriented. 1 + 2h = n − p. p ω n−p p (9. gr ) so that 1 µ0 (M. n − p ≡ 1 mod 2. Observe that for r and ε sufficiently small we have Tε (Mr ) = Tr+ε (M ) − Tr−ε (M ) so that. we deduce that if the codimension of M is odd then r →0 lim µp (Mr . r − ε).16). if codim M is odd. 9. and the second fundamental form will be a scalar valued form. (b) By invoking the Nash embedding theorem. Assume therefore that m is even. Then µ0 (M. Suppose M is a closed. Denote by g the induced metric. compact submanifold of an Euclidean space Rn . Proof. (a) We want emphasize again that in the above theorem we did not require that M be orientable which is the traditional assumption in the Gauss-Bonnet theorem. . Y ∈ Vect (∂D). with smooth boundary ∂D. and the theorem follows from (9. Let us record for later use the following corollary of the above proof. The next definition formalizes this observation. smooth submanifold M of an Euclidean space V such that dim V − dim M is odd we have χ(M ) = 2χ( ∂ Tr (M ) ). we define the co-oriented second fundamental form of D to be the symmetric bilinear map SD : Vect (∂D) × Vect (∂D) → C ∞ (∂D). Definition 9.3. For any relatively compact open subset D of an Euclidean space V . ∀0 < r 1.3. orientable or not. If the dimension n of the ambient space is odd. If m = dim M is odd. g ) = χ(M ). then the normal bundle admits a canonical trivialization.3.16. then the Poincar´ e duality for the oriented n-dimensional manifold with boundary Tr (M ) implies χ(Mr ) = χ( ∂ Tr (M ) ) = 2χ( Tr (M ) ) = 2χ(M ). If n is even.18. §9.3. CURVATURE MEASURES 405 Theorem 9. we deduce that the tube formula implies the Gauss-Bonnet formula for any compact Riemann manifold. Corollary 9. Remark 9.5 Curvature measures of domains in an Euclidean space The second fundamental form of a submanifold is in fact a bilinear form with values in the normal bundle. we apply the above argument to the embedding M → Rn → Rn+1 .17.17). X. Y ) = (D X Y ) • n.3.15 (Gauss-Bonnet). where n : ∂D → V denotes the outer unit normal vector field along ∂D. where we observe that the metric induced by the embedding M → Rn+1 coincides with the metric induced by the original embedding M → Rn . If the submanifold happens to be the boundary of a domain. then both χ(M ) and µ0 (M ) are equal to zero and the identity is trivial. SD (X.3. For every closed compact.3. . z j trj (B ) = det(V + zB ). for every p ∈ U . Fix p0 ∈ M and a local. . . such that. . . (X. r] . em ) of T M defined in a neighborhood U of p0 in M . j ≥0 Equivalently. . We define the tube of radius r around D to be Tr (D) := x ∈ Rm+1 . We denote by EM the exponential map EM : (T Rm+1 )|M → Rm+1 . . We obtain a dual coframe θ . dist (x. so that vol (Tr (D) ) = vol (D) + ∆r E∗ M dVRm+1 . i. positively oriented local orthonormal frame (e1 . For sufficiently small r the map EM defines a diffeomorphism EM : ∆r → Tr (D) \ D. orthonormal frame of Rn .. For r > 0 we denote by ∆r ⊂ (T Rm+1 )|M the closed set ∆r := (tn(p). the pullback of E∗ M dVRm+1 to ∆r has the description 1 m E∗ M dVRm+1 = det  − tSM dt ∧ dθ ∧ · · · ∧ θ  = m j =1  trj (−SM )tj  dt ∧ dθ 1 ∧ · · · ∧ θ m .406 CHAPTER 9. relatively compact subset with smooth boundary M := ∂D. p). CLASSICAL INTEGRAL GEOMETRY Suppose D ⊂ Rm+1 is an open. . D) ≤ r . . As in the previous section. p) = p + X. For every symmetric bilinear form B on an Euclidean space V we define trj (B ) the j -th elementary symmetric polynomial in the eigenvalues of B . . p ∈ M. X ∈ Tp Rm+1 . θ n . t ∈ [0. θ 1 . e1 (p). em (p)) is a positively oriented. . We denote by n the unit outer normal vector field along M := ∂D. . .e. trj B = tr Λj B : Λj V → Λj V . p) −→ EM (X. and by S = SD the co-oriented second fundamental form of D. the collection (n(p). p ∈ M. 19) Suppose X → Rm+1 is a closed.18) Theorem 9. Then.5 shows that. compact smooth submanifold. +1 the ball of radius r in Rm+1 . The tube formula for X implies that ω j εj µm+1−j (Dr ) = j ≥0 k≥0 ω k r + ε)k µm+1−k (X ). We denote by Dm r +1 +1 Tε (Dm ) = Dm r r+ε . We deduce that µm+1−j (Dr ) = 1 ωj ωk k≥j k k−j r µm+1−k (X ) j . Then the curvature densities of D are the densities |dµj | on ∂D defined by 1 trn−j (−SD )|dV∂D |.3.2. ω m+1−k k k (9. m+1 vol Tr (D) = k=0 ω m+1−k rm+1−k µk (D). dim V = n. so that ω m+1 (r + ε)m+1 = k≥0 +1 ω m+1−k εm+1−k µk (Dm ). and SD denotes the co-oriented second fundamental form of ∂D. Definition 9. just as in the case of submanifolds.3. Then for every sufficiently small r > 0.3. Define µm−j (D) := and 1 σj M trj (−SM )dVM . |dµj | := σ n−j where |dV∂D | denotes the volume density on ∂D induced by the Euclidean metric on V . we have µ0 (D) = χ(D). r We conclude +1 µk (Dm )= r ω m+1 m + 1 k m+1 k r = ωk r .3. Suppose D is a relatively compact domain with smooth boundary of an Euclidean space V . the tube Dr := Tr (X ) is a compact domain with smooth boundary and Tε (Dr ) = Tr+ε (X ).19.9. CURVATURE MEASURES We deduce ∆r 407 E∗ M dVRm+1 = j ≥0 rj +1 j+1 M trj (−SM )dVM . 0 ≤ j ≤ m. (9. µm+1 (D) := vol (D) so that using the equality σ j = (j + 1)ω j we deduce the tube formula for domains. and we obtain the following generalization of the tube formula µp Tr (X ) = = w 1 ω n−p w ω n−w n − w p−w r µw (X ) n−p ω p−w n − w p−w r µw (X ).20 (Crofton Formula). We will prove that there exists a constant ξm. Recall that on Grc we have a natural metric with volume density |dγc | and total volume Vc := n−1 j =0 σ j m−1−c c−1 σj j =0 i=0 σ i · . then p+c µp+c (D) = p µp (L ∩ D)|dνc (L)|.21) §9. we have r→0 lim µp Tr (X ) = µp (X ). If the function f is |dν |-integrable.3. relatively compact subset of the Euclidean space Rn with smooth boundary M = ∂D. p−w (9.1. these two densities produce two invariant densities |dγc | and |dνc | on Graff c which differ by a multiplicative constant. and by Graff c the affine Grassmannian of codimension c affine subspaces of Rn . we set V = Rn .3. c.3.3. Then k − j = p − w. Theorem 9. CLASSICAL INTEGRAL GEOMETRY We set n := m + 1 and we make the change in variables p := n − j. m = n − 1 = dim M .p . Step 1.p µp+c (D) = Graff c . and p such that µp (L ∩ D)|dνc (L)|. f (L) = µp (L ∩ D).c.6 Crofton Formulæ for domains of an Euclidean space Suppose D is an open. depending only on m.22) c Grc As explained in Subsection §9. We will carry out the proof in several steps. and consider the function f : Graff c → R.17) to obtain a new volume density |dνc | with total volume n |dνc | = . ξn. For simplicity.3. We rescale this volume density as in (9. by Grc the Grassmannian of linear subspaces of Rn of codimension c. (9.20) In particular.408 CHAPTER 9.3.1. (9. Graff c Proof. w := n − k. We denote by g the induced metric on M . Let 1 ≤ p ≤ n − c.c. ∀0 ≤ p ≤ dim X. v ∈ S ∩ M . We will show that the constant ξ is equal to p+ by explicitly computing both p m +1 sides of the above equality in the special case D = D . The set I(M )∗ is an open subset of I(M ). S ) −→ (v. S ) ∈ V × Graff c . S ) ∈ V × Graff c . Fortunately. CURVATURE MEASURES 409 c Step 2. [S ]) ∈ V × Grc . We set I(M )∗ := r−1 Graff c (M ) . The set Graff c (M ) is an open subset of Graff c . [S ] = S − S . We introduce the incidence relation I= (v.  r  Graff c (M ) = (v. I with inverse V × Grc (V ) We obtain a double fibration I V and we set I(M ) := −1 (v. and we obtain a double fibration M .)) ) I(M )∗ r (9. Sard’s theorem shows that r fails to be a surjection on a rather thin set. Observe that we have a diffeomorphism I → V × Grc . v ∈ S ⊂ V × Graff c . v + L) ∈ I. and Sard’s theorem implies that its complement has measure zero. . Denote by Graff c (M ) the set of codimension c affine planes which intersect M transversally. Step 1. For every S ∈ Graff c we denote by [S ] ∈ Grc the parallel translate of S containing the origin.3.9. (v.23) c Graff (M ) The fiber of r over L ∈ Graff c is the slice ML := L ∩ M which is the boundary of the domain DL := (L ∩ D) ⊂ L. Again we have a diagram I(M ) M  r  Graff c The map r need not be a submersion. Since dim I = dim V + dim Grc = n + c(n − c) we deduce dim I(M ) = n + c(n − c) − codim M = n + c(n − c). L) −→ (v.3. We will rely on a basic trick in integral geometry. . • The above condition imply that e0 is a normal vector to the boundary of the domain D ∩ L ⊂ L.1. Then we can parametrize a small open neighborhood of (x0 . e0 (S ). L0 ) in I∗ (M ) by a family (x. eh+1 (S ). L0 ) in I is parametrized by the family r. • The collection span {e1 . e1 . . . eh (S ). . . • The collection {e0 . where x ∈ M . e1 (S ). . using the pullback r∗ |dγc | we obtain a density ∗ |dλ| := |dµL p | × r |dγc | on I∗ (M ) satisfying. Suppose (x0 . . em (S )} is an orthonormal frame of Rn . e0 (S ). . . eh (S ). if we set k = dim L − p = m − c − p. where r runs in a neighborhood of x0 in the ambient space V . To complete Step 1 in our strategy it suffices to prove that there exists a constant ξ . . Then. . . and by |dVL∩M | the metric volume density on L ∩ M . where x runs in a small neighborhood of x0 ∈ M . . .410 CHAPTER 9. . A neighborhood of (x0 . the co-oriented second fundamental form of D. . e1 (S ). eh+1 (S ).1. em (S ) . em (S )). We require that e0 is the outer normal. CLASSICAL INTEGRAL GEOMETRY The vertical bundle of the fibration r : I∗ (M ) → Graff c (M ) is equipped with a natural density given along a fiber L ∩ M by the curvature density |dµk | of the domain DL . we deduce |dµL p| = 1 trk (−SL )|dVL∩M |. . eh+1 (S ). eh (S ). eh } is a basis of Tx0 M ∩ S . by SL the co-oriented second fundamental form of DL ⊂ L. |dλ| = I∗ (M ) Graff c (M ) L∩M |dµL p | |dγc (L)| = Graff c µp (L ∩ D)|dγc (L)|. . L0 ) ∈ I∗ (M ). • The collection {e0 (S ). σk In the sequel we will use the following conventions. . We will denote this density by |dµL k |.3. eh } is a basis of S . S runs in a small neighborhood U0 of [L0 ] in Grc so that the following hold for every S . depending only on m and c such that ∗ |dλ| = ξ |dµc |. Set h := (m − c). and L is an affine plane of dimension h + 1. . . As explained in Subsection §9. . . . We denote by SD . . . . . . . L). where the curvature density is described in Definition 9. e1 (S ). . The points in I(M ) are pairs (x. . .13. . r = x0 . f m of e1 .e. em . . . . Then. m}. m}. .. .e. • A. . . . . G∗ x0 (M ) can be identified with the space of linear subspaces S of codimension c such that Tx0 M + S = V . Suppose x0 + S intersects M transversally at x0 . . β. We set eA := eA (S ). Then |dγc | = α D r • eα × |dγc | = α θα × |dγc |. Xm ) = |θ1 ∧ · · · ∧ θm |(X1 .i 411 θαi .. . CURVATURE MEASURES • i. Denote by n a smooth unit normal vector field defined in a neighborhood of x0 in M . . . . .21. Xm ). k denote indices running in the set {0. C denote indices running in the set {0.3. . α (9. . n(x) ⊥ Tx M. Xm ∈ Tx0 M we have |(n • e0 )| · |dVM |(X1 . Lemma 9. i. We set ∗ G∗ x0 := Gx0 ∩ I (M ). |n(x)| = 1. . B. eA (S )) ∈ I(M ). f α = eα − (eα • n)n. .3. We denote by (θA ) the dual coframe of (eA ). . . . . These projections form a basis since S intersects Tx0 M transversally. 1. Then at the point x0 ∈ M we have |(n • e0 )| · |dVM | = |θ1 ∧ · · · ∧ θm |. . and |dλ| = |dµL p | × |dγc | = The fiber of 1 det(−SL∩M )|dVL∩M | × σk θα × |dγc |. j. .e. . h}. the volume density of the natural metric on Grc is |dγc | = α. ∀1 ≤ i ≤ h. . . . . . • α. γ denote indices running in the set {h + 1. . . the affine subspace x0 + S intersects M transversally at x0 . .. It suffices to verify this for one basis X1 . Xm of Tx0 M which we can choose to consists of the orthogonal projections f 1 .9. i. . i.24) : I(M ) → M over x0 is described by Gx0 := (r. and set θAB := eA • (D eB ). Proof. . Observe that f i = ei . for any X1 .3. that is. .B ≤m We observe that f i • f j = δij . j ≤ h < α. CLASSICAL INTEGRAL GEOMETRY |dVM |(f 1 . |θ1 ∧ · · · ∧ θm |(f 1 . f m )2 = det(f A • f B )1≤A. On the other hand. . ∀1 ≤ i. We conclude that det( − A) = |n • e0 |2 =⇒ |dVM |(f 1 . . nα := n • eα .  c u= . . f m )2 = det( − A). where A denotes the c × c symmetric matrix with entries nα nβ . . ej ). then SL = (n • e0 )SD |Tx0 ∩[L] . . . . f m ) = |n • e0 |. .B ≤m |. ∀1 ≤ i. j ≤ 2h < α f α • f β = δαβ − nα nβ. Suppose L ∈ Graff c intersects M transversally. . so that |θ1 ∧ · · · ∧ θm |(f 1 . SL (ei .22 (Euler-Meusnier). . . The lemma is now proved. Lemma 9. . j ≤ 2h. This matrix has a c − 1 dimensional kernel corresponding to vectors orthogonal to u. We deduce |dVM |(f 1 . . . ∈R . ej ) = (n • e0 )SD (ei . β ≤ m = h + c. and the corresponding eigenvalue λ is obtained from the equality λu = |u|2 u =⇒ λ = |u|2 = α 2 2 2 n2 α = |n| − |n • e0 | = 1 − |n • e0 | . eα • f β = δαβ − nα nβ . . ei • f α = 0. f m ) = | det(eA • f B )1≤A. ∀1 ≤ i. . and x0 ∈ L ∩ M . h + 1 ≤ α. . f m ) = |n • e0 |2 . . If we denote by u the vector   nh+1  . then we deduce that A = uu† . If n is a unit vector perpendicular to Tx0 M . We have again ei • f j = δij . . . nh+c which we also regard as a c × 1 matrix. f i • f α = 0.3. The vector u itself is an eigenvector of A.412 Then CHAPTER 9. . From the above lemma we deduce trk (−SL |x0 ) = |n • e0 |k trk (−SD |Tx0 M ∩[L] ). σk If we denote by θ([L]. Suppose that V is an Euclidean space.21) = 1 |n • e0 |k trk (−SD |Tx M ∩[L] ) σk θA × |dγc | A=1 1 |n • e0 |k+1 trk (−SD |Tx M ∩[L] ) |dVM | × |dγc |. We have SL (ei . Tx0 M )|k+1 trk (−SD |Tx0 M ∩[L] ) |dγc |([L]). σk c The map G∗ (x0 . σk This proves that along the fiber G∗ x0 we have |dλ|/|dVM | = 1 |n • e0 |k+1 trk (−SD |Tx0 M ∩[L] ) |dγc |([L]). D ei ej decomposes into two components.23. ej ) = e0 • (D ei ej ). Thus.3. one component parallel to e0 . (Deei ej )τ . H ⊂ V is a hyperplane through the origin. Tx0 M ) the angle between [L] and the hyperplane Tx0 M we deduce |dλ|/|dVM | = 1 · | cos θ([L]. tangent to L ∩ M . L0 ) ∈ I∗ (M ) we have |dλ|(x. Denote by O(H ) the subgroup of orthogonal transformations of V which map H to itself and suppose that f : Grc → R . for any (x0 . Taking the inner product with n we deduce SD (ei .3. Hence h D ei ej = SL (ei . ej )(e0 • n).9. L) = 1 trk (−SL )|dVL∩M | × σk θα × |dγc | α m = (use Lemma 9. ej )e0 + k=1 k Sij ek . 413 Let us now observe that the vector (D ei ej ) is parallel with the plane L because the vectors ei and ej lie in this plane. L) −→ [L] ∈ Grc identifies G∗ x0 x0 with an open subset of Gr whose complement has measure zero. and another component.3. Lemma 9. ej ) = (D ei ej ) • n = SL (ei . L) ∈ I∗ (M ). CURVATURE MEASURES Proof. dim V = m + 1. We now have the following result. In a neighborhood of (x0 . and B : H × H → R a symmetric bilinear map. . B ) = 0. On the other hand. . Grc H Grc Proof. c and k . B ) = Q tr1 (B ). .k .414 CHAPTER 9. for every 0 ≤ k ≤ m − c.c. there exists a constant ξ = ξm. trk−1 (B ) . the map B → Ik (f. for almost all S ∈ Grc H we have dim S ∩ ker B > m − c − k. . we apply the above formula to a symmetric bilinear form B such that dim ker B > m − k. Then. ∀S ∈ Grc H. trk (B ) = ξf trk (B ) + Qf tr1 (B ). . dim ker B > m − k =⇒ Qf = 0. depending only on m. Now choose B to be the bilinear form corresponding to the inner product on H . at least m − k + 1 of the m eigenvalues of B vanish. B ) is an O(H )-invariant homogeneous polynomial of degree k in the entries of B . B ) = Pf tr1 (B ). Thus. Then trk (B ) = and we conclude that m k and trk (B |H ∩S ) = m−c k m−c .c. To do this. . Observe that. k m . CLASSICAL INTEGRAL GEOMETRY is an O(H )-invariant function. . f (S ) = to conclude that ∗ |dλ| 1 | cos θ(S. . H )|k+1 σk = ξ trk (−SD )|dVM | = ξ |dµp+c | . . Hence Ik (f. ∀B. . For such forms we have Ik (f. such that Ik (f. B ) := f (S ) trk (B |H ∩S )|dγc |(S ) = ξm. k Grc f |dγc |(S ) = ξf Now apply the above lemma in the special case H = Tx0 M. B = −SD . We can therefore express it as a polynomial Ik (f. Let us prove that Qf ≡ 0. trk−1 (B ) . S intersects H trasversally . The restriction of B to S ∩ H has m − c eigenvalues. . and from the above inequality we deduce that at least m − c − k of them are trivial. for fixed f . Define c Grc H := S ∈ Gr . . so that trk (B ) = 0.k trk (B ) f |dγc |(S ). c.3.c.p Ic. To determine the constant ξ in the above equality we apply it in the special case D = Dm+1 .p . Using (9.p Ic.p m+1 .9. |x|<1 = µm.p Hence ξ ω m+1 ω m+1−c−p m+1 p+c Grc |dνc (S )| (9.c. p+c ω m+1−c−p p + c Now observe that for L ∈ Graff c we set r = r(L) = dist (L. ω m+1−c−p p c 1 0 Using spherical coordinates on Rc we deduce Ic. and it is a disk of dimension (m + 1 − c) = dim L and radius (1 − r2 )1/2 if r < 1.c. Then L ∩ Dm+1 is empty if r > 1.3.p = (1 − |x|2 )p/2 dVRc = σ c−1 rc−1 (1 − r2 )p/2 dr Rc . 0). Graff c (M ) Step 2. we deduce that there exists a constant ξ depending only on m and c such that ξµp+c (D) = µp (L ∩ D)|dνc |(L). We conclude that µp (L ∩ Dm+1 ) = µp (Dm+1−c ) × We set µm.p Grc [L]⊥ µp Dm+1 ∩ (x + [L]) |dV[L]⊥ |(x) |dνc |([L]) (1 − |x|2 )p/2 |dV[L]⊥ |(x) |dνc |([L]) =:Ic.p := µp (Dm+1−c ) = ω p Using Theorem 9. rescaling |dγc | to |dνc |.3. = ω m+1−c−p p p µp (L ∩ Dm+1 )|dνc |(L) = = µm.19) we deduce ξµp+c (Dm+1 ) = ξ ω p+c ω m+1 m+1 m+1 =ξ .22) = µm. c = ω m+1−c m + 1 − c m+1 Ic.1. m+1−c ω m+1−c m + 1 − c .14 we deduce Graff c (1 − r2 )p/2 0 r<1 p > 1. Graff c (M ) r∗ |dλ| = Graff c (M ) Thus.p Gr c x∈[L]⊥ . CURVATURE MEASURES so that ξµp+c (D) = = M ∗ |dλ| 415 = I∗ (M ) |dλ| µp (L ∩ D)|dγc |(L). We know that there exists a tame structure S such that D ∈ Sn . p = c Γ(1 + 2 + 2 ) ωp = ω m+1−c ω p+c m + 1 ωp c m+1−c p Hence ξ ω m+1 m+1 p+c = ω m+1 ω p+c m + 1 ωpωc c m+1 c m+1 m+1−c p+c p m+1−c . Graff c Proof.. The fiber of π over L is the intersection L ∩ M .1.8) c (9. To see that it has compact support it suffices to observe that since M is compact. p = p+c . +1 2 2 2 (9. we deduce that Graff c is also S-definable. Its range is a definable subset of Z. . From the definability of Euler characteristic we deduce that the map Graff c L → χ(L ∩ M ) ∈ Z is definable. i. L) ∈ M × Graff c .416 = CHAPTER 9. the incidence set I = ((x. p We deduce ξ= ω p+c ωpωc = ω p+c p + c ωpωc p We now describe a simple situation when the function Graff c integrable. CLASSICAL INTEGRAL GEOMETRY s=r2 σ c−1 2 1 s 0 c−2 2 (1 − s)p/2 ds c )Γ(1 + p σ c−1 Γ( 2 2) p c 2 Γ(1 + 2 + 2 ) (9. Hence.3. Since the Grassmannian Graff c is semialgebraic. it is integrable and µc (D) = χ(L ∩ D) |dν |(L).7) = Γ(1 + p ω p+c 2) = Γ(1/2) .24. M = ∂D ∈ Sn . x ∈ L is also S-definable.1.1. then the function Graff c L → χ(L ∩ D) = µ0 (L ∩ D) is bounded and has compact support. In particular. (x.6) = σ c−1 c p B . If the bounded domain D ⊂ Rn with smooth boundary is tame. In particular. an affine plane which is too far from the origin cannot intersect M . L → µ0 (L ∩ M ) is Proposition 9. and so is the map π : I → Graff c . a finite set.e. L) → L. λ) ∈ A . In particular.3. the convex hull of three non collinear points. (a) Prove that χ(L ∩ S ) |dν |(L) = χ(U ∩ i(S ) ) |dν |(U ). ˆ p (T ). Then the function Λ ˆ p (Aλ ) ∈ R is bounded and Borel measurable. so that µ (b) Prove that for every real number λ we have ˆ p (λS ) = |λ|p µ ˆ p (S ). S-definable set. This is a compact.26. Fix tame structure S.L is definable. ∀p ≥ 0. Λ is an arbitrary S-definable set.. and A ⊂ K × Λ is a closed S-definable subset of K × Λ. This proposition also shows that ˆ p (D) = µp (D) µ if D ⊂ Rn is a tame domain with smooth boundary. definable subset of Graff p . Let N > n and suppose i : Rn → RN is an affine. (λ.3. µ (d) Suppose T ⊂ Rn is a triangle. its range must be a finite (!?!) subset of Z. L) → χ(Aλ ∩ L) ∈ Z. Hence. Define F : Λ × Graff p (K ) → Z. For every λ ∈ Λ we set Aλ := a ∈ K .3. S-definable set.25. Suppose K ⊂ Rn is a compact. and for every integer 0 ≤ p ≤ n we define ˆ p (S ) := µ χ(L ∩ D) |dν |(L). isometric embedding. . Fix a tame structure S. For any compact tame subset S ⊂ Rn . µ (c) Suppose S1 .3. Prove that ˆ p (S1 ∪ S2 ) = µ ˆ p (S1 ) + µ ˆ 2 (S2 ) − µ ˆ p (S1 ∩ S2 ).e. λ −→ fp (λ) := µ Proof. ˆ p (S ) depends on the The above definition has a “problem”: a priori. Graff p (Rn ) Graff p (RN ) ˆ p (S ) is independent of the dimension of the ambient space. Compute µ Proposition 9.3. Exercise 9. and suppose S ⊂ Rn is a compact. Graff p (Rn ) The proof of Proposition 9.9. Denote by Graff p (K ) the set of affine planes of codimension p in Rn which intersect K . The next exercise asks the reader to prove that this is not the case.27. Since the family Aλ ∩ L λ. (a. the quantity µ dimension of the ambient space Rn ⊃ S . S2 ⊂ Rn are two compact S-definable sets. this also shows that F is measurable and bounded. i. CURVATURE MEASURES 417 Definition 9.24 shows that the integral in the above definition is well defined and finite. we deduce that the function F is definable. µ Graff c D Graff c c Proof. and p+c µp+c (D) = c µp (L ∩ D) |dν (L)| = ˆ p (L ∩ D)|dν |(L). ˆ p (L ∩ D) is bounded and measurable. the function L → µ This function has compact support because the planes too far from the origin cannot intersect D. We continue to denote by Graff c the Grassmannian of affine planes in Rn of codimension c. §9. Part 1.27. Suppose the bounded domain D ⊂ Rn is tame and has smooth boundary.28. In particular.418 Now observe that CHAPTER 9. µ µ0 (L ∩ M )|dν |(L) = µ0 (L ∩ M )|dν (L)|. 0 ≤ p ≤ n the function Graff c D L → µp (L ∩ D) is bounded. Corollary 9. by Proposition 9. ˆ c (M ) = µc (M ).). The measurability follows from the classical Fubini theorem. compact. The boundedness is obvious.29 (Crofton Formula. L) |dν |(L). S-definable submanifold of dimension m of the Euclidean space Rn . smooth. and we assume that M is a closed. Denote by Graff c D the set of affine planes of codimension c which intersect ∂D transversally.3.3.3. while. Theorem 9. CLASSICAL INTEGRAL GEOMETRY fp (λ) = Graff p (K ) F (λ. Then µc (M ) = that is. Moreover ˆ p (L ∩ D). it is integrable. ∀L ∈ Graff c µp (L ∩ D ) = µ D. The above result has the following immediate corollary. Graff c (M ) Graff c . We want to prove the following result. Then for any integer.7 Crofton formulæ for submanifolds of an Euclidean space In this last subsection we fix a tame structure S.3. measurable. The set Graff c D is a tame open and dense subset of Graff so that its complement has measure zero. and has compact support. Denote by Graff c (M ) the subset of Graff c consisting of affine planes which intersect M transversally. 3. Proof. Lemma 9. ∀r1 ≥ r0 . For uniformity. The sets Xr are measurable in Graff c and Xr = Graff c (M ). r0 ). Let Graff c (Mr ) := L ⊂ Graff c . χ(L ∩ Ds ) = χ(L ∩ M ).27.3. ∀s ∈ (0. ∀L ∈ Graff c .9. For every r > 0 we set Xr = L ∈ Graff c (M ). L ∈ Graff c (Ms ). r →0 The tube Dr is a tame domain with smooth boundary. r ∈ [0.3. To proceed further we need the following technical result. For every r < R consider the tube of radius r. r] Observe that Graff c (M. whose proof will presented at the end of this subsection. fr (L) = χ(L ∩ Dr ).30. CURVATURE MEASURES 419 Proof.3. around M . r1 ) ⊂ Graff c (M. Observe that Graff c (M ) is an open subset of Graff c with negligible complement. We can assume that k = codim M < 1.3. Dr := Tr (M ). Fix R > 0 such that for any x such that d(x) ≤ R there exists a unique point x ¯ ∈ M such that |x − x ¯| = d(x).31. L intersects Mr transversally . R). For every x ∈ Rn set d(x) := dist (x.20 implies µc (Dr ) = Thus it suffices to show that r→0 Graff c Graff c χ(L ∩ Dr )|dν |(L).15) we deduce µc (M ) = lim µc (Dr ). and set Mr = ∂ Tr (M ). lim χ(L ∩ Dr )|dν |(L) = Graff c χ(L ∩ M ). r>0 . M ). There exists C > 0 such that |fr (L)| ≤ C. For r ∈ [0. Lemma 9. From (9. and Theorem 9. This follows from Proposition 9. R) we define fr : Graff c → R. we set D0 = M .3. . and thus it has finite measure. The exponential map on T Rn restricts to a map EL0 ∩M : N 0 → L0 . Proof of Lemma 9. For every r > 0 we set Ny r y 0 The collection (Ny )y∈L0 ∩M forms a vector subbundle N 0 → L0 ∩ M of (T Rn )|L0 ∩M . Yr := Graff ∗ \Xr . We denote by Ny 0 y 0 0 Ny = L0 ∩ Ty (L0 ∩ M ) ⊥ .29. and since vol (Y∗ r ) → 0 we conclude that µc (M ) = lim µc (Dr ) = lim r →0 r→0 X∗ r f0 (L)|dνc | = Graff c f0 (L)|dνc |. This concludes the proof of Theorem 9. We now let r → 0. For 0 < r < R we have µc (Dr ) = = X∗ r Graff c fr (L)|dνc |(L) = Grc ∗ fr (L)|dνc |(L) fr (L)|dνc | fr (L)|dνc | + Y∗ r fr (L)|dνc | = 2 X∗ r f0 (L)|dνc | + Y∗ r Hence µc (Mr ) − X∗ r f0 (L)|dνc | ≤ Y∗ r |fr (L)||dνc | ≤ C vol (Y∗ r ). L ∩ DR = ∅ . 0 the orthogonal complement in L of T (L ∩ M ). The measurability follows from the fact that Xr is described using countably many boolean operations on measurable sets. c The region Graff c ∗ (M ) is a relatively compact subset of Graff .31. Let y ∈ L0 ∩ M .3. For x in M we denote by Nx the fiber of N over x.3. 0 as an affine subspace of Rn containing y . Because L intersects M We think of Ny 0 transversally we have 0 dim Ny = dim Ny = k = codim M.420 Set CHAPTER 9.2. Define c ∗ c ∗ X∗ r := Xr ∩ Graff ∗ . see Figure 9. We will prove that for any given L0 ∈ Graff c (M ) there exists and ρ = ρ(L0 ) such that L0 ∈ Xρ . CLASSICAL INTEGRAL GEOMETRY Graff c ∗ := L ∈ Graff c . 0 (r ) := N 0 ∩ D . Consider the normal bundle N := (T M )⊥ → M. we deduce that L0 is transversal to the level sets { d(x) = r } = Mr . L0 ∩ Dρ = Hence χ(L0 ∩ Dr ) = χ(L0 ∩ Dρ ) = χ(L0 ∩ M ). there exists ρ = ρ(L0 ) > 0 such that the map EL0 ∩M : x ∈ N 0 . δ (x) ≤ ρ → L0 ∩ Dρ is a diffeomorphism. CURVATURE MEASURES 421 M Ny(r) y 0 L0 Dr Figure 9. so that χ(L0 ∩ Dρ ) = χ(L0 ∩ M ). ρ]. 0 of the Hessian of d at y is positive definite. We deduce that we have a natural projection π : L0 ∩ Dρ → L0 ∩ M. The downward gradient flow of the restriction to L0 ∩ Dρ of the distance function d(x) produces diffeomorphisms of manifolds with boundary ∼ L0 ∩ Dr . the restriction to Ny Hence. ∀r ∈ (0. ∀r ∈ (0. δL0 = d ◦ EL0 ∩M : N 0 → R.9. L0 ∩ Dρ is homeomorphic to a tube in L0 around L0 ∩ M ⊂ L0 . 0 (ρ). This proves L0 ∈ Xρ . which is continuous and defines a locally trivial fibration with fibers Ny 0 For every y ∈ L0 ∩ M .2: Slicing the tube Dr around the submanifold M by a plane L0 . Since the restriction to L0 ∩ Dρ of the distance function d(x) has no critical points other than the minima y ∈ L0 ∩ M . M ). ∀r ∈ (0. For every y ∈ L0 ∩ M . Denote by δ the pullback to N 0 of the distance function x → d(x) = dist (x.3. . ρ). ρ]. Thus. the fiber Ny (ρ) is homeomorphic to a disk of dimension k . Theorem 9. ∗ m−2 For every R > 0 we denote by Graff (R) the set of codimension (m − 2) planes n n in R which intersect the disk DR of radius R and centered at 0.33. Let m = dim M . −2 By Sard’s theorem the complement of Graff m (M ) in Graff m−2 (M ) has zero measure.24 and the simple observation that m−2 χ(L ∩ Dn (R). h) = where EM (h) is the Hilbert-Einstein functional EM (h) = s(h)dVM (h). Then the function L → nC (L) belongs to L∞ (Gr1 (R2 ). For every line L ∈ Gr1 (R2 ) = Gr1 (R2 ) we set nC (L) := #(L ∩ C ).32. ∀L ∈ Graff we deduce that c(R) = µm−2 (Dn R) = n ωm−2 Rm−2 .3.1 ). Then µm−2 (M.3.422 CHAPTER 9. We set c(R) = Graff m−2 (R) |dν (L)|.1 |(L). Note that since M is compact. R ) = 1. compact. orientable Riemann surface. Suppose C ⊂ R2 is a smooth. ∀R > R0 . there exists R0 > 0 such that Graff m−2 (M ) ⊂ Graff m−2 (R). m−2 m−2 (R). Suppose M is a tame. For every L ∈ Graff m (M ). Using Crofton’s formula in Proposition 9.29 offers a strange interpretation to the Hilbert-Einstein functional . 2π M Denote by Graff m−2 (M ) = Graff (M ) the set of codimension (m − 2) affine planes in Rn −2 which intersect M along a nonempty subset. closed. has compact support and length (C ) = Gr1 (R2 ) nC (L)|dν2. and denote by h the induced metric on M . Remark 9. c(M ) := Graff m−2 (M ) |dν (L)|. ∗ the intersection L ∩ M is a.3. CLASSICAL INTEGRAL GEOMETRY Corollary 9. |dν |2. 1 EM (h). c(R) 2πc(R) . and we can regard Observe now that c(1 R) |dν (L)| is a probability measure on Graff the correspondence Graff m−2 (R) L −→ χ(L ∩ M ) ∈ Z as a random variable ξR whose expectation is ξR := 1 1 µm−2 (M ) = EM (h). compact tame curve. smooth. possible disconnected. and by Graff m (M ) ⊂ Graff m−2 (M ) the ∗ −2 subset consisting of those planes intersecting M transversally.3. orientable submanifold of the Euclidean space Rn . 3. µ For r ∈ [0. µ Proof.34 (Crofton Formula.3. Proof. R→∞ m−2 m−2 423 Theorem 9. and its complement has measure zero.9. r ∈ [0. Denote by Graff c (M ) the subset of Graff c consisting of affine planes which intersect M transversally. for every 0 ≤ p ≤ m − c we have p+c µp+c (M ) = p Graff c (M ) µp (L ∩ M )|dν |(L) = Graff c ˆ p (L ∩ M )|dν (L)|. R).28 implies p+c µp+c (Dr ) = c µp (L ∩ Dr )|dν |(L) = ˆ p (L ∩ Dr )|dν |(L).29.). gr (L) = µ From Proposition 9. R) we define ˆ p (L ∩ Dr ). Then.3. r (9. gr : Graff c → R.3.3.3. We have to prove that r ˆ p (Sr ) = µ ˆ p (S0 ). µ Graff c Graff c Thus it suffices to show that r→0 Graff c lim ˆ p (L ∩ Dr )|dν (L)| = µ Graff c ˆ p (L ∩ M )|dν (L)|.15) we deduce µp+c (M ) = lim µp+c (Dr ).27 we deduce that ∃C > 0 : |gr (L)| ≤ C.3. ∀L ∈ Graff c .3. Then.25) lim gr (L) = g0 (L). CURVATURE MEASURES We deduce EM (h) = 2π n n ωm−2 Rm−2 ξR = 2π ωm−2 lim Rm−2 ξR . r →0 The set Dr is a tame domain with smooth boundary. The set Graff p (S0 ) is an open subset of Graff p . and Corollary 9. From (9. Part 2. 0 for all L ∈ Graff c (M ). lim µ 0 Denote by Graff p (S0 ) the set of codimension p affine planes in Rn which intersect S0 transversally.35. Lemma 9. We continue to use the same notations we used in the proof of Theorem 9. ˆ (Sr ) = gr (L) = µ χ(U ∩ Sr )|dν p (U )| = χ(U ∩ Sr )|dν p (U )|.26) Graff p Graff p (S0 ) . Let L ∈ Graff c (M ) and set Sr := L ∩ Dr . (9.3. 3. Second.10. we deduce from the dominated convergence theorem that r lim 0 Graff c ˆ p (L ∩ Dr )|dν (L)| = µ = Graff c ˆ p (L ∩ M )|dν (L)| µ Graff c (M ) µp (L ∩ M )|dν (L)|.36. Sect.3.27 we deduce that the function Graff p ×[0. m-dimensional submanifold M ⊂ Rn . Arguing exactly as in the proof of Lemma 9. r) → χ(U ∩ Sr ) ∈ Z is definable. µ Using (9. by intersecting it with a large family of simpler shapes. ∀U ∈ Graff p (S0 ).2. ρ).S.3. (b) The subject of integral geometry owes a great deal to the remarkable work of S.35. and push our understanding of this subject to remarkable heights.25) and Lemma 9. R) (U.3. these other simpler shapes could be affine planes. spheres of a given radius. CLASSICAL INTEGRAL GEOMETRY From Proposition 9. and thus bounded. and they are true in much more general situations.424 CHAPTER 9. we believe that geometers should become more familiar with tame context in which the geometric intuition is not distracted by analytical caveats. the set U ∩ Sr is homotopy equivalent to U ∩ S0 . balls of a given radius.3. we deduce from the dominated convergence theorem that lim gr (L) = 0 r Graff p ˆ p (S0 ). there exists ρ ∈ (0.3. R) such that. compact tame manifold we deduce from Theorem 9.3.29 that ˆ p (S0 ) = µp (S0 ) = g0 (L). Rn . χ(U ∩ S0 )|dν p (U )| = µ Using the fact that S0 is a smooth.26). 0 If we let r go to zero in (9. for every U ∈ Graff p (S0 ). For example. Remark 9. First. In the Euclidean space. It was observed by Chern that the Crofton formulæ that we have proved in this section are only special cases of the so called kinematic formulæ which explain how to recover information about a submanifold of a homogeneous space such as Rn . Hence r lim χ(U ∩ Sr ) = χ(U ∩ S0 ). Chern who was the first to have reformulated the main problems from a modern point of view. we can ask what is the value of the integral χ(M ∩ BR (x) )|dx|. for every r ∈ (0. (a) The tameness assumption in the Crofton formulaæ is not really needed. We have chosen to work in this more restrictive context for two reasons.31 we deduce that. 2. and a radius R. given a compact.3. the tameness assumption eliminates the need for sophisticated analytical arguments. see [33]. n.3. for a more in depth look at this subject. CURVATURE MEASURES 425 where BR (x) denotes the ball BR (x) = { y ∈ Rn .n. where cm. A detailed presentation of these ideas will send us too far. Santal´ o [85]. but the kinematic formula extends beyond the range of applicability of the tube formula. yet very much alive monograph of L.9.i are universal constants. |y − x| ≤ R }. To see the origin of this similarity note that BR (x) ∩ M = ∅ ⇐⇒ x ∈ TR (M ). the above formula does indeed specialize to the tube formula. independent of M . and instead of pursuing further this line of thought. One of the kinematic formulæ states that n Rn χ(M ∩ BR (x) ) = i=0 cm.i µi (M )Rn−i . If R is sufficiently small. we recommend to the reader to open the classical. . The above equality is strikingly similar to the tube formula. 1. Next in line is the exterior derivative ∂i 2 u. ∆ : C ∞ (RN ) → C ∞ (RN ).Chapter 10 Elliptic Equations on Manifolds Almost all the objects in differential geometry are defined by expressions involving partial derivatives. This chapter is an introduction to this vast and dynamic subject which has numerous penetrating applications in geometry and topology. The Laplacian defined as above is sometimes called the geometers’ Laplacian. This situation can be dealt with topologically. it is best that we begin with the simplest of the situations.1 Basic notions We first need to introduce the concept of partial differential operator (p. To understand the forthcoming formalism. topological considerations alone are not sufficient. Among them. for brevity) on a smooth manifold M . This is where analysis comes in. 10. This is a vectorial operator in the sense it acts on vector valued functions.o. ∂i := ∂x i valued functions.d. as usual. ∆u := − i ∂ . we inquired whether on a given vector bundle there exist flat connections. using the Chern-Weil theory of characteristic classes. one is led to the study of partial differential equations. and more specifically. For example. This is a scalar operator in the sense that it acts on scalar where. Perhaps the best known partial differential operator is the Laplacian. d : Ωk (RN ) → Ωk+1 (RN ). the elliptic ones play a crucial role in modern geometry. A degree k form ω on RN can be viewed as a collection of N k smooth functions or equivalently. M = RN . Very often. The curvature of a connection is the most eloquent example. and one has look into the microstructure of the problem.1 Partial differential operators: algebraic aspects §10. Many concrete geometric problems lead to studying such objects with specific properties. as a 426 . Note that our definition of the Laplacian differs from the usual one by a sign. (10.1. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS N smooth function ω : RN → R( k ) . f denotes the C ∞ (M )-module multiplication by f . We can rephrase the equality (10. We set P DO (E. we have T (f u) − f (T u) = 0 or. . For any smooth K = R. f ]u = T (f u) − f (T u) = 0.1. [T. The partial differential operators are elements of Op which interact in a special way with the above C ∞ (M )module structures. in terms of commutators. and f ∈ C ∞ (M ). F ) the space of K-linear operators C ∞ (E ) → C ∞ (F ). F ). f ]. F ) := ker adm+1 = T ∈ Op(E.1. First define P DO 0 (E.10. R(k+1) . 427 It is convenient to think of C ∞ (RN . T ∈ ker ad(f0 ) ad(f1 ) · · · ad(fm ). F ).1) Each f ∈ C ∞ (M ) defines a map ad(f ) : Op (E. [T. F ). Remark 10. R( k ) → C ∞ RN . F ). F ) = {T ∈ Op (E. F ). F ) := m≥0 P DO (m) (E. f ] ∈ P DO (m−1) . ∀fi ∈ C ∞ (M ) . They are modules over the ring of smooth functions C ∞ (M ).1) as P DO 0 (E. This point of view is especially useful in induction proofs. Given T ∈ P DO 0 . F ) → Op (E. E ) is an associative K-algebra. Define P DO (m) (E. The spaces of smooth sections C ∞ (E ) and C ∞ (F ) are more than just K-vector spaces. by ad(f )T := T ◦ f − f ◦ T = [T.1. F ). F ) := Hom(E. ∀T ∈ Op(E. The space Op(E. Thus d can be viewed as an operator N N d : C ∞ RN . The elements of P DO (m) are called partial differential operators of order ≤ m. ad(f )T = 0 ∀f ∈ C ∞ (M )} =: ker ad .1. Above. Rν ) as the space of smooth sections of the trivial bundle Rν over RN . u ∈ C ∞ (E ). ∀f ∈ C ∞ (M )}. Note that we could have defined P DO (m) inductively as P DO (m) = {T ∈ Op . C-vector bundles E . F over a smooth manifold M we denote by Op(E. if f ∈ C ∞ (M ). supp Lu ⊂ supp u. ∀f ∈ C ∞ (M ). u] · 1 + u · L(1) = X · u + µ · u. ∀u ∈ C ∞ (E ).. This concludes the proof of the lemma. in order to analyze the action of a p. For m = 0 the result is obvious. Let E. and consequently (see Exercise 3. f ].o. Proposition 10.9) there exists a smooth vector field X on RN such that σ (f ) = X · f. This is done in the exercises at the end of this subsection. Let µ := L(1) ∈ C ∞ (RN ). G) and Q ∈ P DO (n) (E. Denote by R the trivial line bundle over RN . Thus. The sections of R are precisely the real functions on RN . where Kp . g ]u) = σ (f )g · u + f σ (g ) · u. ∀f ∈ C ∞ (RN ). G). R). i.1. Kq ).d. for all u ∈ C ∞ (RN ). We argue by induction over m. For m + n = 0 the result is obvious. for some smooth function σ (f ) ∈ C ∞ (RN ). In other words. If P ∈ P DO (m) (F. Lemma 10. The proposition is proved.1.4. f ] = [P. f ]u + f Lu. the operators in the right-hand-side have orders ≤ m+n−1. We argue by induction over m + n. and Kq are trivial K-vector bundles over RN . for any f. F ) is a local operator. f ∈ C ∞ (RN ). For any open set O such that O ⊃ supp u we can find f ≡ 1 on supp u and f ≡ 0 outside O so that f u ≡ u). f g ]u = [L. . Then. u. On the other hand.e. Let L ∈ P DO (m+1) . F. Proof. Hence σ (f g ) = σ (f )g + f σ (g ). understanding the structure of an arbitrary p.3. f ]u = σ (f ) · u. f ](gu) + f ([L.o. Any L ∈ P DO (m) (E. ELLIPTIC EQUATIONS ON MANIFOLDS Example 10. The above lemma shows that. one can work in local coordinates. in P DO (m) (Kp .o. boils down to understanding the action of a p. Proof. Let L ∈ P DO (1) .1. By induction. and u ∈ C ∞ (E ). Since [L.1.2. G be smooth K-vector bundles over the same manifold M .428 CHAPTER 10. F ) then P ◦ Q ∈ P DO (m+n) (E. then [P ◦ Q. In general.d. For every f ∈ C ∞ (M ) we have L(f u) = [L. f ] ∈ P DO (m) we deduce by induction supp L(f u) ⊂ supp u ∪ supp f.. we have Lu = L(u · 1) = [L. the map f → σ (f ) is a derivation of C ∞ (RN ). f ] ◦ Q + P ◦ [Q. Then [L. We want to analyze P DO (1) = P DO (1) (R. g ∈ C ∞ (RN ) σ (f g )u = [L.d. Proposition 10.d. . fi . . . f= j . each partial derivative ∂i is a 1st order p.d. at a point x0 ∈ M . . C ∞ (M ). i α = (α1 . . . hj ∈ mx0 . Proof. αN ) ∈ ZN ≥0 .1.6. . . . f (x0 ) = 0 .5.1.. multiple compositions of such operators are again p.2. αN α1 αi . g ∈ C ∞ (M ) ad(f ) · (ad(g )P ) = ad(g ) · (ad(f )P ). . f (x0 ) = 0. 0 Then {ad(f1 ) ad(f2 ) · · · ad(fm )P } |x0 = {ad(g1 ) ad(g2 ) · · · ad(gm )P } |x0 . fm ∈ the bundle morphism ad(f1 ) ad(f2 ) · · · ad(fm )P does not change if we permute the f ’s. Lemma 10.o. The operator L= |α|≤m 429 aα (x)∂ α : C ∞ (RN ) → C ∞ (RN ). F → M be two smooth vector bundles over the smooth manifold M .d. f ] = [[P. and any f. gi ∈ C ∞ (M ) (i = 1. .e. . m) such that. of order ≤ m. g ] = ad(g ) · (ad(f )P ).1.1. f ]. . Let E.o. m. df (x0 ) = 0. ∂ α = ∂1 · · · ∂N .10.8. From the above lemma we deduce that if P ∈ P DO (m) then. Then for any P ∈ P DO (E. According to the above proposition. any function f which vanishes at x0 together with its derivatives can be written as gj hj gj . ∀i = 1.1.o. [f. According to the computation in Example 10. Then Jx0 = m2 x0 . F ). Let P ∈ P DO (m) (E. i. . g ] = [[P. F ). g ] + [P. for any f1 . For each x0 ∈ M consider the ideals of C ∞ (M ) mx0 = Jx0 = f ∈ C ∞ (M ). Proof.’s. |α| = is a p. . ∗ dfi (x0 ) = dgi (x0 ) ∈ Tx M. f ∈ C ∞ (M ). . Lemma 10. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS Corollary 10.7. ad(f ) · (ad(g )P ) = [[P. g ]. In the proof we will use the following technical result which we leave to the reader as an exercise.1. f ]. that is symmetric in the variables ξi . we can write φ= j (10. αj ]βj } |x0 + j {αj [Q. gi ∈ C ∞ (M ) such that dfi (x0 ) = dgi (x0 ) ∀i = 1. We have to show that {ad(φ)Q} |x0 = 0. ξ ). . Given P ∈ P DO (m) (E. and set Q := ad(f2 ) · · · ad(fm )P ∈ P DO (1) . We will show that { ad(f1 ) ad(f2 ) · · · ad(fm )P } |x0 = { ad(g1 ) ad(f2 ) · · · ad(fm )P } |x0 . . Proof. Let P ∈ P DO (m) and fi . .1. . Let φ = f1 − g1 .9. αj . we see that this map is uniquely determined by the polynomial σ (P )(ξ ) := σm (P )(ξ ) = σ (P )(ξ. Iterating we get the desired conclusion. the above expression unambiguously induces a linear map { σ (P )(ξ1 .1. F ). . and hence the proposition. where π ∗ E and π ∗ F denote the pullbacks of E and F to T ∗ M via π . for each P ∈ P DO (m) (E.2) αj βj .2). Note that φ ∈ Jx0 so. . . This proves the equality (10. . ELLIPTIC EQUATIONS ON MANIFOLDS Exercise 10. . m). Prove the above lemma. π ∗ F ). . Using the polarization trick of Chapter 8. we may assume (eventually altering the f ’s and the g ’s by additive constants) that fi (x0 ) = gi (x0 ) ∀i. and fi ∈ C ∞ (M ) (i = 1. . according to the above lemma. . m. Hence. . x0 ∈ M . . We have {ad(φ)Q} |x0 = j ad(αj βj )Q |x0 = j {[Q. for any ξ ∈ T ∗ M (i = depends only on the quantities ξi := dfi (x0 ) ∈ Tx i x0 0 1. the linear map 1 ad(f1 ) · · · ad(fm )P } |x0 : Ex0 → Fx0 m! ∗ M . F ). .430 CHAPTER 10. ξm )(P ) : Ex0 → Fx0 . . . Since ad(const) = 0. βj ∈ mx0 . The proposition we have just proved has an interesting consequence. .1. . Along the fibers of T ∗ M the map ξ → σm (P )(ξ ) looks like a degree m homogeneous “polynomial” with coefficients in Hom (Ex0 . Fx0 ). m If we denote by π : T ∗ M → M the natural projection then. we have a well defined map σm (P )(·) ∈ Hom (π ∗ E. . m). . βj ]} |x0 = 0. o. the map x ∈ M . Hom (Kp . Exercise 10. Then σm+n (Q ◦ P ) = σn (Q) ◦ σm (P ).1. Show that L − |α|=m aα is a p. Definition 10.10.1. Let L ∈ P DO (m) (E. Show that σm (P )(ξ ) = |α|≤m aα (x)ξ α = |α|≤m αN α1 . Kq ) has the form L= |α|≤m (x)∂ α f −→ [L. and any ξ ∈ Tx σm (P )(ξ ) : Ex → Fx is a linear isomorphism. .1. The following exercises provide a complete explicit description of the p.o.d. Kq )) for any α . of order m. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS Proposition 10.o. Let Kp and Kq denote the trivial K-vector bundles over RN of rank p and respectively q .14. Let P ∈ P DO (m) (E.d. Show that any L ∈ P DO (m) (Kp .d.’s of order m will be denoted by P DO m . Consider the scalar p.1. Hint: Use the previous exercise to reduce the problem to the case p = 1. F ) and u ∈ C ∞ (E ).11. The operator P ∈ P DO m (E. Let C : symbol has the form C ∞ (RN ) → C ∞ (RN ) be a p. G). for any ∗ M \ {0}. Exercise 10.10.5. Show the operator C ∞ (M ) belongs to P DO (m) (RM .’s on RN .o. of order ≤ m − 1.13.o. F ). Prove the above proposition.-s on R are those indicated in Corollary 10.1. The set of p. In this case σm (P ) is called the (principal) symbol of P .1. aα (x)ξ1 · · · ξN Exercise 10. Exercise 10.1.1. where Aα ∈ C ∞ (RN . Definition 10. and conclude that the only N scalar p.12. F ) and Q ∈ P DO (n) (F.o.5 L= |α|≤m aα (x)∂ α .d. on RN described in Corollary 10. P ∈ P DO (m) is said to have order m if σm (P ) ≡ 0.d.d.16.d. A p.15.1.17. 431 Exercise 10.1.o. Its principal σm (L)(ξ ) = |α|≤m aα (x)ξ α . f ]u ∈ C ∞ (F ) Aα (x)∂ α . F ) is said to be elliptic if.1. we included in this subsection a couple of classical examples which hopefully will ease this process. this shows that the Laplacian ∆ is an elliptic operator. (Covariant derivatives).d. Its symbol can be read from ad(f )∇.d.) Let M be a smooth manifold. ∆ : C ∞ (RN ) → C ∞ (RN ). Example 10.1. we will compute the principal symbols of some p.o. the notions introduced so far may look too difficult to “swallow”. If we set ξ := df in the above equality.2 Examples At a first glance. We can view ∇ as a p.1. ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ). To help the reader get a friendlier feeling towards them. ∂i = ∂ . ELLIPTIC EQUATIONS ON MANIFOLDS §10. i = (∆f ) · −2 Hence ad(f )2 (∆) = ad(f )(∆f ) · −2 i fxi · ad(f ) (∂i ) = −2 i (fxi )2 · = −2|df |2 · .d. More specifically. In particular.. the symbol is the tensor multiplication by ξ .o. f ∈ C ∞ (M ).18. we will use the notation “ f · ” to denote the operation of multiplication by the smooth scalar function f . Then ad(f )(∆) = − i ad(f ) (∂i )2 = − i {ad(f )(∂i ) ◦ ∂i + ∂i ◦ ad(f )(∂i )} {fxi · ∂i + fxi xi · +fxi · ∂i } i =− i {fxi · ∂i + ∂i (fxi ·)} = − fxi · ∂i . By setting ξ = df we deduce σ1 (∇)(ξ ) = ξ ⊗. Example 10. This is the second order p.o. In the sequel.432 CHAPTER 10.’s that we have been extensively using in this book..19.e. Consider a vector bundle E → M over the smooth manifold M . and a connection ∇ on E .1.20. ∂xi Let f ∈ C ∞ (RN ).1. Example 10. The exterior derivative d : Ω• (M ) → Ω•+1 (M ) . ∆ = − i ∂i 2 . For any u ∈ C ∞ (E ) (ad(f )∇)u = ∇(f u) − f (∇u) = df ⊗ u. i. we deduce σ2 (∆)(ξ ) = −|ξ |2 · . (The Euclidean Laplacian). (The exterior derivative. k = nk + n + 1 is the exponent introduced in Subsection 4.1. If 1 ∈ I . .1. g ). . then σ1 (ξ 1 )ω = 0.d. Then ad(f )d ω = d(f ω ) − f dω = df ∧ ω.4) (10. . and denote by ξ ∗ ∈ T M its metric dual. For simplicity we assume Fix ξ ∈ Tx x ∗ ∗M ..k δ . . then ∗e(ξ )(∗ω ) = ∗(ξ 1 ∧ ξ k+1 ∧ · · · ∧ ξ n ) = (−1)(n−k)(k−1) ξ 2 ∧ · · · ∧ ξ k = −(−1)νn.3) where νn.1. Note that if 1 ∈ I . where ∗ is the Hodge ∗-operator ∗ : Ω• (M ) → Ωn−• (M ).1.o. n-dimensional Riemann manifold (M. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS 433 is a first order p. . i ) denotes as usual an Consider a monomial ω = dξ I ∈ Λk Tx 1 k ordered multi-index. (The Hodge-DeRham operator). . g ) be as in the above example. This description can be further simplified.o. ∗ M .1.1. Example 10.21. To compute its symbol. ξ1 = ξ. where d∗ = (−1)νn. and moreover σ1 (δ ) = ∗σ1 (d)∗ = ∗e(ξ ) ∗ . and moreover. |ξ | = |ξ | = 1. e. we can produce an operator δ = ∗d∗ : Ω• (M ) → Ω•−1 (M ). Include ξ in an oriented orthonormal basis (ξ 1 . and f ∈ C ∞ (M ).1. Let (M. As in Chapter 4. where I = (i .d. . The operator δ is a first order p. we deduce σ1 (d) = e(ξ ) = the left exterior multiplication by ξ .22.o. I = (1. Putting together (10. . . consider ω ∈ Ωk (M ). .d. . ∗ M . (10.g. Hence d + d∗ is a first order p.5 while i(ξ ∗ ) denotes the interior derivative along ξ ∗ .k i(ξ ∗ ). The Hodge-DeRham operator is d + d∗ : Ω• (M ) → Ω• (M ). .1.4) we deduce σ1 (δ )(ξ ) = −(−1)νn. Consider an oriented. σ (d + d∗ )(ξ ) = σ (d)(ξ ) + σ (d∗ )(ξ ) = e(ξ ) − i(ξ ∗ ).3) and (10.10. ξ n ) of Tx and denote by ξi the dual basis of Tx M .1.k i(ξ ∗ )ω. k )..5) Example 10.. If we set ξ = df . (10. . = − e(ξ )i(ξ ∗ ) + i(ξ ∗ )e(ξ ) = −|ξ |2 g ·. Notice that d + d∗ is elliptic since the square of its symbol is invertible. Definition 10. A second order p.23. We call it Laplacian since σ (d + d∗ )2 (ξ ) = while Exercise 2.55 shows e(ξ ) − i(ξ ∗ ) 2 σ (d + d∗ )(ξ ) 2 = e(ξ ) − i(ξ ∗ ) 2 .o. L : C ∞ (E ) → C ∞ (E ) is called a generalized Laplacian if σ2 (L)(ξ ) = −|ξ |2 g. g ). .434 CHAPTER 10. Let E → M be a smooth vector bundle over the Riemann manifold (M.2.1.d. ELLIPTIC EQUATIONS ON MANIFOLDS The Hodge Laplacian is the operator (d + d∗ )2 . E2 → M be vector bundles over a smooth oriented manifold. . The formal adjoint of a p. ∀u ∈ C0 1 2 0 P u. volume form on M defined by the metric g . 2. we conclude compactly supported then. Let P ∈ P DO (E1 .1. and ∀v ∈ C ∞ (E ) we have be a formal adjoint of P if. M M Lemma 10. Qv 1 dVg .1. whose existence is not yet guaranteed. choosing a partition of unity (α) ⊂ C0 using the locality of Q = Q1 − Q2 that Qv = αQ(αv ) = 0. Finally. i = 1. E2 ).25. (Q1 − Q2 )v 1 dVg = 0 ∀u ∈ C0 (E1 ). C0 i compactly supported sections of Ei . We denote by dVg = ∗1 the ∞ (E ) denotes the space of smooth. v 2 dVg = u. It is worth emphasizing that P ∗ depends on the choices of g and ·. If now v ∈ C ∞ (E ) is not necessarily This implies (Q1 − Q2 )v = 0 ∀v ∈ C0 2 2 ∞ (M ). is denoted by P ∗ .1. · i on Ei . §10. ∞ (E ). M ∞ (E ).24. ∀v ∈ C0 2 ∞ u. · i . Observe that all the generalized Laplacians are elliptic operators. E2 ). unless otherwise indicated. Any P ∈ P DO (E1 . E2 ) admits at most one formal adjoint. P ∈ P DO (E1 .o. we have Proof.3 Formal adjoints For simplicity. Definition 10. Fix a Riemann metric g on M and Hermitian metrics ·. Let Q1 . Then. Q2 be two formal adjoints of P .d. The operator Q ∈ P DO (E2 . E1 ) is said to ∞ (E ). Let E1 . all the vector bundles in this subsection will be assumed complex. 10.1. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS 435 Proposition 10.1.26. (a) Let L0 ∈ P DO (E0 , E1 ) and L1 ∈ P DO (E1 , E2 ) admit formal adjoints L∗ i ∈ P DO (Ei+1 , Ei ) (i = 0, 1), with respect to a metric g on the base, and metrics ·, · j on Ej , j = 0, 1, 2. Then L1 L0 admits a formal adjoint, and ∗ (L1 L0 )∗ = L∗ 0 L1 . (b) If L ∈ P DO (m) (E0 , E1 ), then L∗ ∈ P DO (m) (E1 , E0 ). ∞ (E ) we have Proof. (a) For any ui ∈ C0 i M L1 L0 u0 , u2 2 dVg = M L0 u0 , L∗ 1 u0 1 dVg = M ∗ u0 , L∗ 0 L1 u2 0 dVg . (b) Let f ∈ C ∞ (M ). Then (ad(f )L)∗ = (L ◦ f − f ◦ L)∗ = −[L∗ , f ] = − ad(f )L∗ . Thus ad(f0 ) ad(f1 ) · · · ad(fm )L∗ = (−1)m+1 (ad(f0 )ad(f1 ) · · · ad(fm )L)∗ = 0. The above computation yields the following result. Corollary 10.1.27. If P ∈ P DO (m) admits a formal adjoint, then σm (P ∗ ) = (−1)m σm (P )∗ , where the ∗ in the right-hand-side denotes the conjugate transpose of a linear map. Let E be a Hermitian vector bundle over the oriented Riemann manifold (M, g ). Definition 10.1.28. A p.d.o. L ∈ P DO (E, E ) is said to be formally selfadjoint if L = L∗ . The above notion depends clearly on the various metrics Example 10.1.29. Using the integration by parts formula of Subsection 4.1.5, we deduce that the Hodge-DeRham operator d + d∗ : Ω• (M ) → Ω• (M ), on an oriented Riemann manifold (M, g ) is formally selfadjoint with respect with the metrics induced by g in the various intervening bundles. In fact, d∗ is the formal adjoint of d. Proposition 10.1.30. Let (Ei , ·, · i ), i = 1, 2, be two arbitrary Hermitian vector bundle over the oriented Riemann manifold (M, g ). Then any L ∈ P DO (E1 , E2 ) admits at least (and hence exactly) one formal adjoint L∗ . 436 CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS Sketch of proof We prove this result only in the case when E1 and E2 are trivial vector bundles over RN . However, we do not assume that the Riemann metric over RN is the Euclidean one. The general case can be reduced to this via partitions of unity and we leave the reader to check it for him/her-self. Let E1 = Cp and E2 = Cq . By choosing orthonormal moving frames, we can assume that the metrics on Ei are the Euclidean ones. According to the exercises at the end of Subsection §10.1.1, any L ∈ P DO (m) (E1 , E2 ) has the form L= |α|≤m Aα (x)∂ α , where Aα ∈ C ∞ (RN , Mq×p (C)). Clearly, the formal adjoint of Aα is the conjugate transpose t A∗ α = Aα . To prove the proposition it suffices to show that each of the operators ∂i ∈ P DO 1 (E1 , E1 ) admits a formal adjoint. It is convenient to consider the slightly more general situation. Lemma 10.1.31. Let X = X i ∂i ∈ Vect (RN ), and denote by ∇X the first order p.d.o. ∞ ∇X u = X i ∂i u u ∈ C0 (Cp ). Then ∇X = −∇X − div g (X ), where div g (X ) denotes the divergence of X with respect to the metric g , i.e., the scalar defined by the equality LX (dVg ) = div g (X ) · dVg . ∞ (RN , Cp ). Choose R Proof. Let u, v ∈ C0 0 such that the Euclidean ball BR of radius R centered at the origin contains the supports of both u and v . From the equality X · u, v = ∇X u, v + u, ∇X v , we deduce RN ∇X u, v dVg = X · u, v dVg − BR ∇X u, v dVg u, ∇X v dVg . (10.1.6) = BR BR ∞ (RN , C), and denote by α ∈ Ω1 (RN ) the 1-form dual to X with Set f := u, v ∈ C0 respect to the Riemann metric g , i.e., (α, β )g = β (X ) ∀β ∈ Ω1 (RN ). Equivalently, α = gij X i dxj . 10.1. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS The equality (10.1.6) can be rewritten ∇X u, v dVg = = BR 437 BR BR df (X )dVg − (df, α)g dVg − RN u, ∇X v dVg u, ∇X v dVg . RN The integration by parts formula of Subsection 4.1.5 yields (df, α)g dVg = (df ∧ ∗g α) |∂BR + f d∗ αdVg . BR ∂BR BR Since f ≡ 0 on a neighborhood of ∂BR , we get (df, α)g dvg = u, v d∗ α dVg . BR BR Since d∗ α = − div g (X ), (see Subsection 4.1.5) we deduce X · u, v dVg = − u, v div g (X )dVg = u, −div g (X )v dVg . BR BR BR Putting together all of the above we get ∇X u, v dVg = u, (−∇X − div g (X ))v dVg , 1 |g | i RN RN i.e., ∇∗ X = −∇X − div g (X ) = −∇X − ∂i ( |g |X i ). (10.1.7) The lemma and consequently the proposition is proved. Example 10.1.32. Let E be a rank r smooth vector bundle over the oriented Riemann manifold (M, g ), dim M = m. Let ·, · denote a Hermitian metric on E , and consider a connection ∇ on E compatible with this metric. The connection ∇ defines a first order p.d.o. ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ). The metrics g and ·, · induce a metric on T ∗ M ⊗ E . We want to describe the formal adjoint of ∇ with respect to these choices of metrics. As we have mentioned in the proof of the previous proposition, this is an entirely local issue. We fix x0 ∈ M , and we denote by (x1 , . . . , xm ) a collection of g -normal coordinates on a neighborhood U of x0 . We set ∇i := ∇∂xi . Next, we pick a local synchronous frame of E near x0 , i.e., a local orthonormal frame (eα ) such that, at x0 , we have ∇i eα = 0 ∀i = 1, . . . , m. 438 The adjoint of the operator CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS dxk ⊗ : C ∞ (E |U ) → C ∞ (T ∗ U ⊗ E |U ), is the interior derivative (contraction) along the vector field g -dual to the 1-form dxk , i.e., (dxk ⊗)∗ = C k := g jk · i∂xj ). Since ∇ is a metric connection, we deduce as in the proof of Lemma 10.1.31 that ∇k ∗ = −∇k − div g (∂xk ). Hence, ∇∗ = k ∇k ∗ ◦ C k = k −∇k − div g (∂xk ) ◦ C k (10.1.8) =− k (∇k + ∂xk log( |g |) ◦ C k . In particular, since at x0 we have ∂xk g = 0, we get ∇∗ |x0 = − k ∇k ◦ C k . The covariant Laplacian is the second order p.d.o. ∆ = ∆∇ : C ∞ (E ) → C ∞ (E ), ∆ = ∇∗ ∇. To justify the attribute Laplacian we will show that ∆ is indeed a generalized Laplacian. Using (10.1.8) we deduce that over U (chosen as above) we have     dxj ⊗ ∇j ∆=− ∇k + ∂xk log |g | ◦ C k ◦   k j =− k ∇k + ∂xk log |g | ◦ g kj · ∇j |g | ◦ ∇j =− k,j g kj ∇k + ∂xk g kj + g kj ∂xk log g kj ∇k ∇j + k,j =− 1 |g | ∂xk |g |g kj · ∇j . The symbol of ∆ can be read easily from the last equality. More precisely, σ2 (∆)(ξ ) = −g jk ξj ξk = −|ξ |2 g. Hence ∆ is indeed a generalized Laplacian. In the following exercise we use the notations in the previous example. 10.1. PARTIAL DIFFERENTIAL OPERATORS: ALGEBRAIC ASPECTS Exercise 10.1.33. (a) Show that g k Γi k =− 1 |g | ∂xk ( |g |g ik ), 439 where Γi k denote the Christoffel symbols of the Levi-Civita connection associated to the metric g . (b) Show that T ∗ M ⊗E E ∆∇ = − tr2 ∇ ), g (∇ where ∇T M ⊗E is the connection on T ∗ M ⊗ E obtained by tensoring the Levi-Civita connection on T ∗ M and the connection ∇E on E , while ∞ ∗ ⊗2 tr2 ⊗ E ) → C ∞ (E ) g : C (T M ∗ denotes the double contraction by g , ij tr2 g (Sij ⊗ u) = g Sij u. Proposition 10.1.34. Let E and M as above, and suppose that L ∈ P DO 2 (E ) is a generalized Laplacian. Then, there exists a unique metric connection ∇ on E and R = R(L) ∈ End (E ) such that L = ∇∗ ∇ + R. The endomorphism R is known as the Weitzenb¨ ock remainder of the Laplacian L. Exercise 10.1.35. Prove the above proposition. Hint Try ∇ defined by ∇f grad(h) u = f {(∆g h)u − (ad(h)L)u} f, h ∈ C ∞ (M ), u ∈ C ∞ (E ). 2 Exercise 10.1.36. (General Green formula). Consider a compact Riemannian manifold (M, g ) with boundary ∂M . Denote by n the unit outer normal along ∂M . Let E, F → M be Hermitian vector bundles over M and suppose L ∈ PDOk (E, F ). Set g0 = g |∂M , E0 = E |∂M and F0 = F |∂M . The Green formula states that there exists a sesquilinear map BL : C ∞ (E ) × C ∞ (F ) → C ∞ (∂M ), such that Lu, v dv (g ) = M ∂M BL (u, v )dv (g0 ) + u, L∗ v dv (g ). M Prove the following. (a) If L is a zeroth order operator, then BL = 0. (b) If L1 ∈ P DO (F, G), and L2 ∈ P DO (E, F ), then BL1 L2 (u, v ) = BL1 (L2 u, v ) + BL2 (u, L∗ 1 v ). (c) BL∗ (v, u) = −BL (u, v ). 440 CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS (d) Suppose ∇ is a Hermitian connection on E , and X ∈ Vect (M ). Then B∇X (u, v ) = u, v g (X, n), B∇ (u, v ) = u, in v E, where in denotes the contraction by n. (e) Denote by ν the section of T ∗ M |∂M g -dual to n. Suppose L is a first order p.d.o., and set J := σL (ν ). Then BL (u, v ) = Ju, v F . (f) Using (a)-(e) show that, for all u ∈ C ∞ (E ), v ∈ C ∞ (F ), and any x0 ∈ ∂M , the quantity BL (u, v )(x0 ) depends only on the jets of u, v at x0 of order at most k − 1. In other words, if all the partial derivatives of u up to order (k − 1), with respect to some connection, vanish along the boundary, then BL (u, v ) = 0. 10.2 Functional framework The partial differential operators are linear operators in infinite dimensional spaces, and this feature requires special care in dealing with them. Linear algebra alone is not sufficient. This is where functional analysis comes in. In this section we introduce a whole range of functional spaces which are extremely useful for most geometric applications. The presentation assumes the reader is familiar with some basic principles of functional analysis. As a reference for these facts we recommend the excellent monograph [18], or the very comprehensive [30, 103]. §10.2.1 Sobolev spaces in RN Let D denote an open subset of RN . We denote by L1 loc (D ) the space of locally integrable real functions on D , i.e., Lebesgue measurable ∞ (RN ), the function αf is Lebesgue integrable, functions f : RN → K such that, ∀α ∈ C0 1 N αf ∈ L (R ). 1 Definition 10.2.1. Let f ∈ L1 loc (D ), and 1 ≤ k ≤ N . A function g ∈ Lloc (D ) is said to be the weak k -th partial derivative of f , and we write this g = ∂k f weakly, if gϕdx = − D D ∞ f ∂k ϕdx, ∀ϕ ∈ C0 (D). Lemma 10.2.2. Any f ∈ L1 loc (D ) admits at most one weak partial derivative. The proof of this lemma is left to the reader. Not all locally integrable functions admit weak derivatives. Exercise 10.2.3. Let f ∈ C ∞ (D), and 1 ≤ k ≤ N . Prove that the classical partial derivative ∂k f is also its weak k -th derivative. Exercise 10.2.4. Let H ∈ L1 loc (R) denote the Heaviside function, H (t) ≡ 1, for t ≥ 0, H (t) ≡ 0 for t < 0. Prove that H is not weakly differentiable. 10.2. FUNCTIONAL FRAMEWORK 441 1 Exercise 10.2.5. Let f1 , f2 ∈ L1 loc (D ). If ∂k fi = gi ∈ Lloc weakly, then ∂k (f1 + f2 ) = g1 + g2 weakly. The definition of weak derivative can be generalized to higher order derivatives as follows. Consider a scalar p.d.o. L : C ∞ (D) → C ∞ (D), and f, g ∈ L1 loc (D ). Then we say that Lf = g weakly if gϕdx = D D ∞ f L∗ ϕdx ∀ϕ ∈ C0 (D). Above, L∗ denotes the formal adjoint of L with respect to the Euclidean metric on RN . Exercise 10.2.6. Let f, g ∈ C ∞ (D). Prove that Lf = g classically ⇐⇒ Lf = g weakly. Definition 10.2.7. Let k ∈ Z+ , and p ∈ [1, ∞]. The Sobolev space Lk,p (D) consists of all the functions f ∈ Lp (D) such that, for any multi-index α satisfying |α| ≤ k , the mixed partial derivative ∂ α f exists weakly, and moreover, ∂ α f ∈ Lp (D). For every f ∈ Lk,p (D) we set  1/p f if p < ∞, while if p = ∞ f When k = 0 we write f p k,∞ k,p := f k,p,D = |α|≤k D |∂ α f |p dx , = |α|≤k 0,p . ess sup |∂ α f |. instead of f Definition 10.2.8. Let k ∈ Z+ and p ∈ [1, ∞]. Set Lk,p loc (D ) := k,p ∞ f ∈ L1 loc (D ) ; ϕf ∈ L (D ) ∀ϕ ∈ C0 (D ) . Exercise 10.2.9. Let f (t) = |t|α , t ∈ R, α > 0. Show that for every p > 1 such that 1 α>1− p , we have ,p f (t) ∈ L1 loc (R). Theorem 10.2.10. Let k ∈ Z+ and 1 ≤ p ≤ ∞. Then (a) (Lk,p (RN ), · k,p ) is a Banach space. ∞ (RN ) is dense in Lk,p (RN ). (b) If 1 ≤ p < ∞ the subspace C0 (c) If 1 < p < ∞ the Sobolev space Lk,p (RN ) is reflexive. The proof of this theorem relies on a collection of basic techniques frequently used in the study of partial differential equations. This is why we choose to cover the proof of this theorem in some detail. We will consider only the case p < ∞, leaving the p = ∞ situation to the reader. 442 CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS Proof. Using the Exercise 10.2.5, we deduce that Lk,p is a vector space. From the classical Minkowski inequality, ν 1/p ν 1/p ν 1/p |xi + yi | i=1 p ≤ i=1 |xi | p + i=1 |yi | p , we deduce that · k.p is a norm. To prove that Lk,p is complete, we will use the well established fact that Lp is complete.  To simplify the notations, throughout this chapter all the extracted subsequences will be denoted by the same symbols as the sequences they originate from. Let (fn ) ⊂ Lk,p (RN ) be a Cauchy sequence, i.e., m,n→∞ lim fm − fn k,p = 0. In particular, for each multi-index |α| ≤ k , the sequence (∂ α fn ) is Cauchy in Lp (RN ), and thus, Lp ∂ α fn → gα , ∀|α| ≤ k. Set f = limn fn . We claim that ∂ α f = gα weakly. ∞ (RN ) we have Indeed, for any ϕ ∈ C0 ∂ α fn ϕdx = (−1)|α| fn ∂ α ϕ. RN RN Since ∂ α fn → gα , fn → f in Lp , and ϕ ∈ Lq (RN ), where 1/q = 1 − 1/p, we conclude gα · ϕ dx = lim n RN RN ∂ α fn · ϕ dx f · ∂ α ϕ dx. RN = lim(−1)|α| n RN fn · ∂ α ϕ dx = (−1)|α| Part (a) is proved. ∞ (RN ) is dense we will use mollifiers. Their definition uses the To prove that C0 operation of convolution. Given f, g ∈ L1 (RN ), we define (f ∗ g )(x) := RN f (x − y )g (y )dy. We let the reader check that f ∗ g is well defined, i.e., y → f (x − y )g (y ) ∈ L1 , for almost all x. Exercise 10.2.11. (Young’s Inequality). If f ∈ Lp (RN ), and g ∈ Lq (RN ), 1 ≤ p, q ≤ 1 1 ∞, then f ∗ g is well defined, f ∗ g ∈ Lr (RN ), 1 r = p + q − 1, and f ∗g r ≤ f p · g q. (10.2.1) 10.2. FUNCTIONAL FRAMEWORK ∞ (RN ), such that To define the mollifiers one usually starts with a function ρ ∈ C0 443 ρ ≥ 0, supp ρ ⊂ |x| < 1 , and RN ρ dx = 1. Next, for each δ > 0, we define ρδ (x) := δ −N ρ(x/δ ). Note that supp ρδ ⊂ |x| < δ and RN ρδ dx = 1. The sequence (ρδ ) is called a mollifying sequence. The next result describes the main use of this construction. N 1 Lemma 10.2.12. (a) For any f ∈ L1 loc (R ) the convolution ρδ ∗ f is a smooth function! (b) If f ∈ Lp (RN ) (1 ≤ p < ∞) then ρδ ∗ f → f as δ → 0. Lp Proof. Part (a) is left to the reader as an exercise in the differentiability of integrals with parameters. ∞ (RN ) is dense in Lp (RN ). Fix ε > 0, To establish part (b), we will use the fact that C0 ∞ (RN ) such that, and choose g ∈ C0 f −g We have ρδ ∗ f − f p p ≤ ε/3. + ρδ ∗ g − g + g−f p. ≤ ρδ ∗ (f − g ) p p Using the inequality (10.2.1) we deduce ρδ ∗ f − f p ≤2 f −g p + ρδ ∗ g − g p . (10.2.2) We need to estimate ρδ ∗ g − g p . Note that ρδ ∗ g (x) = Hence |ρδ ∗ g (x) − g (x)| ≤ Since supp ρδ ∗ g ⊂ x + z ; x ∈ supp g ; z ∈ supp ρδ , there exists a compact set K ⊂ RN such that, supp(ρδ ∗ g − g ) ⊂ K ∀δ ∈ (0, 1). 1 ρ(z )g (x − δz ) dz and g (x) = RN RN ρ(z )g (x)dz. ρ(z )|g (x − δz ) − g (x)|dz ≤ δ sup |dg |. RN This explains the term mollifier: ρδ smoothes out the asperities. 444 We conclude that ρδ ∗ g − g p CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS ≤ K δ p (sup |dg |)p dx 1/p = vol (K )1/p δ sup |dg |. If now we pick δ such that vol (K )1/p δ sup |dg | ≤ ε/3, we conclude from (10.2.2) that ρδ ∗ f − f The lemma is proved. The next auxiliary result describes another useful feature of the mollification technique, especially versatile in as far as the study of partial differential equations is concerned. N Lemma 10.2.13. Let f, g ∈ L1 loc (R ) such that ∂k f = g weakly. Then ∂k (ρδ ∗ f ) = ρδ ∗ g . More generally, if aα ∂ α L= |α|≤m p ≤ ε. is a p.d.o with constant coefficients aα ∈ R, and Lf = g weakly, then L(ρδ ∗ f ) = ρδ ∗ g classically. Proof. It suffices to prove only the second part. We will write Lx to emphasize that L acts via derivatives with respect to the variables x = (x1 , . . . , xN ). Note that L(ρδ ∗ f ) = Since (Lx ρδ (x − y ))f (y )dy. (10.2.3) RN ∂ ∂ ρδ (x − y ) = − i ρδ (x − y ), ∂xi ∂y and ∂i ∗ = −∂i , we deduce from (10.2.3) that L(ρδ ∗ f ) = (L∗ y ρδ (x − y ))f (y )dy = ρδ (x − y )Ly f (y ) dy RN RN = RN ρδ (x − y )g (y )dy = ρδ ∗ g. Remark 10.2.14. The above lemma is a commutativity result. It shows that if L is a p.d.o. with constant coefficients, then [L, ρδ ∗]f = L(ρδ ∗ f ) − ρδ ∗ (Lf ) = 0. This fact has a fundamental importance in establishing regularity results for elliptic operators. p RN (∂i ηR )ϕ + ηR · ∂i ϕdx = RN ∂i (ηR ϕ)f = − RN ηR ϕ∂i f dx. we return to the proof of Theorem 10. so that by the dominated convergence theorem we conclude (∂i ηR ) · f → 0 as R → ∞. which implies ηR ∂i f → ∂i f in Lp . we have ∂ α (ρδ ∗ f ) = ρδ ∗ (∂ α f ) ∀|α| ≤ k.e. Let f ∈ Lk. Truncation.p (RN ). the space of compactly supported Lk. Note that ∂i ηR ≡ 0 for |x| ≤ R and |x| ≥ R + 1.p as δ 0. This confirms our claim.p using two basic We will construct fn ∈ C0 n techniques: truncation and mollification (or smoothing ).2. FUNCTIONAL FRAMEWORK 445 After this rather long detour. Similarly ηR ∂i f → ∂i f a.e. Hence. and |ηR ∂i f | ≤ |∂i f |. ∀R ≥ 0. Clearly (∂i ηR )f + ηR ∂i f ∈ Lp (RN ). Let f ∈ Lk. Note that each ρδ ∗ f is a smooth function.2.2. it suffices to show that any such function can be arbitrarily well approximated in the Lk.p (RN ).p (RN ).. Proof. In particular. We first prove that ∂i (ηR f ) exists weakly and as expected ∂i (ηR · f ) = (∂i ηR ) · f + ηR · ∂i f. The lemma is proved. The sequence ρδ ∗ f converges to f in the norm of Lk.12. According to Lemma 10. Clearly |(∂i ηR )f | ≤ 2|f (x)|. Consider for each R > 0 a smooth function ηR ∈ ∞ (RN ) such that η (x) ≡ 1 for |x| ≤ R.2. Lk. Lp . compactly supported functions.15. The general situation can be proved by induction. Mollification. According to the above lemma.13. Lemma 10. η (x) ≡ 0 for |x| ≥ R + 1.p -norm by smooth. we deduce (∂i ηR )f → 0 a. ∞ (RN ). Let f ∈ ∞ (RN ) such that f → f in Lk.p (RN ). Since η ϕ ∈ C ∞ (RN ). We consider only the case k = 1. and assume ess supp f ⊂ {|x| ≤ R}.p (RN ).p (RN ).10. and moreover ηR · f −→ f as R → ∞. The essentials of this technique are contained in the following result. C0 R R Then ηR · f ∈ Lk. we have Let ϕ ∈ C0 R 0 Lk. and |dη (x)| ≤ 2 ∀x.10. supported in { |x| ≤ R + δ }. The desired conclusion now follows using Lemma 10.p -functions is dense in Lk. so that ηR f ∈ L1.2. Intuitively. . . where p = p/(p − 1). ELLIPTIC EQUATIONS ON MANIFOLDS To conclude the proof of Theorem 10.2 (RN ) are in fact Hilbert spaces.10 is proved.2.2. The above statement should be taken with a grain of salt since at this point it is not clear how one can define u |∂D when u is defined only almost everywhere. arguing as in the proof of completeness. provided that the boundary of D is sufficiently regular. The larger space C ∞ (D) ∩ Lk. k − 1. Lk. Hence (fα ) = T f0 .2.p (D ). (a) For p = 2 the spaces Lk. and has closed range. . . Exercise 10. 1. Indeed.p ( D ) consists of the functions u ∈ Lk. f → (∂ α f )|α|≤k .17. v k = RN  |α|≤k ∂ α u · ∂ α vdx dx.10. ∀j = 0. Lp which is continuous.p L0 (D).p (D ) is denoted by C0 0 k. . (b) If D ⊂ RN is open. The closure of C ∞ (D ) in Lk. We refer again to [3] for details. (b) There exists a constant C > 0 such that.p (RN ) is reflexive as a closed subspace of a reflexive space. then Lk. N.p (RN ) → |α|≤k Lp (RN ). Prove that the following statements are equivalent.p is reflexive if 1 < p < ∞.p (RN ). we have u RN ∂ϕ ≤C ϕ ∂xi Lp . Remark 10. We refer to [3] for a way around this issue. then. where f0 = lim fn . Theorem 10. ∂ν j where ∂/∂ν denotes the normal derivative along the boundary. reflexive if 1 < p < ∞.2.p (D) is dense in Lk. is no longer dense in Lk. .p (RN ) can be viewed as a closed subspace of the direct product Lp (RN ). Note first that Lk.p (D). . if ∂ α fn → fα . . We will use the fact that Lp is reflexive for p in this range.16. . one-to-one. T : Lk. we need to show that Lk. for all ϕ ∈ C ∞ (RN ).446 CHAPTER 10. ∀i = 1. |α|≤k via the map.p (D) is a Banach space. (a) u ∈ L1. The inner product is given by   u. However ∞ (D ).p (D) such that 0 ∂j u = 0 on ∂D. we deduce that fα = ∂ α f0 weakly. We now conclude that Lk. Then φ(f ) ∈ L1.p (RN ) as the quantity σ (k. p) := σN (k. p) can be explained by using the notion of conformal weight. Let f ∈ L1.p as ε → 0. such that |dφ| ≤ const. More generally. Hint: Show that fε := (ε2 + f 2 )1/2 converges to f in L1. Exercise 10. a mixed partial ∂ α u has conformal weight −|α|.20. on {f < 0} Lp 447 ≤ C |h|. the more regular are the functions in that space. Its conformal weight is 0.18. for all h ∈ RN . The origin of the quantities σN (k. and thus it consists of “fewer” functions. Let f ∈ L1. A function u : RN → R can be thought of as a dimensionless physical quantity. For example.p (RN ). In particular the quantity   |α|=k 1/p |∂ α u|p dx .2.p (RN ). The “strength” is a measure of the size of a Sobolev size. and φ(f ) ∈ Lp . and they have conformal weight −1. FUNCTIONAL FRAMEWORK (c) There exists C > 0 such that. Remark 10.e.e.2. §10. p) = k − N/p. The volume form dx is assigned conformal weight N : the volume is measured in meterN . Loosely speaking.19. the bigger the strength.2 Embedding theorems: integrability properties The embedding theorems describe various inclusions between the Sobolev spaces Lk.2. and ∂i φ(f ) = φ (f ) · ∂i f. on {f ≥ 0} a.p (RN ).2. The integral of a quantity of conformal weight w is a quantity of conformal weight w + N .p (RN ). Define the “strength” of the Sobolev space Lk.10. Its partial derivatives ∂i u are physical quantities measured in meter−1 = variation per unit of distance. Show that |f | ∈ L1. The quantities |∂ α |p have conformal weight −p|α|. Exercise 10.p (RN ).2. the quantity |∂ α u|p dx RN has conformal weight N − p|α|. and φ ∈ C ∞ (R). we have ∆h u where ∆h u(x) = u(x + h) − u(x). and ∂i |f | = ∂i f −∂i f a. m. q ) < 0 and k > m then. . ELLIPTIC EQUATIONS ON MANIFOLDS has conformal weight (N − kp)/p = −σN (k. . . but ingenious lemma. and the natural inclusion is continuous. Lk. . . . . If we replace the Euclidean metric g on Rn with the metric gλ = λ2 g . . def . Theorem 10. . Then f (x) = f1 (ξ1 )f2 (ξ2 ) · · · fN (ξN ) ∈ L1 (RN ). . p) = σN (m. ∀u ∈ C ∞ (RN ). |du|gλ = λ−1 |du|g . p = 1. Exercise 10. . i. and moreover. dx1 ∧ · · · ∧ dxN = λN dy 1 ∧ · · · ∧ dy N . For each x ∈ RN .22 (Gagliardo-Nirenberg). N f 1 ≤ i=1 fi N −1 . . .p (RN ) → Lm. .2. xN ) ∈ RN −1 . which means m = 0. .e.448 CHAPTER 10. p. t. where |du|2 = |∂1 u|2 + · · · + |∂N u|2 . and q = N/(N − 1). Proof. xi+1 . . Let N ≥ 2 and f1 . Geometrically. k. the conformal weight is captured by the behavior under the rescalings x = λy . then ∂xi = λ−1 ∂yi . there exists C = C (N. xN )| ≤ x1 −∞ |∂i u(x1 . Prove Lemma 10. . We will show that ∃C > 0 : u N/(N −1) ≤ C |du| 1 ∞ ∀u ∈ C0 (RN ). x ˆi . xi−1 . .q ≤ C f k. . This result then extends by density to any u ∈ L1. dVgλ = λn dVg .2.p ∀f ∈ Lk. p). We have |u(x1 . . . . λ > 0.p (RN ).1 (RN ). . fN ∈ LN −1 (RN −1 ). q ) > 0 such that f m. then. and 1 ≤ i ≤ N define ξi = (x1 . If σN (k. . Equivalently. We first prove the theorem in the case k = 1. .23. Lemma 10. xN )|dt = gi (ξi ).2.21 (Sobolev).p (RN ).2.22. if we introduce the new linear variables y i related to the canonical Euclidean variables xi by xi = λy i .. We follow the approach of [81] which relies on the following elementary. Now choose r such that rN/(N − 1) = q (r − 1). . we conclude u N/(N −1) ≤ 1 N N ∂i u 1 1 ≤ const. where p∗ = Np . i.10.22 that u(x) ∈ LN/(N −1) (RN ).p (RN ) embeds continuously in Lp (RN ). p) |du| p.e. where r > 1 will be specified later. We have to show ∗ that L1. N 1/N r −1 q (r−1) ∂i u p . If q = p/(p − 1) is the conjugate exponent of p. N −p ∞ (RN ).2. so that fi (ξi ) = gi (ξi )1/(N −1) ∈ LN −1 (RN −1 ). N u p∗ ≤r 1 ∂i u p ≤ C (N. Using the classical arithmetic-geometric means inequality. p) = 1 − N/p < 0. This gives r = p∗ and we get N 1/N N −1 . and N 1/N N 1/N 449 u N/(N −1) ≤ 1 gi (ξi ) 1 =( 1 ∂i u 1 .2. Since |u(x)|N/(N −1) ≤ f1 (ξ1 ) · · · fN (ξN ). we conclude from Lemma 10.1 (RN ) embeds continuously in LN/(N −1) (RN ). FUNCTIONAL FRAMEWORK Note that gi ∈ L1 (RN −1 ). then using the H¨ older inequality we get |u|r−1 ∂i u ≤ u Consequently. Now let 1 < p < ∞ such that σN (1. u r rN/(N −1) ≤r u r −1 q (r−1) 1 ∂i u p . N N |du| 1 1. The inequality Let u ∈ C0 N 1/N v implies N/(N −1) ≤ 1 ∂i v 1 N 1/N u r rN/(N −1) ≤r 1 |u| r−1 ∂i u 1 . We have thus proved that L1.. p < N . Set v = |v |r−1 v . q (RN ). We will prove that. we will use the Arz´ ela-Ascoli theorem. To prove this. ELLIPTIC EQUATIONS ON MANIFOLDS This shows L1. ∀|x| ≤ R + 1.δn .p (by Sobolev embedding theorem). Indeed. the mollified sequence un.δ (x)| < C ∀n. and a subsequence δn 0 such that.p (RN ) such that ess supp un ⊂ {|x| ≤ R} ∀n. |ρδ ∗ u (x)| ≤ δ −N ρ |y −x|≤δ x−y δ p∗ |un (y )|dy ≤ δ −N · vol (Bδ )(p ∗ −1)/p∗ Bδ (x) |un (y )|dy ≤ C (N. Theorem 10. Let (k. BR+1 |ρδ (x1 − y ) − ρδ (x2 − y )| · |un (y )|dy 1.2. for every 1 ≤ q < p∗ = N p/(N − p).δ (x1 ) − un. BR+1 |un (y )|dy ≤ C (δ ) · |x1 − x2 | · un Step 2: Conclusion. We let the reader fill in the details. Let (un ) be a bounded sequence in L1. q ) ∈ Z+ × [1. Then any bounded sequence (un ) ⊂ Lk. p) > σN (m.p → Lm.p → Lp if 1 ≤ p < N . p)δ −N un ≤ C (δ ) un Similarly. the following hold. The proof will be carried out in several steps. Using the diagonal procedure.q if σN (k.p . p) = σN (m.24 (Rellich-Kondrachov). R > 0 has a subsequence strongly convergent in Lm. p) < 0 imposes 1 ≤ p < N . if we set gn := un. q ) < 0 k > m follows easily by induction over k .δ (x2 )| ≤ C |x1 − x2 | ∀n. q ). p). Proof. |un. |un. still denoted by (un ). so that the condition σN (1. we can extract a subsequence of (un ).p (RN ) supported in a ball BR (0). 1. Step 1.δ (x2 )| ≤ ≤ C (δ ) · |x1 − x2 | Step 1 is completed. The general case Lk. such that |un. We discuss only the case k = 1.δ (x1 ) − un. the sequence (un ) contains a subsequence convergent in Lq . (m. |x2 | ≤ R + 1. ∀|x1 |. and we will show there exists C = C (δ ) > 0.δ = ρδ ∗ un admits a subsequence which is uniformly convergent on the ball {|x| ≤ R + 1}.450 ∗ CHAPTER 10. We have to show that. for every 0 < δ < 1. ∞) such that k > m and 0 > σN (k. . BR = BR |un (x) − gn (x)|dx 1/r ( H¨ older inequality) ≤ BR |un (x) − gn (x)|q dx · vol (BR )(r−1)/r (r = p∗ /q ) → 0. x. Set u and define C 0. FUNCTIONAL FRAMEWORK (i) The sequence gn is uniformly convergent on BR . we denoted by osc (u. (ii) limn gn − un 1.RN 451 = 0. and even differentiable. However..α. x∈D := u ∞ + [u]α < ∞ . . The middle term. BR Hence the (sub)sequence (un ) is Cauchy in Lq (BR ).D := sup |u(x)|. then the functions of Lk. and thus it converges.10. The first and the third term.for all n.RN ≤ {sup |gn (x) − gm (x)|}q · vol (BR ) → 0. m un − um q. We now examine separately each of the three terms in the right-hand-side.p.BR ≤ un − gn q.BR + gm − um q.z ∈D R−α osc (u. u 0. §10. The compactness theorem is proved. Let α ∈ (0.e. A function u : D ⊂ RN → R is said to be α-H¨ older continuous if [u]α := sup 0<R<1. gn − gm q q.p.D ∞. ≤ C (R) un − gn B. namely the spaces of H¨ older continuous functions. y ∈ S }.BR q p∗ . A. S ) the oscillation of u on S .BR . i.α (D) := u:D→R. To formulate our next results we must introduce another important family of Banach spaces. un − gn q q.p are only measurable. if the strength σN (k. for all 1 ≤ q < p.RN (Sobolev) ≤ C (R) un − gn 1. where for any set S ⊂ D.2.p have a built-in regularity: each can be modified on a negligible set to become continuous. the functions in the Sobolev spaces Lk.2.3 Embedding theorems: differentiability properties A priori. BR (z ) ∩ D) < ∞. p) is sufficiently large. 1). osc (u. and are defined only almost everywhere. We claim the subsequence (un ) as above is convergent in Lq (BR ).BR + gn − gm q. S ) := sup {|u(x) − u(y )| . Indeed . α (D). Lemma 10. then dy = ρN −1 dρdω . Proof.2. The proof relies on the following elementary observation. and dω denotes the Euclidean “area” form on the unit round sphere.1 (BR ) and set u := Then |u(x) − u| ≤ 1 vol (BR ) 2N σ N −1 u(x)dx. ∂r If we set |∂r u(x + rω )| = 0 for |x + rω | > R. α) := k + α.2. ∀|β | ≤ k. p) = σ (k.4) In the above inequality σ N −1 denotes the “area” of the unit (N − 1)-dimensional round sphere S N −1 ⊂ RN . α) > 0.D + |β |=k [∂ β u]α. Proof. ω ) centered at x. The space C k.α := |β |≤k ∂β u ∞. |x−y | vol (BR )(u(x) − u) = − dy BR 0 ∂ u(x + rω )dr.26. Then Lm. BR BR |du(y )| dy. Define the strength of the H¨ older space C k. and (necessarily) m = 0. u k. ∞] and (k. If we use polar coordinates (ρ. We deduce ∞ 2R vol (BR )|u(x) − u| ≤ dr 0 S N −1 dω 0 |∂r u(x + rω )|ρN −1 dρ .D . 1) such that m > k and σN (m. Let u ∈ C ∞ (BR ) ∩ L1. Theorem 10.p (RN ) embeds continuously in C k.α (D) is a Banach space with respect to the norm. Consider (m. ∂r |x − y | Integrating the above equality with respect to y we get. α) ∈ Z+ × (0. |x − y |N −1 (10. for every integer k ≥ 0.α as the quantity σ (k. ∂ β u ∈ C 0. where ρ = |x − y |.α (RN ). define C k. p) ∈ Z+ × [1. We consider only the case k = 1.25 (Morrey). |x−y | u(x) − u(y ) = − 0 x−y ∂ u(x + rω )dr (ω = − ).2. .452 CHAPTER 10. ELLIPTIC EQUATIONS ON MANIFOLDS More generally.α (D) := u ∈ C m (D). then ∞ vol (BR )|u(x) − u| ≤ dy |x−y |≤2R 0 |∂r u(x + rω )|dr. R := u(y )dy. In the sequel for any R > 0. L∞ -estimates. The inequality (10. p) > 0.p (RN ) ∩ C 1 (RN ) u ∞ ≤ C u 1. Step 1. Step 2. vol (BR (x0 )) BR (x0 ) 2. and set ux := ux. We will show there exists C > 0 such that. q ) is a universal constant depending only on N . 1.4) we deduce |u(x)| ≤ |ux | + CN |du(y )| dy ≤ C |x − y |N −1 u + B (x) B (x) p |du(y )| dy . We will complete the proof of Morrey’s theorem in three steps. the function y → |x − y |−(N −1) lies in Lq (B (x)). FUNCTIONAL FRAMEWORK = = (2R)N N ∞ 0 ∞ 453 |∂r (x + rω )|dωdr 1 S N −1 dr 0 S N −1 (2R)N N rN −1 dr r N −1 |∂r u(x + rω )|dω |du(z )| dy. q ). |x − y |N −1 (10.5) we conclude |u(x)| ≤ C u 1. Using the H¨ older inequality in (10. and 1 dy ≤ C (N. Oscillation estimates. Using (10.1 (BR ).1 .p ∀x.2. We will show there exists C > 0 such that for all u ∈ L1. .10.2. we can replace the round ball BR centered at origin by any other ball. centered at any other point.5) Since σN (1. For each x ∈ RN . ∀u ∈ L1.p (RN ) ∩ C 1 (RN ) [u]α ≤ C u 1.p . p−1 N −1 In particular.2. and any x0 ∈ RN we set 1 ux0 . In the above lemma. and q .2. so that its conjugate exponent q satisfies q= N p < . we deduce that p > N .4) can be extended by density to any u ∈ L1. denote by B (x) the unit ball centered at x.p .2. |x − y |q(N −1) B (x) where C (N. |x − z |N −1 (z = x + rω ) = (2R)N N BR |∂r u(z )| (2R)N ≤ |x − z |N −1 N BR We want to make two simple observations. 0 < r < 2. where the constant C2 depends only on N. ν and C1 .2. Thus |x| |u(x) − u(0)| ≈ C 0 tα−1 dt ≈ C |x|α . for any ball BR (x0 ). where 0 ≤ ν < q and q > 1.p and almost everywhere. ν= Hence 1 N N − q (N − 1) = 1 − = α. BR (x0 )) ≤ CRα . If u satisfies the (q. ELLIPTIC EQUATIONS ON MANIFOLDS Indeed. Exercise 10.2. q p |u(x) − ux0 . from the inequality (10. q.α -norm to a function v ∈ C 0. Let u ∈ L1. it “explodes” no worse that rα−1 as r → 0.4). . Remark 10. and any x ∈ BR (x0 ). i.4) we deduce that.27. In fact. osc (u.2. Show that (up to a change on a negligible set) the function u is α-H¨ older continuous.R | ≤ C | x − y |N −1 BR (x0 ) (q = p p−1 ) 1/q (H¨ older) ≤ C |du| where p · BR (x0 ) 1 dy |x − y |q(N −1) ≤ C |du| pR ν . ν )-energy estimate. on average. ∃C > 0 : 1 rN Br (x) |du(y )|q dy ≤ C1 r−ν ∀x ∈ RN .1 (RN ) satisfy a (q.α (RN ) which agrees almost everywhere with u. The energy estimate is a very useful tool in the study of nonlinear elliptic equations. ν )-energy estimate then although |du| may not be bounded.R | ≤ CRα ∀x ∈ BR (x0 ). and consequently. Hint: Prove that |du(y )| ≤ Crα . The estimates established at Step 1 and 2 are preserved as n → ∞.28.2. ∞ (RN ) such that Step 3:. Step 2 is completed.p (RN ) we can find (un ) ∈ C0 un → u in L1. α = 1 − ν/q . The result in the above exercise has a suggestive interpretation. and moreover [u]α ≤ C2 .e. these estimates actually show that the sequence (un ) converges in the C 0. Conclusion. Given u ∈ L1.454 CHAPTER 10. |du(y )| |u(x) − ux0 . N −1 | x − y | Br (x) and then use the inequality (10. ∞] and (m.10.p.α. •) defines two important objects. 1). p) ∈ Z+ × [1. .p (RN ) admits a subsequence which converges in the C m. In particular. ε > 0 and for all u ∈ Lm. For each R > 0 choose a smooth. but we chose not to include their long but elementary proofs since they do not use any concept we will need later. The results in this.α norm on any bounded open subset of RN . dVg ) (modulo the equivalence relation of equality almost everywhere). where H is a finite dimensional complex Hermitian space.4 Functional spaces on manifolds The Sobolev and the H¨ older spaces can be defined over manifolds as well.RN ≤ Cε ηR u m. R.2.2. there exists C = C (r. and for all u ∈ C k. (i) The Levi-Civita connection ∇g . They play an important part in applications. p) ∈ Z+ × [1.α (RN ). To define these spaces we need three things: an oriented Riemann manifold (M. and the previous section extend verbatim to slightly more general situations. ∞).α. 1) such that σN (k. The last results we want to discuss are the interpolation inequalities. Fix (m.2.Br . there exists C = C (r. and a connection ∇ = ∇E compatible with h. (a) For every 0 < r ≤ R +1. cutoff ∞ (RN ) such that function ηR ∈ C0 ηR ≡ 1 if |x| ≤ R. α) ∈ Z+ × (0. ε > 0. p) such that. α) ∈ Z+ × (0. g ). α) k > m. we have ηR u j. K) the space of K-valued p-integrable functions on (M. namely to functions f : RN → H . ηR ≡ 0 if |x| ≥ R + 1. and (k.Br .RN + Cε−j (m−j ) ηR u 0. and |dηR (x)| ≤ 2 ∀x ∈ RN . for every 0 ≤ j < k .RN ≤ Cε ηR u k. dVg defines a Borel measure on M . m. Theorem 10. The metric g = (•. (ii) A volume form dVg = ∗1. FUNCTIONAL FRAMEWORK 455 The Morrey embedding theorem can be complemented by a compactness result. k. §10. a K-vector bundle π : E → M endowed with a metric h = •. p) > σ (m.29 (Interpolation inequalities).RN + Cε−j (m−j ) ηR u p. we have ηR u j.p. Then a simple application of the Arzela-Ascoli theorem shows that any bounded sequence in Lk. R. (b) For every 0 < r ≤ R + 1. Let (k. The interested reader may consult [3] or [12] for details. α) such that.p (RN ). • . for every 0 ≤ j < m.α. We denote by Lp (M. then for every p1 . q ∈ [1. for almost all x ∈ M except possibly a negligible set.e.E This follows immediately from the Cauchy inequality |h(ψ (x).. ∞].456 CHAPTER 10. k ) be vector bundles with metrics and consider a multilinear bundle map Ξ : E1 × · · · × Ek → KM .e. . (b) The function x → |ψ (x)|p h is integrable with respect to the measure defined by dVg . .2. i.2. on M.31. j = 1. K). ess supx |ψ (x)| if p = ∞ 1/p Note that if p. and the usual H¨ older inequality. such that 1 − 1/p0 = 1/p1 + · · · + 1/pk . . . Exercise 10. (a) π ◦ ψ (x) = x. and ∀ψj ∈ Lpj (Ej ). .30.32. M ψ. Let p ∈ [1. . Lp (E ). then the metric h : E × E → KM defines a continuous pairing •. i. k Ξ(ψ1 . 1/p +1/q = 1. Lp (E ) is a Banach space with respect to the norm ψ = M p. ELLIPTIC EQUATIONS ON MANIFOLDS Definition 10. . An Lp -section of E is a Borel measurable map ψ : M → E . ψk ) dVg ≤ Ξ p0 M · ψ1 p1 · · · ψk pk . . . . φ(x)| ≤ |ψ (x)| · |φ(x)| a. ψ −1 (U ) is Borel measurable for any open subset U ⊂ E ) such that the following hold. .E if p < ∞ |ψ (x)|p dvg (x) . If We regard Ξ as a section of E1 k ∗ ∗ Ξ ∈ Lp0 (E1 ⊗ · · · ⊗ Ek ).. ∗ ⊗ · · · ⊗ E ∗ . φ dVg ≤ ψ p.E · φ q. Let Ei → M (i = 1. ∞] are conjugate.e. . . . .2. . The space of Lp -sections of E (modulo equality almost everywhere) is denoted by We let the reader check the following fact. ∞]. pk ∈ [1. • : Lp (E ) × Lq (E ) → L1 (M. Proposition 10. . we can define the spaces Lp (T ∗ M ⊗m ⊗ E ). .10. define ∇m as the composition ∇m : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ) ∇E ∇T 457 −→ ∗ M ⊗E · · · → C ∞ (T ∗ M ⊗m ⊗ E ). . We say that Definition 10. FUNCTIONAL FRAMEWORK For each m = 1. We set p u m. .2.p (E ) as the space of sections u ∈ Lp (E ) such that. and in particular. . (a) Let u ∈ L1 loc (E ) and v ∈ Lloc (T M m ∇ u = v weakly if M v. 1 ∗ ⊗m ⊗ E ).E = j =1 ∇j u p . φ dVg = ∞ u.. (b) Define Lm. such that ∇j u = vj weakly. . ∇ where we used the symbol ∇ to generically denote the connections in the tensor products T ∗ M ⊗j ⊗E induced by ∇g and ∇E The metrics g and h induce metrics in each of the tensor bundles T ∗ M ⊗m ⊗ E . 2. (∇m )∗ φ dVg ∀u ∈ C0 (T ∗ M ⊗m ⊗ E ).2. m.p := u m. .33. . ∀j = 1.p. . there exist vj ∈ Lp (T ∗ M ⊗j ⊗ E ). p. we have dx⊗α ⊗ ∂ α u.2.p (E ) introduced above depends on several choices: the metrics on M and E and the connection on E .34. The space Lm. and m ∈ Z+ . Proposition 10.35. The length of Dm u(x) is   |α|=m 1/2 |∂ α u(x)|2  . When M is non-compact this dependence is very dramatic and has to be seriously taken into consideration. for every u ∈ C ∞ (M ).p (RM ) coincides as a set with the Sobolev space Lk.2. Lp (RM ) = Lp (M. A word of warning.RM is equivalent with the norm • m. Dm u = |α|=m where for every multi-index α we denoted by dx⊗α the monomial dxα1 ⊗ · · · ⊗ dxαN .RN introduced in the previous sections.p. R). Denote by D the Levi-Civita connection. The Sobolev space Lm. (Lk. The trivial line bundle E = RM is naturally equipped with the trivial metric and connection. Then. The norm • m.p (RN ). g ) be the space RN endowed with the Euclidean metric. Let (M. k. Then.p. Example 10. · ∞.p (E ).E ) is a Banach space which is reflexive if 1 < p < . On the other hand. We assume that the injectivity radius ρM of M is positive. For each ξ ∈ Ex . we define osc (u. Set ρ0 := min{1. if g1 is a different metric on M . g ) be a compact.e.p0 (E ) admits a subsequence convergent in the Lk1 . ∇1 ) as sets of sections. Then the following are true. S ) := sup |u(x) − u(y )|.T ∗ M ⊗m ⊗E .y η |x = |η − Ty. then g canonically defines a metric space structure on M .p (E ). ∇) as before. If u : M → E is a section of E . (b) If 1 ≤ p < ∞.p0 (E ) embeds continuously in Lk1 .36. 0 < r < ρ0 . 1) are such that k0 ≥ k1 and σN (k0 . i. p1 ). ELLIPTIC EQUATIONS ON MANIFOLDS The proof of this result is left to the reader as an exercise. (c) If (ki . and the H¨ older spaces C k.p1 -norm. y ∈ S .y . ∞) (i = 0. . then Lk0 . More precisely.y : Ey → Ex the ∇E -parallel transport along γx. If moreover. p0 ) = k0 − N/p0 ≥ k1 − N/p1 = σN (k1 . Let (E. Let (M. If x. We denote by Tx.p1 (E ) is compact. defining the oscillation of a section of some bundle over M requires a little more work. g ) is a Riemann manifold.E + [∇m u]α.p (E.y starting at x. x ∈ M . we can talk about the oscillation of a function u : M → K. For any k ≥ 0.p (E. h. then C ∞ (E ) is dense in Lk. g1 .α. The H¨ older spaces can be defined on manifolds as well. h.x ξ |y . oriented Riemann manifold. x. and η ∈ Ey we set |ξ − η | := |ξ − Tx. Finally set [u]α. Br (x)).E := sup r−α osc (u. and E a vector bundle over M equipped with a metric h. and the identity map between these two spaces is a Banach space is continuous. and S ⊂ M has the diameter < ρ0 .p0 (E ) → Lk1 . ρM }..2. g. N -dimensional. (see Chapter 4) and in particular. y ) ≤ ρ0 .α (E ) := u ∈ C k (E ). and ending at y . h1 .458 CHAPTER 10. k0 > k1 and k0 − N/p0 > k1 − N/p1 . we define k u and we set k.α (E ) do not depend on the metrics g. and ∇1 is another connection on E compatible with some metric h1 then Lm. C k.α <∞ . (a) The Sobolev space Lm. Theorem 10. u k. then they can be joined by a unique minimal geodesic γx. and compatible connection ∇. h and on the connection ∇. y ∈ M are two points such that distg (x. ∇) = Lm.E := j =0 ∇j u ∞.p1 (E ). A similar statement is true for the H¨ older spaces. then the embedding Lk0 . If (M.p (E ). pi ) ∈ Z+ × [1. any bounded sequence of Lk0 . 3 Elliptic partial differential operators: analytic aspects This section represents the analytical heart of this chapter.3. it may not be C 2 . then 0= d |t=0 E (u0 + tv ) = dt S2 (du0 .1) where ∆ denotes the Laplace-Beltrami operator on the round sphere. (10. then the embedding is also compact. 1) and m − N/p ≥ k + α. and if it does. More precisely. ∞). (10.3. There are a few grey areas in this approach. This avenue was abandoned until the dawns of the twentieth century. ∀u..α (E ). ∆ = d∗ d. and emphasized the need to deal with these issues. S2 so that necessarily. 10. so that the integration by parts is illegal etc. 2 If u0 is a minimum of E . Riemann suggested that one should look for the minima of the energy functional u −→ E (u) := S2 1 (|du|2 + u2 ) − f u dVg . . R).10. If moreover m − N/p > k + α.2) Integrating by parts we get (d∗ du0 + u0 − f ) · v dVg = 0 ∀v. we will introduce the notion of weak solution. Suppose we want to solve the partial differential equation ∆u + u = f ∈ L2 (S 2 .e. p) ∈ Z+ × [1. α) ∈ Z+ × (0. and we leave this task to the reader. i. and a priori estimates. dv )g + u0 · v − f · v dVg ∀v. We will see them at work in the next section. then Lm. when Hilbert reintroduced them into the spotlight. ∆u0 + u0 = f. Consider the following simple example.3. The method can be briefly characterized by two key phrases: partition of unity and interpolation inequalities. We developed all the tools needed to prove this theorem. E (u0 ) ≤ E (u).p (E ) embeds continuously in C k. We discuss two notions that play a pivotal role in the study of elliptic partial differential equations. ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS: ANALYTIC ASPECTS459 (d) If (m. (k. and Weierstrass was quick to point them out: what is the domain of E . u0 may not exist. R) > 0 .p + u p . and R > 0. one asks when this extended notion of solution coincides with the classical one. Throughout the proof we will use the same letter C to denote various constants C = C ( L k+1. (10. the general result is deduced from the special case using perturbation techniques in which the interpolation inequalities play an important role. • the natural Hermitian metric on E . In the second part.3.p ≤C Lu k. Consider an elliptic operator of order m L= |α|≤m α Aα (x)∂x : C ∞ (E ) → C ∞ (E ). Naturally. ∞). Then. Riemann’s idea was first rehabilitated in Weyl’s famous treatise [98] on Riemann surfaces. Theorem 10. The proof consists of two conceptually distinct parts. ∀u ∈ C0 BR (0) ).α >0 ≤C Lu k. 1). Denote by •.460 Elliptic equations on manifolds His important new point of view was that the approach suggested by Riemann does indeed produce a solution of (10.R .4) Proof. and any R > 0 define L k.2) be at least C 2 . It took the effort of many talented people to materialize Hilbert’s program formulated as his 19th and 20th problem in the famous list of 27 problems he presented at the Paris conference at the beginning of the 20th century. For any k ∈ Z+ .R . The norm of Lk.1 (Interior elliptic estimates). This section takes up the issues raised in the above simple example. there exists C = C ( L k+1. k. α. ∀u ∈ C0 BR (0) ). R) > 0 such that. ∞ (E | Then. it suffices that u of (10. N. N. R) (b) Let (k. we have u k+m. Clearly. the trivial connection. (10.α .α + u 0. so that everything boils down to a question of regularity.3. The key fact which will allow us to legitimize Riemann’s argument is the ellipticity of the partial differential operator involved in this equation. p. there exists C = C ( L ∞ (E | such that. (a) Let (k.3.|β |≤k+m−|α| BR (0) β sup ∂x (Aα (x)) .1 Elliptic estimates in RN Let E = Cr denote the trivial vector bundle over RN . We refer the reader to [1] for more details. p) ∈ Z+ × (1. and R > 0.p (E ) (defined using the Euclidean volume) will be denoted by • k. k.3. N.3. and by ∂ . The suitable notion of solution is precisely described in (10. p. In the first part we establish the result under the supplementary assumption that L has constant coefficients.2).3.1) “provided if need be that the notion of solution be suitable extended ”.R = |α|≤m.p . k. In this subsection we will establish the following fundamental result. we have u k+m. α) ∈ Z+ ×(0.R .3.3) k+1. §10. 461 where Aα are r × r complex matrices . C) denote by f ˆ(ξ ) := f 1 (2π )N/2 exp(−ix · ξ )dx. independent of x ∈ RN . ∞ (RN ).∀u ∈ C0 (BR ) [T u]α ≤ C u 0. such that (a1 ) S N −1 ΩdvS N −1 = 0. m Tu p > 0 such that. the limit (a2 ) For any u ∈ C0 (T u)(x) = lim ε 0 |y |≥ε Ω(y ) u(x − y )dy |y |N exists for almost every x ∈ RN . RN Theorem 10. Let m : S N −1 → C be a smooth function. and define m(ξ ) : RN \ {0} → C by m(ξ ) = m ξ |ξ | . Then the following hold.α . We will rely on a very deep analytical result whose proof goes beyond the scope of this book. (c)(Korn-Lichtenstein). . C).2 (Calderon-Zygmund). there exists C = 0. ˆ(ξ ) its Fourier transform For each f ∈ L1 (RN .α C (α. For every 0 < α < 1. ≤C u p ∞ ∀u ∈ C0 (RN ). We set L := Aα . We will prove the conclusions of the theorem hold in this special case.3. and moreover cu(x) + T u(x) = 1 (2π )N/2 exp(ix · ξ )m(ξ )ˆ u(ξ )dξ. R) > 0 such that . and c ∈ C. RN ∞) (b) For every 1 < p < ∞ there exists C = C (p. and any R > 0. m 0. We assume L has the form L= |α|=m α Aα ∂x .α. Constant coefficients case.BR . (a) There exist Ω ∈ C ∞ (S N −1 .Analytic aspects Step 1. Chap II §4.3. Thus. and ην ≥ 0.4. This proves (10. and we get (−i)m |α|=m Aα ξ α ∂ β u(ξ ) = (−i)m ξ β v ˆ( ξ ). .462 Elliptic equations on manifolds For a proof of part (a) and (b) we refer to [94]. we can find functions mij (ξ ) ∈ C ∞ (Rn \ {0}). . Note that B (ξ ) is homogeneous of degree −m. Suppose now that L is an arbitrary elliptic operator of order m. ur (x)). The proof of (10.4) is entirely similar and is left to the reader. Cover BR by finitely many balls ∞ (B (x )). We Fourier transform the above equality. . (10.3.3. If we set v = Lu. ην = 1. and consider ην ∈ C0 r ν R 10N of these balls. Assuming L has the above special form we will prove (10. and such that ∂ β ui (ξ ) = j mij (ξ )ˆ v j (ξ ). which are homogeneous of degree 0. Part II. ∞ (E | Let u ∈ C0 BR ).5) we deduce ˆ ∂ β u(ξ ) = ξ β B (ξ )v ( ξ ) ∀ξ = 0. the operator σ (L)(ξ ) = A(ξ ) = |α|=m Aα ξ α : Cr → Cr is invertible for any ξ = 0.5. of this inequality can be found in [12]. In any case a proof.3. Define u ˆ(ξ ) := (ˆ u1 (ξ ).3) when L has this special form. so that M (ξ ) = ξ β B (ξ ) is homogeneous of degree 0. such that each point in B is covered by at most Br (xν ). Part (c) is “elementary” and we suggest the reader to try and prove it.5) Note that since L is elliptic. The general case. Using Theorem 10. then for any multi-index β such that |β | = m.3. where B (ξ ) = A(ξ )−1 .3.2 (a) and (b) we deduce ∂β u p ≤C v p = C Lu p . . . of smooth functions compactly supported in BR . because L has constant coefficients.3). We discuss first the case k = 0. . Let r > 0 sufficiently small (to be specified later). . Let us now return to our problem. From (10. ν . Step 2. we have L∂ β u = ∂ β Lu = ∂ β v. . The function u can be viewed as a collection u(x) = (u1 (x). u ˆr (ξ )). ≤ C ( Lu p.Br (xν ) p. p.Br (xν ) ≤ Cr−(m−1) u m−1. Using (10. where C = C ( L uν m. We use the equality vν = L(ην u) = ην Lu + [L.Br (xν ) + vν p.R .Br (xν ) p.p ≤ C ( uν p.BR + r−(m+N −1) u m−1.Br (xν ) ). Hence uν We need to estimate vν m.BR ).Br (xν ) + u m−1.Br (xν ) .p ≤ CRN (r−N Lu 1.p. m−1. ≤ C (r uν p.Br (xν ) + r−(m−1) u + r−(m−1) u m−1.3) we deduce uν m.p.p.ν (x) = Aα (xν ) − Aα (x). and taking into account that the number of spheres Br (xν ) is O((R/r)N ) we deduce u m.p.Analytic aspects ∂ β ην ≤ Cr−|β | ∀|β | ≤ m.BR so that p. .o. Set uν := ην u.Br (xν ) + vν p. of order m − 1.ν (x)| ≤ Cr on Br (xν ).d.ν (x)∂ α u 1. and Lν := |α|=m Aα (xν )∂ α .BR ).ν (x)∂ α uν − |β |<m Aβ (x)∂ β uν + vν .Br (xν ) + uν p.Br (xν ) ). then If u ∈ C0 R 463 v = Lu = L ν ην u = ν L(ην u).Br (xν ) ). ην ]u. ∞ (B ). so that [L. ≤ C ( ην Lu p.BR We sum over ν .p.BR ≤ ν uν m.Br (xν ) p. Note that r depends only on R. ≤ C ( uν p.R ).Br (xν ) ). p.Br (xν ) we deduce + vν p.p. ην ]u Hence vν so that uν m. in which [L.p.Br (xν ) + |α|=m εα. ην ] = ad(ην )L is a p.BR ≤ C ( Lu + u m−1. where εα. p.3.p. Since |εα. We rewrite the equality vν = Luν as Lν uν = (Lν − L)uν + vν = |α|=m def εα.p.p.BR ).Br (xν ) We can now specify r > 0 such that Cr < 1/2 in the above inequality.p. L u m. To establish this inequality for arbitrary k we argue by induction. + u can be estimated from above by +C u p. R. and α ∈ (0.BR + ε−(m−1) u p. ∂ β ]) = [σm (L).p. α.1 is proved.BR ).p.p.BR + ∂β u p.BR . ∂ β ]u p.BR + u 0. If now we choose ε > 0 sufficiently small we deduce the case k = 0 of the inequality (10. Exercise 10.o.p. ∂ β ] is a p. We still need to deal with the term u m−1.p. and u k+m. ∂ β ]u = ∂ β v + [L. 1 < p < ∞.3.BR + u m+k−1. Let L as in Theorem 10. Theorem 10.α. p.BR + u p.p.Br ≤ C Lu k. σm+k ([L. 1).p.464 Elliptic equations on manifolds where C is as in the statement of Theorem 10. there exists C = C (k. Prove the above corollary.BR + [L. It is left to the reader as an exercise. and Lu = v then L(∂ β u) = ∂ β Lu + [L.3) is completely proved.p. σk (∂ β )] = 0.3.BR ≤ C( ∂β v k. we deduce from the interpolation inequalities that there exists C > 0 such that u Hence u m.3) with k = 0.BR . Using the truncation technique and the interpolation inequalities we deduce the following consequence.p. .BR +ε u m. L k+1. The H¨ older case is entirely similar.α. ε u m+k. r) > 0 such that.p.α.BR in the above inequality.BR k.BR ≤ε u m.d.3.3. ∂ β ]u.R . Consider a multi∞ (E | index |β | = k .BR .p.BR using the interpolation inequalities as before.p.BR p. If we pick η ∈ C0 (B2R ) such that η ≡ 1 on BR .3. ∀u ∈ C ∞ (E ) u k+m. The crucial observation is that [L.R + Cε−(m−1) u p. ≤ C ( Lu p. It is precisely at this point that the interpolation inequalities enter crucially. Indeed.1.BR + u k.BR ) ≤ C( v The term u m+k−1.BR .3.1.3. Using (10. of order ≤ m + k − 1.3.BR m−1. The inequality (10. we deduce ∂β u m. Then. If u ∈ C0 BR ).Br ≤C Lu k.BR ).4. N. and fix 0 < r < R.3.p.3). for every k ∈ Z+ . ∞ (E | ∞ View u as a section of C0 B2R ). Corollary 10. 3.e. elliptic) combination of mixed partial derivatives can be defined weakly. with smooth coefficients. v ∈ Cloc (RN ).6) holds everywhere. Theorem 10. Let p ∈ (1. v ∈ Lp loc (R ) and Aα (x)∂ α u (x) = v (x) (10. The essential ingredient in the proof is the technique of mollification. φ dx.p loc (E ).α m. v : RN → Cr be measurable functions.3.6) u. Then any Lp -weak solution u of Lu = v ∈ Lp loc (E ) is an Lp strong solution .7.α if there exists α ∈ (0.3.3. Loosely speaking the above theorem says that if a “clever” (i. For each δ > 0 set uδ = ρδ ∗ u ∈ C ∞ (E ) vδ = ρδ ∗ v ∈ C ∞ (E ).6) if u ∈ Lm. 465 In this subsection we continue to use the notations of Definition 10.6. and (10.e. N (b) The function u is said to be an Lp -strong solution of (10. ∀φ ∈ C0 (E ). u ∈ Lm. 1) such that. φ dx = 0.3. Remark 10. L∗ φ dx = RN RN ∞ v.3.) N (c) The function u is said to be an Lp -weak solution if u. when L is elliptic. u ∈ Cloc (RN ). .3..e. (The partial derivatives of u should be understood in generalized sense..3.Analytic aspects §10.6) hold almost everywhere.5. (a) The function u is a classical solution of the partial differential equation Lu = |β |≤m 0. v ∈ p N Lloc (R ). then any mixed partial derivative (up to a certain order) can be weakly defined as well.3.i.p loc (R ). The principal result of this subsection will show that. ∞ (E ) Lemma 10. Note the following obvious inclusion {Lp weak solutions} ⊃ {Lp strong solutions}. then the above inclusion is an equality.8. and (10. Then for every φ ∈ C0 δ →0 RN lim wδ . The decisive result in establishing the regularity of u is the following. ∞) and suppose that L : C ∞ (E ) → C ∞ (E ) is an elliptic operator of order m. Let u.2 Elliptic regularity the previous subsection. wδ converges weakly to 0 in Lp loc . i. Let wδ = Luδ − vδ ∈ C ∞ (E ). we deduce that a subsequence of (uδ ) converges weakly to some u ∈ Lm. Since uδ → u. Hence u m. because Lm. we deduce that. . L∗ x φ (x) dy dx β ρδ (x − y )u(y ).p. In particular.3. On the other hand. vδ p. we deduce that it must be bounded in this norm.p (E |B2r ) is reflexive.3. Bβ (x)∂x φ (x) dy dx = β B RN β u(y ).p loc (E ).6 is proved.p u ∈ Lloc .3. we deduce uδ p.3.Br ≤ C.B3r + vδ p.B3r ≤ C. ρδ (x − y )Bβ (x)∂x φ (x) dy dx.p. Proof of Lemma 10.p (E | ).3.B3r + wδ p. Using the elliptic estimates of Corollary 10. L∗ x φ (x) dx = B RN ρδ (x − y )u(y ). Assume supp φ ⊂ B = B . and vδ → v in Lp (E |B3r ). we deduce uδ m. since wδ is weakly convergent in Lp (E |B3r ). uδ → u strongly in Lp (E |Br ). We analyze each of the above terms separately.8 to prove u ∈ Lm. Lemma 10. φ dx = 0. Theorem 10. Assume the formal adjoint of L has the form β L∗ = Bβ (x)∂x . we deduce u |Br = u ∈ L m. ρδ ∗] → 0 as δ → 0. φ dx = = β B RN B B uδ (x).3. using the Rellich-Kondrachov compactness theorem.8 show δ →0 ∞ (E ).466 Elliptic equations on manifolds Remark 10. m. Note that uδ is a classical solution of Luδ = vδ + wδ . |β |≤m We have Luδ . This shows Since uδ → u in Lp Br loc (as mollifiers).8 says that [L. because r is arbitrary. We first show how one can use Lemma 10. Moreover.3.B2r ≤ C ( uδ p.p (E |B2r ). Fix 0 < r < R. We have to Pick φ ∈ C0 R lim B Luδ .9. on a subsequence.B3r ). Roughly speaking. φ dx − B vδ .B3r . Bβ (y )ρδ (x − y )∂x φ(x) dydx. Bβ (x)∂x φ(x) dx ∗ β ρδ (x − y )Bβ (y )u(y )dy . We will examine separately each term in the above sum. (Bβ (x) − Bβ (y ))∂x φ(x) dydx β ρδ (x − y )u(y )dy . φ dx − B vδ . φ(x) dx = = B RN B RN v (y ). Bβ (y )ρδ (x − y )∂x φ(x) dxdy (switch back the order of integration) = β B RN β u(y ). Hence B Luδ . ρδ (x − y )φ(x) dydx u(y ). Bβ (y )(∂x ρδ (x − y ))φ(x) dxdy (integrate by parts in the interior integral) = β RN B β u(y ). L∗ y ρδ (x − y )φ(x) dydx = β B RN β u(y ). ρδ (x − y ) (Bβ (x) − Bβ (y )) ∂x φ(x) dydx B RN = B RN β ρδ (x − y ) u(y ). ρδ (x − y ) (Bβ (x) − Bβ (y )) ∂x φ(x) dydx. ∂x φ(x) dx = B RN − B RN . φ dx = = β B RN β u(y ). Bβ (y )∂y (ρδ (x − y )φ(x)) dydx (switch the order of integration) = β RN B β u(y ). β u(y ).Analytic aspects Similarly B 467 vδ (x). Bβ (y )∂y (ρδ (x − y )φ(x)) dxdy (∂x ρδ (x − y ) = −∂y ρδ (x − y )) = β (−1)β RN B β u(y ). p. ∂x φ(x) dx. p Corollary 10. on manifolds. and for every 0 < r < R we have u where as usual C = C ( L m+k. The results in this.BR + Lun k. loc Then +m.p.3. oriented Riemann manifold and E. If u ∈ Lp loc (E ) is an L -weak solution of Lu = v .BR ). . ∂x φ(x) dx = 0. then u ∈ Lloc (E ).10.3.p Lun → Lu strongly in Lk (E ).p.11 (Weyl Lemma).p v ∈ Lloc (E ).p k+m. B where ∗ u) (Bβ δ denotes the mollification of As δ → 0 ∗ ∗ uδ → u and (Bβ u)δ → Bβ u in Lp loc .3. Bβ (x)∂x φ(x) dx − ∗ u.p ∀k . Bβ Elliptic equations on manifolds ∗ β (Bβ u)δ (x).p ∞ (E ) such that Proof. then u must be smooth. They take a particularly nice form for operators on compact manifolds. loc and un m+k. Hence u ∈ Lloc m.p. R.p. Lemma 10. Using loc ∀k ∈ Z+ .d. and so extend to the more general case of p.8 is proved. Hence δ →0 B lim RN β u(y ) . F ) an elliptic operator of order m.BR + u p.Br ≤ C ( un 0. p Corollary 10. and k. r.p un → u strongly in Lk (E ). and v is smooth. If u ∈ Lp loc (E ) is weak L -solution of Lu = v . The desired estimate is obtained by letting n → ∞ in the above inequality.α Morrey embedding theorem we deduce that u ∈ Cloc ∀m ≥ 0. Since v is smooth we deduce v ∈ Lk.p Proof.BR ). ρδ (x − y ) (Bβ (x) − Bβ (y )) ∂x φ(x) dydx = B β u(x). F → M be two metric vector bundles with compatible connections. Denote by L ∈ P DO m (E. Pick a sequence un ∈ C0 loc +m. g ) be a compact.Br ≤ C( v k. k+m. Bβ (x)∂x φ(x) dx − B ∗ β Bβ (x)u(x). and the previous subsection are local.R . +m. k+m+1. . We already know that u ∈ Lk (E ).o. 1 < p < ∞.468 = B β uδ (x). . . Let (M.). (Kato’s inequalities). Remark 10. where C = C (L.E ≤ C ( u 0.F + u p. L∗ φ E dVg ∞ ∀φ ∈ C0 (F ). u E − 2|∇u|2 .13. φ M F dvg 469 = M u. and any p ∈ (1. and ∇i . the function x → |u(x)| is in L1. Suppose (M. Prove that for any compact subset K ⊂ M . for all u ∈ L2. we have ∆M (|u|2 ) = 2 ∆E u.p (E ). Exercise 10. then u ∈ C m+k. i = 0. k. for almost all x ∈ M . (a) Show that for every u ∈ L1. and boundary value problems. Let (M. and moreover |d |u(x)| | ≤ |∇u(x)|. and u k+m. Conclude that ∀φ ∈ C ∞ (M ) such that φ ≥ 0. (a) Let u ∈ Lp (E ) and v ∈ Lk.3.p. For generalized Laplacians these topics are discussed in great detail in [36] and [79]. we have (d|u|.α.α. E0 . there exists a positive constat C .α (E ). any relatively compact open neighborhood U of K . Suppose that L : C ∞ (E0 ) → C ∞ (E1 ) is an elliptic operator of order ν with smooth coefficients. and u k+mp. p). M M i.. ∈ Z≥0 .2 (M ).3. E1 → M are two smooth hermitian vector bundles on M . where C = C (L. The regularity results and the a priori estimates we have established so far represent only the minimal information one needs to become an user of the elliptic theory.F ). g ) denote a compact oriented Riemann manifold without boundary. such that v. u(x) E weakly.15. 1 are Hermitian connection on Ei . 1 < p < ∞. k.14.E + v k. g ) is a connected. We refer to [16] for a very nice presentation of this subject.e.α. α). Exercise 10. u E φ dVg .3.12.α (E ) and v ∈ C k. ∞).2 (E ). oriented. The boundary value problems for first order elliptic operators require a more delicate treatment. (b) If u ∈ C m. we have mentioned nothing about two important topics: equations with non-smooth coefficients. |u(x)|∆M (|u(x)|) ≤ ∆E u(x). Regrettably. Then u ∈ Lk+m. (b) Set ∆E = ∇∗ ∇.p (F ). d(φ|u|) )g dVg ≤ ∆E u.α (F ) (0 < α < 1) are such that Lu = v .2 (E ). Riemann manifolds of dimension m.Analytic aspects Theorem 10. and denote by ∆M the scalar Laplacian. not necessarily.E ). (Local estimates). Show that. Consider a metric vector bundle E → M equipped with a compatible connection ∇.E ≤ C( v k.3. (C2 ). g ) be a compact. ∇0 . §10.E1 ) + u Lp (U.16. and the coefficients of L such that. Set F (u) = 0 u (10. We assume for simplicity that volg (M ) = M dVg = 1. (C1 ) The function f is smooth and strictly increasing.p (K. Let (M. and s ∈ C ∞ (M ) satisfy the following conditions. such that f (t) ≥ at + b ∀t ∈ R. connected. where s denotes the average of s. ∇1 . where the above Sobolev norms are defined in terms of the connections ∇i .3. Let f and s(x) satisfy the conditions (C1 ). oriented Riemann manifold of dimension N . Theorem 10. We will consider a slightly more general situation than the one required by the uniformization theorem. s= M s(x)dVg . U. ∀u ∈ C ∞ (E0 ). . . ∆u + f (u) = s(x).3 An application: prescribing the curvature of surfaces In this subsection we will illustrate the power of the results we proved so far by showing how they can be successfully used to prove an important part of the celebrated uniformization theorem. Then there exists a unique u ∈ C ∞ (M ) such that ∆u (x) + f (u(x)) = s(x) ∀x ∈ M. we have u L +ν.470 Elliptic equations on manifolds depending only on K. where f : R → R.E0 ) . We assume (C3 ) lim|t|→∞ (F (t) − st) = ∞. p.E0 ) ≤ C Lu L . C3 ). Denote by ∆ = d∗ d : C ∞ (M ) → C ∞ (M ) the scalar Laplacian.3. We will study the following partial differential equation.3. (C2 ) There exist a > 0 and b ∈ R. so that the average of any integrable function ϕ is defined by ϕ= M ϕ(x)dVg (x).p (U.7) f (t)dt. In the process we will have the occasion to introduce the reader to some tricks frequently used in the study of nonlinear elliptic equations. e. dφ) + f (u)φ}dVg = s(x)φ(x)dVg (x).2 (M ). and {(du.7). Let h : R → R be a continuous. v ∈ L1. Consider the energy functional I : L1. 2 This functional is not quite well defined. Proof of Step 1. dφ) + h(u)φ}dVg ≥ {dv.. 0) = {(u − v ) − |u − v |}. We will circumvent the trouble with the possible non-integrability of F (u) by using a famous trick in elliptic partial differential equations called the maximum principle.2 (M ) → R.Analytic aspects Proof. dφ) + f (u)φ − s(x)φ(x)}dVg = 0.3. so that hφ (0) = 0 ∀φ. Hence t = 0 is a minimum of hφ (t) = I (u+tφ). ∀v ∈ L1.2 (M ) such that (i) h(u). I (u) ≤ I (v ).2 (M ).17. Regularity. Leaving this issue aside for a moment.e.7) is a function u ∈ L1. We will use the direct method of the calculus of variations outlined at the beginning of the current section. strictly increasing function and u. M M Step 2. Anyway. Lemma 10. ∀φ ∈ L1. so that a minimizer of I is a weak solution provided that we deal with the integrability issue raised at the beginning of this discussion. M . Proof. since there is no guarantee that F (u) ∈ L1 (M ) for all u ∈ L1.2 (M ) such that φ ≥ 0 a. We show that a weak solution is in fact a classical solution. The proof of this theorem will be carried out in two steps. on M . A simple computation shows that hφ (0) = M {(du.2 (M ) I (u) ≤ I (u + tφ) ∀t ∈ R.e. I (u) = M 1 |du(x)|2 + F (u(x)) − g (x)u(x) dVg (x). the lesson we learn from this formal computation is that minimizers of I are strong candidates for solutions of (10. A weak solution of (10. h(v ) ∈ L2 (M ). we can perform a formal computation ` a la Riemann. Let 1 (u − v )− = min{u − v.3.8) M M ∀φ ∈ L1. Assume u is a minimizer of I . 471 Step 1.3.3.e.2 (M ). Thus. 2 .2 (M ) such that f (u(x)) ∈ L2 (M ). {(du. Existence of a weak solution. for all φ ∈ L1. (ii) ∆u + h(u) ≥ ∆v + h(v ) weakly i. i. dφ) + h(v )φ}dVg (10. Then u ≥ v a. using φ ≡ 1 in (10.e. To proceed further we need the following a priori estimate.3.8). On the other hand. Clearly d(u − v ). then (u − v ) ≡ (u − v )− ≡ c.3.19 we have (u − v )− ∈ L1. on {u < v } 0 a.7) has at most one weak solution. so that by the maximum principle u ≥ (≤) v .3. we deduce h(u)dVg ≥ h(v )dVg ! M M Hence c cannot be negative. on {u ≥ v } Using φ = −(u − v ) in (10. This shows the equation (10.3. d(u − v )− = |d(u − v )− |2 .8) we deduce − M d(u − v ). this means (u − v )− ≡ c ≤ 0. and d(u − v )− = d(u − v ) a. so that |d(u − v )− | ≡ 0. Since h is strictly increasing we conclude M h(u) dVg < M h(v )dVg . Show the conclusion of Lemma 10.3. . Exercise 10. Since M is connected. If c < 0.17 continues to hold. The maximum principle is proved. so that u = v + c < v .472 Elliptic equations on manifolds According to the Exercise 10.2 (M ).7).2. so that (u − v )− ≡ 0 which is another way of saying u ≥ v .e. then ∆u + f (u) ≥ (≤) ∆v + h(v ) weakly. We now return to the equation (10. Assume h in the above lemma is only non-decreasing. d(u − v )− + h(u) − h(v ) u−v − dVg ≥ 0.3. and since h is nondecreasing. Note first that if u and v are two weak solutions of this equation. Hence.18. but u and v satisfy the supplementary condition u ≥ v. (h(u) − h(v ))(u − v )− ≥ 0. M |d(u − v )− |2 dVg ≤ − M ( h(u) − h(v ) )(u − v )− dVg ≤ 0. 3.9) (10. We will use the direct method of the calculus of variations on a new functional 1 ˜(u) = ˜ (u(x)) − s(x)u(x) dVg (x).21. B > 0 such that ˜(u)| ≤ A|u| + B. .3. on M . the sublevel sets J c = {x . If C is a positive constant such that f (C ) ≥ sup s(x).e.3. i. Then J admits a minimizer.3. f The conditions (C1 ) and (C2 ) imply that there exist A. We deduce as in the proof of Lemma 10. weakly closed and bounded in X . f The advantage we gain by using this new functional is clear.10). M then u(x) ≤ C a.9) shows ˜ has at most quadratic growth. weakly lower semi-continuous.3..7).Analytic aspects 473 Lemma 10. we should try to find a weak solution of (10. Let u be a weak solution of (10. so that F ˜ (u) ∈ L1 (M ).20. Fix C0 > 0 such that f (C0 ) ≥ sup s(x). there exists x0 ∈ X such that J (x0 ) ≤ J (x) ∀x ∈ X.19 that u ≤ C0 .e.3.3. J (x) ≤ c} are respectively convex.7).3.3.10) ˜(t)dt.3.e. Proof. range where f coincides with f The above lemma shows that instead of looking for a weak solution of (10.. Consider a strictly increasing C 2 -function ˜ : R → R such that f ˜(u) = f (u) for u ≤ C0 .19. i. Proposition 10.7). I |du(x)|2 + F 2 M where ˜ (u) := F 0 u (10. |f Lemma 10. The existence F of a minimizer is a consequence of the following fundamental principle of the calculus of variations. for all u ∈ L2 (M ). The inequality (10. The equality f (C ) ≥ sup s(x) implies ∆C + f (C ) ≥ s(x) = ∆u + f (u) weakly so the conclusion follows from the maximum principle. If u is a weak solution of ˜(u) = s(x) ∆u + f then u is also a weak solution of (10. Proof.3. f ˜(u) is linear for u ≥ C0 + 1. which is precisely the ˜. coercive functional. Let X be a reflexive Banach space and J : X → R a convex. weakly lower semi-continuous and coercive (with respect to Lemma 10. We need to show the functional u→ M ˜ (u)dVg F is also weakly lower semi-continuous. F If un → u strongly in L1. The convexity is clear since F The functionals u→ 1 2 |du|2 dVg and u → − s(x)u(x)dVg (x) M M are clearly weakly lower semi-continuous. ˜ is convex. ˜ (un )dVg . we can find α > 0. ˜ is strictly increasing. β ∈ R such that Since F ˜ (u) − αu − β ≥ 0. we deduce that J c is weakly compact. The next result will conclude the proof of Step 1. n Since X is reflexive. F which shows that M ˜ (u)dVg ≤ lim inf F n This means that the functional L1.22. and is convex. I 1 . lim n M αun + βdVg = M αu + βdVg .2 (M ) we deduce using the Fatou lemma that ˜ (u) − αu − β }dVg ≤ lim inf {F n→∞ M M ˜ (un ) − αun − β }dVg . {F On the other hand. The proposition is proved. ν Jc Hence x0 is a minimizer of J . Hence a (generalized) subsequence (xν ) of xn converges weakly to some x0 ∈ J c .2 (M ) u→ M ˜ (u)dVg F . ˜ is convex. Using the lower semi-continuity of J we deduce J (x0 ) ≤ lim inf J (xν ) = inf J. ˜ is convex on account that f Proof. 2 the L -norm). weakly closed and bounded in the norm of X .474 Proof. Note that inf J (x) = inf J (x) c X J Elliptic equations on manifolds Consider xn ∈ J c such that lim J (xn ) = inf J.3. 3. (Rellich-Kondrachov) so that. There exists C > 0 such that |du|2 dVg ≥ C |u⊥ |2 dVg . the sublevel sets u. M M Proof. The inclusion L1.2 (M ) v. weakly in L1. ε | dVg = 1. we deduce that.2 (M ). . Since L1. We have thus established that ˜ is convex and weakly lower semi-continuous. ∀u ∈ L1. The coercivity will require a little more work. Assume that for any ε > 0 there exists uε ∈ L1. and the convexity conditions are very closely related. on a subsequence u⊥ ε (u⊥ ε) is bounded in L1.2 (M ) → L2 (M ) is compact. This choice of notation is motivated by the fact that u⊥ is perpendicular (with respect to the L2 (M )-inner product) to the kernel of the Laplacian ∆ which is the 1-dimensional space spanned by the constant functions.24 (Poincar´ e inequality). We refer to [24] for a presentation of the direct method of the calculus of variations were the lower semicontinuity issue is studied in great detail. We first need to introduce some more terminology. The key ingredient will be a Poincar´ e inequality. Note the average of u⊥ is 0. strongly in L . This implies v = 0. M 475 ˜ (u)dVg ≤ c . since |v |2 dVg = lim ε 2 |u⊥ ε | dVg = 1. and M M The above two conditions imply that the family is reflexive. We see that the lower semi-continuity.23. Lemma 10. In some sense they are almost equivalent. on a subsequence 2 u⊥ ε → v strongly in L (M ). 2 duε = du⊥ ε → 0.2 (M ).2 (M ). The Hahn-Banach separation principle can now be invoked to conclude these level sets are also weakly closed. We argue by contradiction.Analytic aspects is strongly lower semi-continuous. F are both convex and strongly closed.3. M M On the other hand. Now set u⊥ (x) = u(x) − u.2 (M ). Thus. I Remark 10. For any u ∈ L2 (M ) we denoted by u its average. such that 2 |u⊥ |duε |2 dVg ≤ ε. the boundedness of |du⊥ |2 dVg M implies the boundedness of u⊥ 2. so that we should concentrate only on u and du⊥ 2 .476 We conclude M Elliptic equations on manifolds (dv. 2 M The Poincar´ e and Cauchy inequalities imply C u⊥ 2 2 − s⊥ 2 · u⊥ 2 + M ˜ (u)dVg − s · u ≤ κ. |du(x)|2 + F 2 |du⊥ |2 dVg and |u|2 = |u|2 + M M (10.3.11) can be rewritten as 1 ⊥2 ˜ |du | + F (u) − s⊥ u⊥ dVg − s · u ≤ c.M . M |u⊥ |2 dVg . Moreover.11) M Since M |du|2 dVg = M |u⊥ |2 dVg . In view of the Poincar´ e inequality.3. The inequality (10. it suffices to show that the quantities u. Let κ > 0. F ˜ is convex. F so that C u⊥ 2 2 − s⊥ 2 · u⊥ 2 ˜ (u) − s · u ≤ κ.3.12) . dφ) dVg = 0 ∀φ ∈ L1. ˜(u). and F ˜ F M udVg ≤ M ˜ (u)dVg .2 (M ) such that We can now establish the coercivity of I 1 ˜ (u(x)) − s(x)u(x) dVg (x) ≤ κ. we have a Jensen-type inequality Since volg (M ) = 1. In particular. so that dv ≡ 0. dv ) dVg = 0.2 (M ). dφ) dVg = lim ε M (duε . Since M is connected. c= M v (x)dVg (s) = lim ε M u⊥ ε dVg = 0. M |du⊥ |dVg are bounded. M (dv. we deduce v ≡ c = const. and u ∈ L1. This contradicts the fact that v = 0. The Poincar´ e inequality is proved. +F (10. ˜(u(x)) − s(x). (a) s = M s(x)dVg > 0 (b) For every λ > 0 there exists a unique u = uλ ∈ C ∞ (M ) such that ∆u + λeu = s(x). We will use a technique called bootstrapping. From the theorem we have just proved we deduce immediately the following consequence. Using Sobolev f (or Morrey) embedding theorem we can considerably improve the integrability of u. Let (M. (b) ⇒ (a) follows by multiplying (10.13) Proof.12).3. (10. Using again elliptic regularity we deduce u ∈ L2. M .25. to gradually improve the regularity of the weak solution.10). This forces u⊥ 2 to be ˜ is coercive. and then integrating by parts so that s ¯= λ eu(x) dVg (x) > 0. Then u(x) is a weak L2 solution of ∆u = h(x) on M. Feed this point it is convenient to work with f . This implies h(x) ∈ L2.for all q > 1.q (M ) .3.Analytic aspects 477 Set P (t) := Ct2 − s⊥ 2 t. and Lemma 10. From the inequality (10.q (M ). The regularity of the minimizer.16 is complete. oriented Riemann manifold and s(x) ∈ C ∞ (M ).3. connected. Feed this information back in (10.3.2 (M ). The proof of Theorem 10. (At this ˜ which was only C 2 ). this shows u(x) ∈ Lq1 (M ).22 is proved. Thus I Step 2. Corollary 10. g ) be a compact. The elliptic regularity theory implies that u ∈ L2. for all q > 1. We deduce that (i) either u ∈ Lq (M ) if −N/q ≤ 2 − N/2 ≤ 0 (dim M = N ).16.3. which blends the elliptic regularity theory. Elliptic regularity implies u ∈ L2. (ii) or u is H¨ older continuous if 2 − N/2 > 0.12) we deduce ˜ (u) − s · u ≤ κ − m.q (M ). and the Sobolev embedding theorems.q1 and Sobolev inequality implies that u ∈ Lq2 (M ) for some q2 > q1 . which implies h(x) ∈ Lq1 (M ). (a) ⇒ (b) follows from Theorem 10. Let u be the weak solution of (10.13) with v (x) ≡ 1. and regularity theory improves the regularity of u two orders at a time.3. Assume volg (M ) = 1. In view of Morrey embedding theorem the conclusion is clear: u ∈ C ∞ (M ). for any q > 1.3. Then the following two conditions are equivalent. bounded. and let m = inf P (t).3. Note that since the growth of f ˜ is at most linear. In any case.3. F Using condition (C3 ) we deduce that |u| must be bounded. for all q > 1. for some q1 > 2. we have where h(x) = −f 2 ˜(u(x)) ∈ L (M ). We conclude that P ( u⊥ 2 ) must be bounded. rather than with f back in h(x). Invoking elliptic regularity again we deduce that u ∈ L4. After a finite number of steps we conclude that h(x) ∈ Lq (M ). One can formulate a more general question than the Yamabe problem.Parker. g ) is a compact oriented Riemann manifold.27. g ) be an oriented Riemann manifold of dimension N and f ∈ C ∞ (M ). does there exist a metric conformal to g whose scalar curvature is constant? ” In dimension 2 this problem is related to the uniformization problem of complex analysis. which was only relatively recently solved. The higher dimensional situation dim M > 2 is far more complicated. Let (M. where s(x) is the scalar curvature of the metric g . ∆u + eu = −s(x). The case dim M = 2 is completely solved in [56].28. Exercise 10. Theorem 10. Assume volg (Σ) = 1. We look for g ˜ of the form g ˜ = eu g . The solution of the Yamabe problem in its complete generality is due to the combined efforts of T. Aubin.3. Denote by g ˜ the conformal metric g ˜ = ef g .26. If s(x) is the scalar curvature of g and s ˜ is the scalar curvature of g ˜ show that s ˜(x) = e−f {s(x) + (N − 1)∆g f − (N − 1)(N − 2) |df (x)|2 g} 4 where ∆g denotes the scalar Laplacian of the metric g while | · |g denotes the length measured in the metric g . This problem is known as the Kazdan-Warner problem.e.3. Two Riemann metrics g1 and g2 are said to be conformal if there exists f ∈ C ∞ (M ). Given a compact oriented Riemann manifold (M. For a very beautiful account of its proof we recommend the excellent survey of J. Let (Σ.3. Let M be a smooth manifold. Remark 10. i. oriented Riemann manifold of dimension 2. Lee and T.478 Elliptic equations on manifolds Although the above corollary may look like a purely academic result. such that g2 = ef g1 .29 (Uniformization Theorem). A long standing problem in differential geometry. We will use it to prove a special case of the celebrated uniformization theorem. is the Yamabe problem : “If (M. Schoen [86]. g ) decide whether a smooth function s(x) on M is the scalar curvature of some metric on M conformal to g . Definition 10.3. it has a very nice geometrical application. [8. g ) be a compact. If χ(Σ) < 0 (or equivalently if its genus is ≥ 2) then there exists a unique metric g ˜ conformal to g such that s(˜ g ) ≡ −1. [63].27 we deduce that u should satisfy −1 = e−u {s(x) + ∆u}. The Gauss-Bonnet theorem implies that s = 4πχ(Σ) < 0 . Using Exercise 10. 9] and R. both topologically and analytically.3. Proof. 3.3. which is an “inverse” of the universal covering map R θ → eiθ ∈ S 1 . Fix c > 0.32.3. Exercise 10. The uniformization theorem implies that the compact oriented surfaces of negative Euler characteristic admit metrics of constant negative sectional curvature. U0 (x)) ≥ f (x. weakly in L1.2 (M )∩ L∞ (M ) such that U0 (x) ≥ u0 (x). Assume that the equation ∆g u = f (x. .2 (M ). On a manifold of dimension 2 the scalar curvature coincides up to a positive factor with the sectional curvature. U0 ∈ L1. In the following exercises (M. Consider a smooth function f : M × R → R such that for every x ∈ M the function u → f (x. g ) denotes a compact. i. oriented Riemann manifold without boundary. u0 (x)) ≥ ∆g u0 . Exercise 10. and define (un )n≥1 ⊂ L2.e. Remark 10.14) admits a pair of comparable sub/super-solutions.e.14) satisfying u0 ≤ u ≤ U0 . u) is increasing.3. Show that for every f ∈ L2 (M ) the equation ∆u + cu = f has an unique solution u ∈ L2. The universal cover of a compact.30.3. The “inverse” of the covering projection π : C → Σ is classically called a uniformizing parameter.31. (b) Show that un converges uniformly on M to a solution u ∈ C ∞ (M ) of (10. using the Cartan-Hadamard theorem we deduce the following topological consequence. and ∆g U0 ≥ f (x. (a) Show that u0 (x) ≤ u1 (x) ≤ u2 (x) ≤ · · · ≤ un (x) ≤ · · · ≤ U0 (x) ∀x ∈ M. on M.2 (M ).Analytic aspects 479 so that the existence of u is guaranteed by Corollary 10. The uniformization theorem is proved.2 (M ) inductively as the unique solution of the equation ∆un (x) + cun (x) = cun−1 (x) + f (x. a. Now. Fix c > 0. un−1 (x)). there exist functions u0 .3..33. even if the monotonicity assumption on f is dropped. (c) Prove that the above conclusions continue to hold. oriented surface Σ of negative Euler characteristic is diffeomorphic to C. u) (10. Corollary 10. This is a uniformizing parameter on the circle. It is very similar to the angular coordinate θ on the circle S 1 .3.25. into a less painful task. z1 . C. (a) In more concrete terms. y ). · : Z × Z → K denotes the inner product2 in a Hilbert space Z . z2 .4. not necessarily continuous. 2 When K = C. where ·. is a closed subspace in X × Y . the operator T : D(T ) ⊂ X → Y is closed if. densely defined operator T : D(T ) ⊂ X → X is said to be selfadjoint if T = T ∗. The operator T is said to be closed if its graph. The operator T is said to be densely defined if its domain D(T ) is dense in X . The main reason why these operators are so “friendly” is the existence of a priori estimates. Also note that if T : D(T ) ⊂ X → Y is a closed. z1 .4. · is conjugate linear in the second ¯ z1 . densely defined operator T : D(T ) ⊂ X → X is selfadjoint if the following two conditions hold. §10. coupled with the Rellich-Kondratchov compactness theorem. is a closed. x∗ ) ∈ Y × X .1 The Fredholm theory Throughout this section we assume that the reader is familiar with some fundamental facts about unbounded linear operators.4 Elliptic equations on manifolds Elliptic operators on compact manifolds The elliptic operators on compact manifolds behave in many respects as finite dimensional operators.2. (d) The closed.e. λz2 = λ . Y be two Hilbert spaces over K = R. ∀λ ∈ C.1. It is the goal of this last section to present the reader some fundamental analytic facts which will transform the manipulation with such p. and (ii) y = T x.d. are the keys which will open many doors. We refer to [18] Ch. (b) Let T : D(T ) ⊂ X → Y be a closed. Remark 10. An exhaustive presentation of this subject can be found in [55]. (b) If T : X → Y is a closed operator.II for a very concise presentation of these notions. it follows that (i) x ∈ D(T ).4. x = y ∗ . ΓT := (x. i. then T is bounded (closed graph theorem). then ker T is a closed subspace of X . (c) One can show that the adjoint of any closed. Definition 10. defined on the linear subspace D(T ) ⊂ X . densely defined operator. (a) Let X . T x) ∈ D(T ) × Y ⊂ X × Y . and T : D(T ) ⊂ X → Y be a linear operator. These estimates. densely defined operator. for any sequence (xn ) ⊂ D(T ). we use the convention that the inner product ·. The djoint of T is the operator T ∗ : D(T ∗ ) ⊂ Y → X defined by its graph ΓT ∗ = {(y ∗ . (c) A closed. densely defined operator. T xn ) → (x..480 10. variable. densely defined linear operator. T x ∀x ∈ D(T ). such that (xn . x∗ .o. z2 ∈ Z . 2 (E ) such that un → u strongly in L2 (E ). Definition 10. v dVg ≤ C u . ∀u ∈ Lk. 481 If only the first condition is satisfied the operator T is called symmetric. y ∈ D(T ) and • D(T ) = y ∈ X . and Lun → v strongly in L2 (F ). F ).d. Let (M.o. ∃C > 0. | T x. Hence (un ) is a Cauchy sequence in Lk. D(La ) = Lk.2 (F ) we need to show that if v ∈ L2 (F ) is such that ∃C > 0 with the property that Lu. (b) If L∗ : C ∞ (F ) → C ∞ (E ) is the formal adjoint of L then (L∗ )a = (La )∗ . y = x. so that un → u in Lk.2 (E ). v ∈ Lk. F two metric vector bundles with compatible connections.2 (E ). (a) The analytical realization La of L is a closed.2 (E ).2 (E ). The analytical realization of L is the linear operator La : D(La ) ⊂ L2 (E ) → L2 (E ).3. ∀x ∈ D(T ) . n → ∞.2 (E ) is dense in L2 (E ) . ∀u ∈ Lk. E.2 -norms by · k . oriented Riemann manifold. g ) be a compact. It is now clear that v = Lu. and (La )∗ = (L∗ )a on Lk. (a) Since C ∞ (E ) ⊂ D(La ) = Lk. y | ≤ C |x|. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS • T x.2 (F ).10.2 (E ). From the elliptic estimates we deduce un − um k ≤C Lun − Lum + un − um → 0 as m. T y for all x. M . a k -th order elliptic p. given by u → Lu for all u ∈ Lk.4. consider a sequence (un ) ⊂ Lk.2 (E ). Proof. and the Lk.4. and L ∈ P DO k = P DO k (E.4.4. Proposition 10. To prove that La is also closed. densely defined linear operator. we deduce that La is densely defined. v dVg = u. To prove that D (La )∗ ⊂ D (L∗ )a = Lk. M M we deduce D (La )∗ ⊃ D (L∗ )a = Lk. (b) From the equality Lu.2 (F ). L∗ v dVg .2 (F ). We will denote the various L2 norms by · . 2 (F ). v is a L2 -weak solution of the elliptic equation L∗ v = φ. The above terminology has its origin in the work of Ivar Fredholm at the twentieth century. Remark 10. v dVg extends to a continuous linear functional on L2 (E ). Consider again the elliptic operator L of Proposition 10.5.8. (i) dim ker T < ∞ and (ii) The range R (T ) of T is closed.4. Hence. Proof.4. In this case. Using elliptic regularity theory we deduce v ∈ Lk. The operator La : D(La ) ⊂ L2 (E ) → L2 (F ) is Fredholm. Definition 10. states that if K : H → H is a compact operator from a Hilbert space to itself. The proposition is proved. . the above inequality shows that the functional u→ M Lu. Indeed. The operator T is said to be semi-Fredholm if the following hold.2 (E ) such that bounded contains a subsequence strongly convergent in L2 (E ).6.4.2 (E ) embeds compactly in L2 (E ). φ dVg ∀u ∈ Lk. On the other hand. Lemma 10. Proof.4. (b) The operator T is called Fredholm if both T and T ∗ are semi-Fredholm. densely defined linear operator.7.4. Using elliptic estimates we deduce that un k un + Lun n≥ 0 is ≤ C ( Lun + un ) ≤ const.2 (M ).4.482 Elliptic equations on manifolds then v ∈ Lk.2 (E ). since M is compact. His result. (L∗ )∗ u. Following the above result we will not make any notational distinction between an elliptic operator (on a compact manifold) and its analytical realization. the integer ind T := dimK ker T − dimK ker T ∗ is called the Fredholm index of T . then H + K is a Fredholm operator of index 0. The lemma is proved. (a) Let X and Y be two Hilbert spaces over K = R. later considerably generalized by F. C and suppose that T : D(T ) ⊂ X → Y is a closed. there exists φ ∈ L2 (E ) such that u.2 (E ). Any sequence (un )n≥0 ⊂ Lk. Theorem 10. Riesz. v dVg = M M In other words. The Fredholm property is a consequence of the following compactness result. the space Lk. Hence (un ) is also bounded in Lk. In the proof we will rely on the classical result of F. we deduce u ∈ D(L) and Lu = 0.4. For each vn we can find a unique un ∈ X = (ker L)⊥ such that Lun = vn . Since L is closed.8 since un + Lun = un is bounded. Lemma 10.e.2 (a) ker L is closed in L2 (E ). Riesz which states that a Banach space is finite dimensional if and only if its bounded subsets are precompact (see [18].2 (E ) so that un → u in Lk.. To prove that the range R (T ) is closed. Lun → 0.9 (Poincar´ e inequality). notice that ker L is a Banach space with respect to the L2 -norm since according to Remark 10. This contradicts the condition u = 1. Denote by X ⊂ Lk. When we couple this inequality with the elliptic estimates we get un − um k ≤ C ( Lun − Lum + un − um ) ≤ C vn − vm → 0 as m. Clearly Lu = v .4.4. a special case of which we have seen at work in Subsection 9. for all u ∈ Lk. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS 483 We will first show that dim ker L < ∞. u.4.2 (E ) the subspace consisting of sections L2 -orthogonal to ker L. We can now conclude the proof of Theorem 10.2 (E ). Consider a sequence (vn ) ⊂ R (L) such that vn → v in L2 (F ). i. Assume that for any n > 0 there exists un ∈ X such that un = 1 and Lun ≤ 1/n.4.3.10. Note that u = 1. We get a sequence (un .2 (E ) which are L2 -orthogonal to ker L. Note first that. 0). Next.7. n → ∞. Chap. we will rely on the following very useful inequality.8 we deduce that a subsequence of (un ) is convergent in L2 (E ) to some u. Using Lemma 10. φ dVg = 0 ∀φ ∈ ker L. Thus. There exists C > 0 such that u ≤ C Lu . Hence (un ) is a Cauchy sequence in Lk.3. This follows immediately from Lemma 10. Using the Poincar´ e inequality we deduce u n − u m ≤ C vn − vm . VI). We will show that any sequence (un ) ⊂ ker L which is also bounded in the L2 -norm contains a subsequence convergent in L2 . Lun ) ⊂ ΓL = the graph of L. Hence u ∈ ker L ∩ X = {0}. and in particular. Lun ) → (u. We want to show v ∈ R (L).4. . M Proof. according to Weyl’s lemma ker L ⊂ C ∞ (E ). such that (un . It is not difficult to see that in fact u ∈ X . so that v ∈ R (L). un + Lun is bounded. We will argue by contradiction. 484 Elliptic equations on manifolds We have so far proved that ker L is finite dimensional. Corollary 10. It is only a cone in the linear space P DO (m) . We stick to the notations used so far. Unfortunately. Corollary 10. but it carries a natural topology which we now proceed to describe. L2 ∈ Ellk (E. We set δ (L1 . it is not an affine space. A nonexistence hypothesis implies an existence result! This partially explains the importance of the vanishing results in geometry. Using the closed range theorem of functional analysis we deduce the following important consequence. L2 ) = sup Define ∗ d(L1 . The last corollary is really unusual.4. F ) the space of elliptic operators C ∞ (E ) → C ∞ (F ) of order k .e. Then Lψ 2 = M LψLψ dVg = M L∗ Lψ.7. Over a compact manifold ker L = ker L∗ L and ker L∗ = ker LL∗ . in the remaining part of this subsection we will try to unveil some of the natural beauty of elliptic operators. Conversely. L2 ) = max δ (L1 . We will show that the index of an elliptic operator has many of the attributes of a topological invariant. let ψ ∈ C ∞ (E ) such that L∗ Lψ = 0. ψ dVg = 0. δ (L∗ 1 . . It states the equation Lu = v has a solution provided the dual equation L∗ v = 0 has no nontrivial solution. Let L1 .4. L2 ). Denote by Ellk (E. Clearly ker L ⊂ ker L∗ L. With an existence result in our hands presumably we are more capable of producing geometric objects. The Fredholm property of an elliptic operator has very deep topological ramifications culminating with one of the most beautiful results in mathematics: the Atiyah-Singer index theorems. If ker L∗ = 0 then for every v ∈ L2 (F ) the partial differential equation Lu = v admits at least one weak L2 -solution u ∈ L2 (E ). .e. this would require a lot more extra work to include it here. By using the attribute space when referring to Ellk we implicitly suggested that it carries some structure. La is semi-Fredholm. This completes the proof of Theorem 10. It is not a vector space. Since (La )∗ = (L∗ )a . In the next chapter we will describe one powerful technique of producing vanishing theorems based on the so called Weitzenb¨ ock identities. u k = 1.11. F ). i. However.. L2 (E ) = ker L ⊕ R (L∗ ) and L2 (F ) = ker L∗ ⊕ R (L).10 (Abstract Hodge decomposition).4. it is not even a convex set. Corollary 10. i. we deduce (La )∗ is also semi-Fredholm.4. L1 u − L2 u . and R (L) is closed. .12. Proof. and L∗ is also an elliptic operator. L2 ). the results to the effect that ker L∗ = 0. Any k -th order elliptic operator L : C ∞ (E ) → C ∞ (F ) over the compact manifold M defines natural orthogonal decompositions of L2 (E ) and L2 (F ). Let X . we write RL = RL (L). The index map ind : Ellk (E. We will call RL0 (L1 ) is the regularization of L1 at L0 .15. and their derivatives up to order k depend continuously upon λ. Lemma 10.4. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS 485 We let the reader check that (Ellk .10. F ) satisfying d(L0 . .13. L) ≤ r. d) is indeed a metric space. Exercise 10. If Li : D(Li ) ⊂ X → Y (i = 0. 1) are two Fredholm operators define ∗ RL0 (L1 ) : D(L1 ) ⊕ ker L∗ 0 ⊂ X ⊕ ker L0 → Y ⊕ ker L0 L → ind (L) by RL0 (L1 )(u.13 Let L0 ∈ Ellk (E. The result below lists the main properties of the regularization. the operator RL0 (L1 ) is given by the block decomposition RL0 (L1 ) = L1 ıL∗ 0 PL0 0 . where Λ is a topological space. Y be two Hilbert spaces. Theorem 10. For simplicity. Prove the above lemma. For any Fredholm operator L : D(L) ⊂ X → Y denote by ıL : ker L → X (respectively by PL : X → ker L) the natural inclusion ker L → X (respectively the orthogonal projection X → ker L). ∗ ∗ (b) R∗ L0 (L1 ) = RL0 (L1 ). is continuous.4. is then a continuous map Λ λ → Lλ ∈ Ellk . (c) RL0 is invertible (with bounded inverse). It is a very good “routine booster”. Roughly speaking.4. F ) → Z. A continuous family of elliptic operators (Lλ )λ∈Λ . (a) RL0 (L1 ) is a Fredholm operator. 0 In other words. Proof of Theorem 10.14. u ∈ D(L1 ). We have to find r > 0 such that. PL0 u). when L0 = L1 = L. ∀L ∈ Ellk (E. We strongly recommend the reader who feels less comfortable with basic arguments of functional analysis to try to provide the no-surprise proof of the above result. φ ∈ ker L∗ 0. F ). we have ind (L) = ind (L0 ). this means that the coefficients of Lλ .4. φ) = (L1 u + ıL∗ φ. The proof relies on a very simple algebraic trick which requires some analytical foundation. The operator L0 is called the center of the regularization.4. 2) we deduce Ln un = −φn . n → ∞. Since RL0 (L) is Fredholm. and RL∗ (L∗ ) 0 are injective. Using (10. (10..2 to some u.3) now gives L0 un − Ln un ≤ 1/n. Moreover. From (10. the sequence (Ln un ) is strongly convergent to −φ in L2 (F ). then ind (L) = ind (L0 ). if L is sufficiently close to L0 .4. Assume that there exists a sequence (un .4. We will do this only for RL0 (L).2 (E ) × ker L∗ 0 .2 (E )×ker L∗ 0. Set φ := lim φn . and in fact in any Sobolev norm.4. Ln ) ≤ 1/n. L) < r. Step 1. Elliptic equations on manifolds Step 1.e.3) RL0 (Ln )(un .4. We will conclude that if d(L. Hence the sequence un strongly converges in Lk. L0 ) < r. n n Since un ⊥ ker L0 (by (10. lim Ln un = lim L0 un = −φ ∈ L2 (F ). φn ) ⊂ Lk.486 We will achieve this in two steps. L0 u = −φ and u ⊥ ker L. (10. Hence it contains a subsequence strongly convergent in L . such that u k + φ = 1. We will find r > 0 such that.4) implies u = 0 and φ = 0. 0).2)) we deduce from the Poincar´ e inequality combined with the elliptic estimates that un − um k ≤ C L0 un − L0 um → 0 as m. . i.4) This contradicts the abstract Hodge decomposition which coupled with (10. the regularization of L at L0 is invertible. i. φn ) = (0.. for any L satisfying d(L0 . The condition (10.1) (10. lim un n k = u k and u k + φ = 1. and a sequence (Ln ) ⊂ Ellk (E. it suffices to show that both RL0 (L).4. F ) such that un k + φn = 1. where r > 0 is determined at Step 1. with bounded inverse. Note that φ = limn φn .4.4. φ) ∈ Lk. We argue by contradiction.e.2) (10.4. since the remaining case is entirely similar.4.1) we deduce that (φn ) is a bounded sequence in the finite dimensional space 2 ker L∗ 0 . Putting all the above together we conclude that there exists a pair (u. Step 1 is completed. Step 2. and d(L0 . 4.13 is proved. We claim the map v → u = u(v ) is injective. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS Step 2. T A B C . u v This means T φ + Au = 0 and Bφ + Cu = v. Let r > 0 as determined at Step 1. L1 ∈ Ellk (E. then T φ = 0.. and L0 replaced by L∗ 0 . The operator T is invertible and its inverse is bounded. We will use the invertibility of this operator to produce an injective operator ker L∗ ⊕ ker L0 → ker L ⊕ ker L∗ 0. we orthogonally decompose L2 (E ) = (ker L)⊥ ⊕ ker L and L2 (F ) = (ker L∗ )⊥ ⊕ ker L∗ . Corollary 10. and thus finish the proof of Theorem 10. ∗ Set U := ker L ⊕ ker L∗ 0 .4. As such. and L ∈ Ellk (E. We can view both φ and u as (linear) functions of v . if u(v ) = 0 for some v .10. Theorem 10. will produce the opposite inequality. From the equality v = Bφ + Cu we deduce v = 0. . We mention only one of them. The theorem we have just proved has many topological consequences. for any v ∈ V we can find a unique pair (φ.2 (E ) ∩ (ker L)⊥ ⊂ (ker L)⊥ → (ker L∗ )⊥ = Range (L) denotes the restriction of L to (ker L)⊥ . such that φ 0 RL0 (L) = . Since RL0 (L) is invertible. F ) if σk (L0 ) = σk (L1 ) then ind (L0 ) = ind (L1 ). Now let us provide the details. We will regard RL0 (L) as an operator RL0 (L) : (ker L)⊥ ⊕ U → (ker L∗ )⊥ ⊕ V. u) ∈ (ker L)⊥ ⊕ U . First.16.4. i.13.e. Let L0 . with L replaced by L∗ . and since T is injective φ must be zero.4. ind (L0 ) ≤ ind(L). φ = φ(v ) and u = u(v ). A dual argument. Indeed. and V := ker L ⊕ ker L0 . F ). This implies dim ker L∗ + dim ker L0 ≤ dim ker L + dim ker L∗ 0 . it has a block decomposition RL0 (L) = where T : Lk. We have thus produced the promised injective map V → U . Hence RL0 (L) = L ıL∗ 0 0 PL0 487 is invertible. F ). There aren’t that many polynomials around.V (u ⊕ v ) = Lu + v is surjective.488 Elliptic equations on manifolds Proof. our deformation freedom is severely limited by the “polynomial” character of the symbols. F ) a continuous family of elliptic operators. we have an isomorphism of line bundles det ker SL. Two polynomial-like elliptic symbols may be homotopic in a larger classes of symbols which are only positively homogeneous along the fibers of the cotangent bundle). (a) Show that any subspace V ⊂ L2 (F ) containing ker L∗ is a stabilizer of L.18. there exists a finite dimensional subspace V ⊂ L2 (F ). At this point one should return to analysis.o. Such objects exist. then we have a natural isomorphism vector bundles ker SL. Let L ∈ Ellk (E. but we believe the reader who reached this point can complete this journey alone. and try to conceive some operators that behave very much like elliptic p. i.e. (a) Show that the family L admits an uniform stabilizer. 90] for a very efficient presentation of this subject. Exercise 10.V1 ⊗ det V1∗ ∼ = det ker SL.4. (Look at the symbols). . The deformation invariance of the index provides a very powerful method for computing it by deforming a “complicated” situation to a “simpler” one. It is called the determinant line bundle of the family L and is denoted by det ind(L).d. For every t ∈ [0. Thus the line bundle det ker SL.V2 ⊗ det V2∗ . Thus ind (Lt ) is an integer depending continuously on t so it must be independent of t.17. such that V is a stabilizer of each operator Lλ in the family L. More generally. 1] Lt = (1 − t)L0 + tL1 is a k -th order elliptic operator depending continuously on t.V ⊕ V ∼ = ker SL. then ind L = dim ker SL.4..V defines a vector bundle over Λ. 1 2 2 In particular. and have more general symbols.V − dim V. This corollary allows us to interpret the index as as a continuous map from the elliptic symbols to the integers. then the family of subspaces ker SLλ . There is one (major) difficulty. Unfortunately.V ⊗ det V ∗ → Λ is independent of the uniform stabilizer V . Conclude that if V is a stabilizer. We will not follow this path. A finite dimensional subspace V ⊂ L2 (F ) is called a stabilizer of L if the operator SL. and are called pseudo-differential operators. (c) Show that if V1 and V2 are two uniform stabilizers of the family L.V : Lk. Consider a compact manifold Λ and L : Λ → Ellk (E.V ⊕ V1 . Exercise 10. We refer to [62. These symbols are “polynomials with coefficients in some spaces of endomorphism”. The analysis has vanished! This is (almost) a purely algebraictopologic object. any finite dimensional subspace of L2 (F ) containing a stabilizer is itself a stabilizer. (b) Show that if V is an uniform stabilizer of the family L.2 (E ) ⊕ V → L2 (F ) SL. (a) We only need to show that ker(λ − L) consists of smooth sections. The main result of this subsection states that one can find an orthonormal basis of L2 (E ) which diagonalizes La . ε. discrete. unbounded set. the operator λ − La is a Fredholm.2 (E ) ⊂ L2 (E ) → L2 (E ). is a selfadjoint.4. g ). Theorem 10. . so its spectrum spec(L) is an unbounded closed subset of R. ∀|λ − λ0 |. The first identity is true over the entire L2 (E ) while part (d) of the theorem shows the domain of validity of the second equality is precisely the domain of L. countable. (a) The spectrum spec(L) is real. Proof.19. and a complex vector bundle E → M endowed with a Hermitian metric •. λ = λ0 . and in particular. Then Lk.2 Spectral theory We mentioned at the beginning of this section that the elliptic operators on compact manifold behave very much like matrices. Then the following are true. Perhaps nothing illustrates this feature better than their remarkable spectral properties. • and compatible connection.4. Throughout this subsection L will denote a k -th order. a compact. Consider as usual. Part (c) of this theorem allows one to write = λ Pλ and L = λ λPλ . and for each λ ∈ spec(L) .d. so that λ ∈ spec(L) ⇐⇒ ker(λ − L) = 0. oriented Riemann manifold (M. Assume for simplicity λ0 = 0. Its analytical realization La : Lk.2 (E ) = D(La ) = ψ ∈ L2 (E ) . (c) There exists an orthogonal decomposition L2 (E ) = λ∈spec(L) ker(λ − L). formally selfadjoint elliptic operator L : C ∞ (E ) → C ∞ (E ). More precisely. Note that for any λ ∈ R the operator λ − La is the analytical realization of the elliptic p. λE − L. elliptic operator. This is the subject we want to address in this subsection. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS 489 §10. Let L ∈ Ellk (E ) be a formally selfadjoint elliptic operator. (b) The spectrum spec(L) is a closed.4. given λ0 ∈ spec(L). λ λ2 Pλ ψ 2 <∞ . In view of Weyl’s lemma this is certainly the case since λ − L is an elliptic operator.o. the spectrum of L consists only of eigenvalues of finite multiplicities. (d) Denote by Pλ the orthogonal projection onto ker(λ − L). we will find ε > 0 such that ker(λ − L) = 0. the subspace ker(λ − L) is finite dimensional and consists of smooth sections. spec(L) ⊂ R. Thus. (b)&(c) We first show spec(L) is discrete.10. then Lψ ∈ L2 (E ). We have to show un = T vn admits a subsequence which converges in L2 (E ).e. bounded. We can now use the spectral theory of such well behaved operators as described for example in [18]. Chap. This proves (b)&(c). Thus t0 − L has a bounded inverse T = (t0 − L)−1 . Note that un is a solution of the partial differential equation (t0 − L)un = t0 un − Lun = vn . Using the elliptic estimates we deduce un k ≤ C ( un + vn ). Thus T is a compact. if λ2 Pλ ψ λ 2 < ∞.. We claim that T is also a compact operator. Obviously T is a selfadjoint operator. L2 (E ) = µ∈spec(T )\{0} ker(µ − T ). The spectrum of T is a closed. so the above inequality implies un k is also bounded. and since ker T = 0.2 (E ). i. Assume (vn ) is a bounded sequence in L2 (E ). Thus spec(L) must be a discrete set. note that if ψ ∈ Lk. so that the Poincar´ e inequality implies 1 = un ≤ C Lun = Cλn → 0. such that Lun = λn un . selfadjoint operator. Now consider t0 ∈ R \ σ (L). 6. µ ∈ spec(T ) \ {0}}.2 (E ) → L2 (E ). consider the sequence of smooth sections φn = |λ|≤n λPλ ψ . un = 1. countable set with one accumulation point.490 Elliptic equations on manifolds We will argue by contradiction. λPλ ψ λ 2 = λ λ2 Pλ ψ 2 < ∞. Conversely. Any µ ∈ spec(T ) \{0} is an eigenvalue of T with finite multiplicity. Clearly un ∈ R (L) = (ker L∗ )⊥ = (ker L)⊥ . µ = 0. The desired conclusion follows from the compactness of the embedding Lk. Obviously un = T vn is bounded in L2 (E ). Thus there exist λn → 0 and un ∈ C ∞ (E ). To prove (d). Using the equality L = t0 − T −1 we deduce σ (L) = {t0 − µ−1 . Using the elliptic estimates we deduce ψn − ψm k ≤ C ( ψn − ψm + φn − φm ) → 0 as n. E = CM and L = −i ∂ : C ∞ (S 1 . C).2 (S 1 ) if and only if (1 + n2 )|un |2 < ∞. u =1 > −∞. . inf M Lu. ∂θ ker(n − L) is the usual Fourier decomposition of periodic functions. ∂θ The operator L is clearly a formally selfadjoint elliptic p. Hence spec(L) = Z and ker(n − L) = spanC {exp(inθ)}. C) → C ∞ (S 1 .e.2 -limit of smooth sections.2 (E ). φn = Lψn .10. u dvg . Let M = S 1 . ELLIPTIC OPERATORS ON COMPACT MANIFOLDS which converges in L2 (E ) to φ= λ 491 λPλ ψ. The theorem is proved. u(0) = u(2π ). m → ∞. where ψn = |λ|≤n Pλ ψ converges in L2 (E ) to ψ .4.20. On the other hand. u ∈ Lk.4. The eigenvalues and the eigenvectors of L are determined from the periodic boundary value problem −i which implies u(θ) = C exp(iλθ) and exp(2πλi) = 1. Hence ψ ∈ Lk. Note that u(θ) = n un exp(inθ) ∈ L1. n∈Z The following exercises provide a variational description of the eigenvalues of a formally selfadjoint elliptic operator L ∈ Ellk (E ) which is bounded from below i.2 (E ) as a Lk. The orthogonal decomposition L2 (S 1 ) = n ∂u = λu.d. Example 10.o. 21. This is a special case of a famous question raised by V. Weyl’s formula shows that the asymptotic behavior of the spectrum of ∆ contains several geometric informations about M : we can read the dimension and the volume of M from it. L(V ) ⊂ V .4. . According to the previous exercise λ0 is an eigenvalue of L. .4.23.2 (E ) ∩ V ⊥ . More precisely H. i. Set V0 = 0. Set Vm+1 = spanC {φ0 .d.2 (E ). u =1 is an eigenvalue of L. Set λ1 = λ(V1⊥ ) and iterate the procedure. . (b) Denote by Grm the Grassmannian of m-dimensional subspaces of Lk. λ0 . Thus “we can hear” the dimension and the volume of a drum. Pick φ0 an eigenvector corresponding to λ0 such that φ0 = 1 and form V1 = V0 ⊕ span {φ0 }. and denote λ0 = λ(V0⊥ ). (a) When L is a formally selfadjoint generalized Laplacian then the result in the above exercise can be considerably sharpened. (V ⊥ denotes the orthogonal complement of V in L2 (E )).o.Weyl showed that rank (E ) · volg (M ) . Use the results in the above exercises to show that if L is a bounded from below. g ) of dimension M . φ1 . Show that λm = V ∈Grm inf max M Lu.do.4. If we think of M as the elastic membrane of a drum then the eigenvalues of ∆ describe all the frequencies of the sounds the “drum” M can produce.2 (E ) be a finite dimensional invariant subspace of L.22. . u ∈ V u =1 . φm } and λm+1 = λ(Vm +1 (a) Prove that the set {φ1 . or [90]. φm corresponding to m + 1 eigenvalues ⊥ ) etc. . (10. . For details we refer to [11]. . . . k -th order formally selfadjoint elliptic p.e. λ1 ≤ · · · λm of L.4. Exercise 10. . u dvg . namely the heat operator ∂t + L on R × M . over an N -dimensional manifold then λm (L) = O(mk/N ) as m → ∞. Exercise 10. and spec(L) = {λ1 ≤ · · · ≤ λm · · · }.5) lim Λ−N/2 d(Λ) = Λ→∞ (4π )N/2 Γ(N/2 + 1) The very ingenious proof of this result relies on another famous p. . Let V ∈ Lk. and d(Λ) = dim ⊕λ≤Λ ker(λ − L) = O(ΛN/k ) as Λ → ∞. (b) Assume L is the scalar Laplacian ∆ on a compact Riemann manifold (M.24. u ∈ Lk..Kac in [51]: can one hear the shape of a drum? In more . u dvg . . . . φm .492 Elliptic equations on manifolds Exercise 10. (b) The quantity λ(V ⊥ ) = inf M Lu.} is a Hilbert basis of L2 (E ).4. . Remark 10. After m steps we have produced m + 1 vectors φ0 . Show that (a) The space V consists only of smooth sections. φ1 . ξ ) −→ (E1 )x → · · · σ (Dm−1 )(x. Denote by ∆S n−1 the scalar Laplacian on the unit sphere S n−1 ⊂ Rn equipped with the induced metric..e. σ (D0 )(x. Di Di−1 = 0.4. (i) (C ∞ (Ei ). n n−2 (d) Prove Weyl’s asymptotic estimate (10. ξ ) ∈ T ∗ M \ {0} the sequence of principal symbols 0 → (E0 )x is exact. . (ii) For each (x. g ) be an oriented Riemann manifold.4. Compute the spectrum of the scalar Laplacian on the torus T 2 equipped with the flat metric.o.d. It is convenient to work in a slightly more general context than Hodge’s original theorem.27.ξ ) −→ (Em )x → 0 . ELLIPTIC OPERATORS ON COMPACT MANIFOLDS 493 rigorous terms this question asks how much of the geometry of a Riemann manifold can be recovered from the spectrum of its Laplacian.25. This is what spectral geometry is all about. . . We assume n ≥ 2. We refer to [38] and the references therein for more details. i.4. §10. (a) Show that if p is a homogenous harmonic polynomial of degree k . Show that dim ker λk − ∆S n−1 = k+n−1 k+n−3 − .4. and then use this information to prove the above Weyl asymptotic formula in this special case. Let (M.4. Exercise 10.3 Hodge theory We now have enough theoretical background to discuss the celebrated Hodge theorem. Di ) is a cochain complex. . A polynomial p = p(x1 .’s 0 1 0 → C ∞ (E0 ) → C ∞ (E1 ) → · · · → C ∞ (Em ) → 0 D D Dm−1 satisfying the following conditions. Definition 10.10. and p ¯ denotes its restriction to the unit sphere S n−1 centered at the origin. then ∆S n−1 p ¯ = n(n + k − 2)¯ p.26. (c) Set λk := n(n + k − 2). (b) Show that a function ϕ : S n → R is an eigenfunction of ∆S n−1 if and only if there exists a homogeneous harmonic polynomial p such that ϕ = p ¯. An elliptic complex is a sequence of first order p. Exercise 10. ∀1 ≤ i ≤ m. It has been established recently that the answer to Kac’s original question is negative.5) in the special case of the operator ∆S n−1 .4. xn ) is called harmonic if ∆Rn p = 0. the DeRham complex is elliptic. D· ) ∼ = ker ∆i ⊂ C ∞ (Ei ).3 where it is shown to be exact. (a) H i (E· .6) This is precisely part (c) of Hodge’s theorem. d) is an elliptic complex. Form the operators about Sobolev spaces and formal adjoints Di ∗ ∗ ∞ ∞ ∆i = Di Di + Di−1 Di −1 : C (Ei ) → C (Ei ). ˆ and ∆ are elliptic.4. ∆ ∈ P DO (E. We can now talk ∗ . E = ⊕Ei .29 (Hodge).) ˆ .28.12 we have ker ∆ = ker D orthogonal decomposition ˆ ). ∆ = D∗ D + DD∗ = (D + D∗ )2 = D We now invoke the following elementary algebraic fact which is a consequence of the exactness of the symbol sequence. D· ) < ∞ ∀i. we deduce D2 = (D∗ )2 = 0 which implies ˆ 2. formally selfadjoint p. D· )) over a compact oriented Riemann manifold (M.2 → L2 .3.1. Hence. Proof.1. Consider an elliptic complex (C ∞ (E· . D∗ . Since Di Di−1 = 0. The DeRham complex (Ω∗ (M ).4. The operators D Use Exercise 7. We will have the occasion to discuss another famous elliptic complex in the next chapter. A priori these may be infinite dimensional spaces. Endow each Ei with a metric and compatible connection.4.d. . D = ⊕Di . L2 (E ) = ker ∆ ⊕ R (D (10.23 of Subsection 7.1. In this case. ∆ = ⊕∆i . We will see that the combination ellipticity + compactness prevents this from happening.o. D∗ = ⊕Di . We can now state and prove the celebrated Hodge theorem. g ).4. ∀i. Assume that M is compact and oriented. Theorem 10.4. There exists an orthogonal decomposition ∗ L2 (Ei ) = ker ∆i ⊕ R (Di−1 ) ⊕ R (Di ).30. D Thus D.494 Elliptic equations on manifolds Example 10. E ). (c) (Hodge decomposition). Then the following are true. D· ). so that we have an Note that according to Corollary 10. Denote its cohomology by H • (E· . Set ∗ ˆ = D + D∗ .1. (Hint: Exercise 10.22 in Subsection 7. the associated sequence of principal symbols is (e(ξ ) = exterior multiplication by ξ ) ∗ ∗ 0 → R −→ Tx M −→ · · · −→ det(Tx M) → 0 e(ξ ) e(ξ ) e(ξ ) is the Koszul complex of Exercise 7. i (b) dim H (E· . and Di as bounded operators L1. where we view both Di−1 . and ∗ denotes the Hodge ∗-operator defined by the Riemann metric g and the fixed orientation on M . Applying Di on both sides of this equality we get ∗ ∗ 0 = Di u = Di Pi u + Di Di−1 u + Di Di w = Di Di w.4.6) we have ˆ ψ ∈ L1. D· ) → ker ∆i .4. Di u = 0 and B i = Di−1 (C ∞ (Ei−1 )). The above claim is a consequence of several simple facts. ker ∆i ⊂ Z i . u = Pi u + Dψ Since u − Pi u ∈ C ∞ (Ei ). .k ∗ d∗. B i ⊂ R (D and we deduce that no two distinct elements in ker ∆i are cohomologous since otherwise their difference would have been orthogonal to ker ∆i . This will complete the proof of Hodge theorem. Hence. u ∈ C ∞ (Ei ). This follows from the equality ker ∆ = ker D Fact 2. using the decomposition (10. D· ).k = nk + n + 1. Hodge theorem is proved. Fact 1. where νn.2 (E ). We know that the formal adjoint of d : Ωk (M ) → Ωk+1 (M ) is d∗ = (−1)νn. Set ∆ = dd∗ + d∗ d. D· ) = Z i /B i . Fact 1 shows it is also surjective so that ker ∆i ∼ = H i (E· . Let us apply Hodge theorem to the DeRham complex on a compact oriented Riemann manifold d d d 0 → Ω0 (M ) −→ Ω1 (M ) −→ · · · −→ Ωn (M ) → 0. Thus. n = dim M. ˆ. If u ∈ Zi then u − Pi u ∈ Bi .10. We conclude that Pi descends to a linear map ker ∆i → H i (E· . Thus ˆ ) = (ker ∆)⊥ . D· ). the induced linear map Pi : ker ∆i → H i (E· . Set Zi = so that H i (E· . we deduce from Weyl’s lemma that ψ ∈ C ∞ (E ). there exist v ∈ C ∞ (Ei−1 ) and w ∈ C ∞ (Ei ) such that ψ = v ⊕ w and ∗ u = Pi u + Di−1 v + Di w. ∗ = ker D D ∗ . D· ) is injective. 495 We claim that the map Pi : Z i → ker ∆i descends to an isomorphism H i (E· . Indeed. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS For each i denote by Pi the orthogonal projection L2 (Ei ) → ker ∆i . the above equalities imply Since ker Di i i u − Pi u = Di−1 v ∈ B i . e.e. g ). then so is ∗ω since d ∗ ω = ± ∗ d ∗ ω = 0. On the other hand.4. ∆[ω ]g = 0 ⇐⇒ d[ω ]g = 0 and d∗ [ω ]g = 0. The Hodge ∗-operator defines a bijection ∗ : Hk (M. and ∗d ∗ (∗ω ) = ± ∗ (dω ) = 0. The Hodge ∗-operator is bijective since ∗2 = (−1)k(n−k) . then d∗ ζ = 0. If ω is g -harmonic. g ) the space of g -harmonic k -forms on M . denotes the natural pairing between a vector space and its dual. η ∈ Ωk−1 (M ). If moreover ω is closed. M M . g ) → Hn−k (M. and [ω ]g ∈ Ωk (M ) is g -harmonic. i.31 (Hodge). i.496 Elliptic equations on manifolds Corollary 10.4. The form [ω ]g is called the g -harmonic part of ω .. the Poincar´ e duality described in Chapter 7 induces another isomorphism PD Hk −→ (Hn−k )∗ . · 0 0 ∗ = M ω ∧ η. g ) with its dual. The above corollary shows Hk (M. Corollary 10. η where ·. ζ ∈ Ωk+1 (M ). Proposition 10.33. ∗ω. g ) ∼ = H k (M ) for any metric g . η g dvg = ω ∧ η ∀ω ∈ Hk η ∈ Hn−k . and thus we can view ∗ as an isomorphism Hk → (Hn−k )∗ . Proof. P D = ∗. defined by P D(ω ).4.32. Denote by Hk (M. and this means any cohomology class [z ] ∈ k H (M ) is represented by a unique harmonic k -form. Any smooth k -form ω ∈ Ωk (M ) decomposes uniquely as ω = [ω ]g + dη + d∗ ζ. Using the L2 -inner product on Ω• (M ) we can identify Hn−k (M. Let G denote a compact connected Lie group equipped with a biinvariant Riemann metric h. .4.4. Exercise 10. We have ∗ω.4. ELLIPTIC OPERATORS ON COMPACT MANIFOLDS Proof. ∗ω g dvg = η ∧ ∗2 ω = (−1)k(n−k) η ∧ ω = ω ∧ η.34. Prove that a differential form on G is h-harmonic if and only if it is bi-invariant. Show that |ω0 |2 g dvg ≤ |ω |2 g dvg ∀ω ∈ Cω0 . Let ω0 ∈ Ωk (M ) be a harmonic k form and denote by Cω0 its cohomology class. η g dvg = η. 497 Exercise 10.10.35. M M with equality if and only if ω = ω0 . Thus C ∞ (E ) can be identified with the space of smooth functions u : S 1 → R2n .o.1. and then we will spend the remaining part discussing some frequently encountered examples.1. and §11.o. Denote by E the trivial vector bundle R2n over the circle S 1 .1 The structure of Dirac operators Consider a Riemann manifold (M. 11. Let J : C ∞ (E ) → C ∞ (E ) denote the endomorphism of E which has the block decomposition 0 −Rn J= 0 Rn 498 . ∗ σ (D2 )(x.. ξ ) = −|ξ |2 g Ex ∀(x. i. The Dirac operator is said to be graded if E splits as E = E0 ⊕ E1 .1. In other words. Note that the Dirac operators are first order elliptic p. A Dirac operator is a first order p. We will first describe their general features.e. D has a block decomposition D= 0 A B 0 . (Hamilton-Floer). such that D2 is a generalized Laplacian.d. D : C ∞ (E ) → C ∞ (E ).-s. ξ ) ∈ T M.2.Chapter 11 Dirac Operators We devote this last chapter to a presentation of a very important class of first order elliptic operators which have numerous applications in modern geometry. Example 11. Definition 11. g ).d. and D(C ∞ (Ei )) ⊂ C ∞ (E(i+1) mod 2 ).1.1 Basic definitions and examples a smooth vector bundle E → M . (Cauchy-Riemann). dθ d Clearly. Now define ˆ : C ∞ (E ˆ⊕E ˆ ) → C ∞ (E ˆ⊕E ˆ ). Then. F2 = − dθ 2 is a generalized Laplacian. g ) be an oriented Riemann manifold. (Hodge-DeRham). the Hodge-DeRham operator d + d∗ : Ω• (M ) → Ω• (M ). 2 Example 11. D by u(t) v (t) = 0 −∂t + D ∂t + D 0 u(t) v (t) . x0 ) − u(t0 .1. A section u ˆ is then a smooth 1-parameter family of pullback of E to M ˆ of E sections u(t) of E . where t denotes the coordinate along the factor R. .1. h→0 where the above limit is in the t-independent vector space Ex0 . g ). D : C2 → C2 defined by u v where z = x + y i. the We have a natural projection π : M ˆ via π . The Cauchy-Riemann operator is the first order p. Let (M. h −→ 2 0 −∂z ∂z 0 ¯ · u v . is a Dirac operator. and ˆ the cylinder R × M . ˆ is a Dirac operator called the suspension of D. 2 2 A simple computation shows that D is a Dirac operator. Example 11. Denote by M ˆ 2 ˆ the cylindrical metric g ˆ = dt + g on M .d. and we set E ˆ := π ∗ E .1. In particular. according to the computations in the previous chapter.5.11.o.1.4. THE STRUCTURE OF DIRAC OPERATORS 499 with respect to the natural splitting R2n = Rn ⊕ Rn . We define the Hamilton-Floer operator du F : C ∞ (E ) → C ∞ (E ). we can define unambiguously ∂t u ˆ(t0 . x) → x. Note that the Cauchy-Riemann Then D operator is the suspension of the Hamilton-Floer operator. and by g D : C ∞ (M ) → C ∞ (M ) is a Dirac operator. ∂z ¯ = (∂x + i∂y ). Fu = J ∀u ∈ C ∞ (E ). and 1 1 ∂z = (∂x − i∂y ).3. Example 11. Suppose E is a vector bundle over the Riemann manifold (M. x0 ) . ˆ → M . t ∈ R. Consider the trivial bundle C2 over the complex plane C equipped with the standard Euclidean metric. (t. x0 ) = lim 1 u(t0 + h. x) is an endomorphism of Ex depending linearly upon ξ . we see that each Dirac operator induces a bundle morphism c : T ∗ M ⊗ E → E (ξ. g ).. x ∈ M we conclude that {c(ξ ). Clifford structure) is called a Clifford bundle. Then. From the equality ∗ c(ξ + η ) = −|ξ + η |2 ∀ξ. η )Ex .e.7. Proposition 11. the Clifford multiplication by α is an odd endomorphism of the superspace C ∞ (E+ ) ⊕ C ∞ (E− ). DIRAC OPERATORS Let D : C ∞ (E ) → C ∞ (E ) be a Dirac operator over the oriented Riemann manifold (M. A pair (vector bundle. c(α)C ∞ (E ± ) ⊂ C ∞ (E ∓ ). Thus. (b) A Z2 -grading of a Clifford bundle E → M is a splitting E = E + ⊕ E − such that. ∀u ∈ C ∞ (E ). η ∈ Tx M. c(η )} = −2g (ξ. e) → c(ξ )e. where for every 1-form α we denoted by c(α) the bundle morphism c(α) : E → E given by c(α)u = c(α. Let E → M be a smooth vector bundle over the Riemann manifold (M. where for any linear operators A. where π : T ∗ M → M denotes ∗ M . g ). (a) There exists a Dirac operator D : C ∞ (E ) → C ∞ (E ). ∀ξ. the operator the natural projection. for any x ∈ M . Definition 11.6. and any ξ ∈ Tx c(ξ ) := σ (D)(ξ. B we denoted by {A. Its symbol is an endomorphism σ (D) : π ∗ E → π ∗ E . such that {c(ξ ). To prove the reverse implication let c : T ∗M ⊗ E → E be a Clifford multiplication. ξ ) = −|ξ |2 g Ex . u). To summarize. . B } := AB + BA. for every connection ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ). B } their anticommutator {A. Since D2 is a generalized Laplacian we deduce that c(ξ )2 = σ (D2 )(x. Proof. The morphism c is usually called the Clifford multiplication of the (Clifford) structure. i.500 CHAPTER 11. (b) The bundle E admits a Clifford structure. (a) A Clifford structure on a vector bundle E over a Riemann manifold (M. g ) is a smooth bundle morphism c : T ∗ M ⊗ E → E .1. c(η )} = −2g (ξ. We have just seen that (a)⇒(b).1. Then the following conditions are equivalent. η ∈ Ω1 (M ). η )E . ∀α ∈ Ω1 (M ). such that c(ξ )2 = −|ξ |2 . 1. and subject to the relations {u. v ∈ V. where E is a K-vector space. If (ei ) is an orthonormal basis of V .1. We will look at the following universal situation. v )E . Exercise 11. ρj } = −2δij E .the metric dual of ξ . The collection (ρi ) generates an associative subalgebra in End (E ). while iξ denotes the interior differentiation along ξ ∗ ∈ Tx M . Euclidean space (V.9. Prove the last assertion in the above example. Definition 11.2. For each x ∈ M . then ρ is completely determined by the linear operators ρi = ρ(ei ) which satisfy the anti-commutation rules {ρi . q ) is the associative R–algebra with unit. where eξ denotes the (left) exterior multiplication by ξ . then the Dirac operator c◦∇ is none other than the Hodge-DeRham operator.10. and ρ : V → End (E ) is an R-linear map such that {ρ(u). Consider a real finite dimensional. If ∇ denotes the Levi-Civita connection on Λ• T ∗ M . g ) is then a pair (E. The Clifford algebra Cl(V.11.1. ρ). with symbol c. and q :V ×V →R a symmetric bilinear form.4 shows that c defines a Clifford multiplication on Λ• T ∗ M . ∇ c 501 ∗M Example 11. Clearly D is a Dirac operator.55 of Section 2. ∀u. v ∈ V.o. and it is natural to try to understand its structure. §11. g ). A Clifford multiplication associated with (V. In the following subsections we will address precisely this issue. ρ(v )} = −2g (u. g ) be a Riemann manifold. generated by V . The above proposition reduces the problem of describing which vector bundles admit Dirac operators to an algebraic-topological one: find the bundles admitting a Clifford structure. Let (M.2. and ξ ∈ Tx define ∗ ∗ c(ξ ) : Λ• Tx M → Λ• Tx M by c(ξ )ω = (eξ − iξ )ω.1.2 Clifford algebras The first thing we want to understand is the object called Clifford multiplication. THE STRUCTURE OF DIRAC OPERATORS the composition D = c ◦ ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ) → C ∞ (E ) is a first order p. . finite dimensional vector space. The Exercise 2.8. v } = uv + vu = −2q (u.d. v ) ·  ∀u. Let V be a real.1. for all admissible S . It is known from linear algebra that the isomorphism classes of such pairs are classified by some simple invariants: (dim V. where I is the ideal generated by u ⊗ v + v ⊗ u + 2q (u. Prove the universality property.1.1. sign q ). and (S ◦ T )# = S# ◦ T# . The map ı is the composition V → A → Cl(V.. The correspondence T → T# constructed above is functorial. qi ).. V ı M Cl(V. Thus when thinking of automorphisms of this structure one should really concentrate only on those automorphisms of R–algebras preserving the distinguished subspace. Let (Vi . Remark 11. q ) = A/I . q ) exists. Exercise 11. any linear map T : V1 → V2 such that q2 (T v ) = q1 (v ). or Cln . v ) · . there exists an unique morphism of algebras Φ : Cl(V. q) A  K Φ ı denotes the natural inclusion V → Cl(V. q ) depends only on the isomorphism class of the pair (V. when q is an Euclidean metric on the n-dimensional space V . where we view Vi as a linear subspace in Cl(Vi . 2) be two real. and T .1. DIRAC OPERATORS Proposition 11. Corollary 11. v ) ⊗ 1.502 CHAPTER 11.e. The above corollary shows that the algebra Cl(V. qi ) (i = 1. We will be interested in the special case when dim V = rank q = sign q = n. (Vi )# = Cl(Vi .12. The Clifford algebra Cl(V. u. If (ei ) is an orthonormal basis of V . rank q. This makes a Clifford algebra a structure richer than merely an abstract R–algebra: it is an algebra with a distinguished real subspace. q ) → A such that the diagram below is commutative. v ∈ V . ∀v ∈ V1 induces a unique morphism of algebras T# : Cl(V1 . the Clifford algebra Cl(V. and {(u). Set Cl(V.13. finite dimensional vector spaces endowed with quadratic forms qi : V → R. 0) is the exterior algebra Λ• V . q1 ) → Cl(V2 . q ) where the second arrow is the natural projection.e. (b) In the sequel the inclusion V → Cl(V. q ) is usually denoted by Cl(V ).11. q ). q2 ) such that T# (V1 ) ⊂ V2 . Then. i. q )= vector space + quadratic form. We let the reader check the universality property. In this case.qi ) . (a) When q ≡ 0 then Cl(V.1. Sketch of proof Let A = ⊕k≥0 V ⊗k (V ⊗0 = R) denote the free associative R–algebra with unit generated by V . q ) will be thought of as being part of the definition of a Clifford algebra.14. (v )} = −2q (u. and is uniquely defined by its universality property: for every linear map  : V → A such that A is an associative R-algebra with unit. i. then we . Cl1 (V ) = ker(α + ).2. ∀ui ∈ V. Note that for any u.e. but its multiplication ∗ is defined by ˜ V ). Note that Cl(V ) = Cl0 (V ) ⊕ Cl1 (V ). For x ∈ Cl(V ) we set x† := (α(x)) = α(x ).1. x ∗ y := y · x ∀x. y ∈ Cl( where “·” denotes the usual multiplication in Cl(V ). so that. so that. v ∈ V u · v + v · u = u ∗ v + v ∗ u. we deduce that the map V → Cl(V ). This may as injection V → Cl( well be regarded as an antimorphism Cl(V ) → Cl(V ) which we call the transposition map. Using the universality property of Cln . Invoking again the Exercise 2. the Clifford algebra Cl(V ) is naturally a super-algebra. Note that (u1 · u2 · · · ur ) = ur · · · u1 . Set Cl0 (V ) := ker(α − ). +. For each v ∈ V define c(v ) ∈ End (Λ• V ) by c(v )ω := (ev − iv )ω ∀ω ∈ Λ• V. v → −v ∈ Cl(V ) extends to an automorphism of algebras α : Cl(V ) → Cl(V ). THE STRUCTURE OF DIRAC OPERATORS 503 can alternatively describe Cln as the associative R–algebra with 1 generated by (ei ). In other words. i. i. .11. ∗) denote the opposite algebra of Cl(V ). Then Cl( ˜ V ) coincides with Let (Cl( Cl(V ) as a vector space.. ˜ V ). The element x† is called the adjoint of x. x → x .55 we deduce c(v )2 = −|v |2 g.. and subject to the relations ei ej + ej ei = −2δij . and moreover Clε (V ) · Clη (V ) ⊂ Cl(ε+η) mod 2 (V ). α2 = . where as usual ev denotes the (left) exterior multiplication by v . the map c extends to a morphism of algebras c : Cl(V ) → End (Λ• V ).e. by the universality property of the Clifford algebras. while iv denotes the interior derivative along the metric dual of v . we conclude that the natural ˜ V ) extends to a morphism of algebras Cl(V ) → Cl( ˜ V ). the automorphism α naturally defines a Z2 -grading of Cl(V ). using the universality property of Clifford algebras. Note that α is involutive. 1. then σ (ei1 · · · eik ) = ei1 ∧ · · · ∧ eik ∀eij . ρ) be a K-Clifford module. The module (E. The real theory is far more elaborate. Exercise 11. ρ : Cl(V ) → EndK (E ). Λ• V is a selfadjoint.C)-vector space E is said to be a K. .Clifford module if there exists a morphism of R–algebras ρ : Cl(V ) → EndK (E ).1. (b) A K-superspace E is said to be a K-Clifford s-module if there exists a morphism of s-algebras ρ : Cl(V ) → EndK (E ). Definition 11. This is an element of Λ• V . where the ∗ in the right-hand-side denotes the adjoint of c(x) viewed as a linear operator on the linear space Λ• V endowed with the metric induced by the metric on V . Example 11.16. (c) Let (E. We now see that what we originally called a Clifford structure is precisely a Clifford module. The resulting linear map Cl(V ) x → σ (x) ∈ Λ• V.17.504 CHAPTER 11. For each x ∈ Cl(V ). called the symbol of x. (a) A K(=R-. ρ) is said to be selfadjoint if there exists a metric on E (Euclidean if K = R. For more information we refer the reader to the excellent monograph [62].1. The inverse of the symbol map is called the quantization map and is denoted by q : Λ• V → Cl(V ).15. If (ei ) is an orthonormal basis. real Cl(V ) super-module. is called the symbol map. we set σ (x) := c(x)1 ∈ Λ• V . Hermitian if K = C) such that ρ(x† ) = ρ(x)∗ ∀x ∈ Cl(V ). This shows that the symbol map is bijective since the ordered monomials {ei1 · · · eik . · · · ik ≤ dim V } form a basis of Cl(V ). In the following two subsections we intend to describe the complex Clifford modules.1. DIRAC OPERATORS Exercise 11. 1 ≤ i1 . Prove that ∀x ∈ Cl(V ) we have c(x† ) = c(x)∗ .18. Show that q(Λeven/odd V ) = Cleven/odd (V ). Let {e1 .) Proof. Let n = 2k .1. where V 1. 1 ≤ j ≤ k 2 is an orthonormal basis of V 1. Alternatively. and consider an n-dimensional Euclidean space (V. V 1. a skew-symmetric operator J : V → V such that J 2 = −V . V 0.1 . The (complex) representation theory of Cln depends on the parity of n so that we will discuss each case separately.19. ek . Such a J exists since V is even dimensional. . =C (V. while the collection 1 εj = √ (ej + ifj ).2. Proposition 11.1 = ker(i + J ). J )).15) of Subsection 2. v ) + ig (u.0 = ker(i − J ) and V 0.0 = spanC (ej − ifj ). J ) ∼ (Above.11. g ) .0 ⊕ V 0. . J ) h(u.5) which allows us to identify V 0. We can now decompose V ⊗ C into the eigenspaces of J V = V 1. . With respect to this Hermitian metric the collection 1 εj := √ (ej − ifj ). . ∀i.3 Clifford modules: the even case In studying complex Clifford modules it is convenient to work with the complexified Clifford algebras Cln = Cln ⊗R C.0 .1. Jv ). There exists a complex Cl(V )-module S = S(V ) such that Cl(V ) ∼ = EndC (S) as C−algebras. v ) = g (u.. 1 ≤ j ≤ k 2 .5. Consider a complex structure on V .2. fk } be an orthonormal basis of V such that Jei = fi . (see (2. The reader may want to refresh his/her memory of the considerations in Subsection 2. THE STRUCTURE OF DIRAC OPERATORS 505 §11. Vc∗ denotes the complex dual of the complex space (V. Extend J by complex linearity to V ⊗R C. The decisive step in describing the complex Cl(V )-modules is the following. it depends on several auxiliary choices. The metric on V defines a Hermitian metric on the complex vector space (V. f1 .2. (The above isomorphism is not natural.1 = spanC (ej + ifj ).1 ∼ =C Vc∗ ∼ =C (V 1.e.1. i.0 )∗ . This boils down to verifying the anticommutation rules {c(ei ).1 .1 can be identified with complex linear functionals on V 1. .0 V. A morphism of algebras c : Cl(V ) → End (S2k ) is uniquely defined by its restriction to V ⊗ C = V 1. Hence.20. they will act by interior differentiation √ c(w)w1 ∧ · · · ∧ w = − 2i(w)(w1 ∧ · · · ∧ w ) = √ 2 j =1 (−1)j gC (wj .0 . ∀ω ∈ Λ•. The anti-commutation rules follow as in the Exercise 2. {c(ei ).0 and V 0. c(fj ) = i(e(εj ) + i(εj )). We have i 1 ei = √ (εi + εi ).2. To check that the above constructions do indeed define an action of Cl(V ) we need to check that. c(fj )} = 0. fj = √ (εj − εj ). and as such.1. w)w1 ∧ · · · ∧ w ˆj ∧ · · · ∧ w .1 . Definition 11. DIRAC OPERATORS S2k := Λ• V 1.0 ⊕ V 0. we have to specify the action of each of the components V 1.506 is an orthonormal basis of V 0.0 will act by exterior multiplication √ √ c(w)ω = 2e(w)ω = 2w ∧ ω. Set CHAPTER 11. c(ej )} = −2δij = {c(fi ). This shows S2k is naturally a Cl(V )-module. The elements w ∈ V 0. This completes the proof of the proposition.1 . 2 2 c(ei ) = e(εi ) − i(εi ).55 using the equalities gC (εi . εj ) = δij . Note that dimC S2k = 2k so that dimC EndC (S) = 2n = dimC Cl(V ).0 = Λ•. A little work (left to the reader) shows the morphism c is injective. ∀v ∈ V .0 V. The Cl(V )-module S(V ) is known as the (even) complex spinor module. so that. where gC denotes the extension of g to (V ⊗ C) × (V ⊗ C) by complex linearity. The elements w ∈ V 1. we have c(v )2 = −S2k . c(fj )}. It is called the chirality operator defined by the orientation. The action of Cl2k on S2k ⊗ W is defined by v · (s ⊗ w) = c(v )s ⊗ w.23.24. Any complex Cl2k -module has the form S2k ⊗ W . and thus defines a Z2 -grading on S S = S+ ⊕ S− (S± = ker(± − Γ)). For any positively oriented orthonormal basis e1 . Exercise 11. where W is an arbitrary complex vector space. Corollary 11. The vector space W is called the twisting space of the given Clifford module.21. Remark 11. g ) is an oriented. .1.1.1. its twisting space can be recovered as the space of morphisms of Clifford modules W = HomCl2k (S2k . and thus it is an element intrinsically induced by the orientation. Let S = S(V ) denote the spinor module of Cl(V ). Assume now that (V. This produces two things: it defines an orientation on V and identifies S = S(V ) ∼ = Λ∗. One can check easily this element is independent of the oriented orthonormal basis.1. Given a complex Cl2k -module E . Γ} = 0.0 V. Note that Γ2 = 1 and Γx = (−1)deg x Γ ∀x ∈ Cl0 (V ) ∪ Cl1 (V ). . THE STRUCTURE OF DIRAC OPERATORS 507 Using basic algebraic results about the representation theory of the algebra of endomorphisms of a vector space we can draw several useful consequences. ∀v ∈ V . Prove that with respect to these data the chiral grading of S is S+/− ∼ = Λeven/odd. . This means that any algebra isomorphism Cl(V ) → End (S(V )) is an isomorphism of Z2 -graded algebras.22. 2k -dimensional Euclidean space.0 V .) Corollary 11. (See [89] for a very nice presentation of these facts. and hence a Z2 -grading of End(S). Let J be a complex structure on V . Since {v.11. . There exists an unique (up to isomorphism) irreducible complex Cl2k module and this is the complex spinor module S2k .1. The chirality operator defines an involutive endomorphism of S. e2k we can form the element Γ = ik e1 · · · e2k ∈ Cl(V ). we deduce the Clifford multiplication by v is an odd endomorphism of S. E ). . in which ˆ S∗ . Then Λ• R V ⊗ C is naturally a Clifford module. However this does not exhaust the family of Clifford s-modules.25.• •. it has the form ∼ Λ• R V ⊗ C = S ⊗ W. oriented. This complex structure produces an isomorphism 0. W complex s − space . The chirality operator introduces a Z2 -grading in any complex Clifford module which we call the chiral grading. Euclidean space. so that Λ• R V ⊗ C can be given two different s-structures: the chiral superstructure. we set y † := α(y ) = α(y ) . The family of s-modules can be completely described as ˆ W . In order to formulate the final result of this subsection we need to extend the automorphism α. and S∗ .0 ∼ Λ• R V ⊗R C = Λ V ⊗ Λ V = S ⊗ (Λ V ) = S ⊗R S . the chirality operator defines Z2 gradings on both S. This can be done via the Hodge ∗-operator.1. and the anti-automorphism to the complexified Clifford algebra Cl(V ). Example 11. More precisely we have Γ · ω = ik+p(p−1) ∗ ω ∀ω ∈ Λp V ⊗ C.508 CHAPTER 11. ∀x ∈ Cl(V ). (11. we pick a complex structure on V whose induced orientation agrees with the given orientation of V .26. g ) be a 2k -dimensional.0 ∗ ∼ ∗ ∼ •. Let (V. As in the real case. This shows the twisting space is S∗ . The automorphism α can be extended by complex linearity α(x ⊗ z ) = α(x) ⊗ z. RV ⊗ C = Λ To understand the chiral grading we need to describe the action of the chiral operator on Λ• R V ⊗ C.1. Using the grading of S∗ is forgotten.1. DIRAC OPERATORS The above considerations extend to arbitrary Clifford modules. while extends according to the rule (x ⊗ z ) = x ⊗ z.1) Exercise 11. Prove the equality 11.24 we deduce that the second grading is precisely the degree grading even Λ• V ⊗ C ⊕ Λodd V ⊗ C. ∀y ∈ Cl(V ). The modules endowed with the chiral grading form where ⊗ the subfamily in which the twisting s-space W is purely even.1.1. and the grading as a super-tensor product S⊗ Exercise 11.1. . On the other hand. S⊗ ˆ denotes the s-tensor product. To find the twisting space. Thus. . Then for every isomorphism of algebras ρ : Cl(V ) → EndC (S(V )) there exists a Hermitian metric on S(V ) such that ρ(y † ) = ρ(y )∗ . e1 .28.4 Clifford modules: the odd case The odd dimensional situation can be deduced using the facts we have just established concerning the algebras Cl2k .29. .1. s2 ) := h(ρ(g )s1 . Pick an orthonormal basis {e0 .1. . ∀s1 . Proof. s2 ∈ S. . this metric is unique up to a multiplicative constant.27. Ψ(x + x ) = x + e0 · x . e2k } of V . . . and some metric on W . Clm ∼ = Cleven m+1 . g ) as above. Moreover. Sketch of proof Choose an orthonormal basis {e1 . ρ(g )s2 ). The sought for metric is a tensor product of the canonical metric on S. Corollary 11. .11.1. for each g ∈ G. Let (V. denote by hg the pulled-back metric hg (s1 . We leave the reader to check that this is the metric we are after. Lemma 11. hG := 1 |G| hg . and ρ : Cl(V ) → EndC (E ) be a complex Clifford module. §11. . . Now define 0 1 0 1 Ψ : Clm → Cleven m+1 . We can now form the averaged metric. Pick a Hermitian metric h on S(V ) and. em } of the standard Euclidean space Rm+1 . and denote by G the finite subgroup of Cl(V ) generated by the basic vectors ei . odd 1 where x0 ∈ Cleven m . for some isomorphism of algebras ∆ : Cl(V ) → End(S). and x ∈ Clm .1. g ). . . We leave the reader to check that this is indeed an isomorphism of algebras. Decompose E as S ⊗ W . Let S(V ) denote the spinor module of the 2k -dimensional Euclidean space (V. em }.1. g ∈G Each ρ(g ) is an unitary operator with respect to this metric. The uniqueness follows from the irreducibility of S(V ) using Schur’s lemma. and ρ as ∆ ⊗ W . Proof. THE STRUCTURE OF DIRAC OPERATORS 509 Proposition 11. We view Clm as the Clifford algebra generated by {e1 . ∀y ∈ Cl(V ). . Then E admits at least one Hermitian metric with respect to which ρ is selfadjoint. . These generate the algebra Clm+1 . The bridge between these two situations is provided by the following general result. e. i.e. ∀x ∈ M . (a) A bundle of Clifford algebras C → M such that Cx ∼ = Cln or Cln . irreducible Cl(V )-modules S+ (V ) and S− (V ) such that Cl(V ) ∼ = EndC (S+ ) ⊕ EndC (S− ) as ungraded algebras. v ). bundles).e. ∀x ∈ M ∗ {ı(u). a vector bundle E → M . For more details we refer to [62]. Denote by S2k+2 the spinor module of Cl(V ⊕ R). (d) A connection on E. ∀u. (b) A fiberwise injective morphism of vector bundles ı : T ∗ M → C such that. To produce a Dirac operator on an n-dimensional Riemann manifold (M. Fix an orientation on V and a positively oriented orthonormal basis e1 . e2k+1 . Then there exist two complex. g ) be a (2k + 1)-dimensional Euclidean space. Choose an isomorphism ρ : Cl(V ⊕ R) → EndC (S2k+2 ).1. where V ⊕ R is given the direct sum Euclidean metric and the orientation or (V ⊕ R) = or ∧ or (R). we can choose Hermitian metrics on S± (V ) such that the morphism ρ : Cl(V ) → EndC S(V ) is selfadjoint. together with a morphism c : C → End (E) whose restrictions to the fibers are morphisms of algebras. The next step in our story is to glue all the pointwise data presented so far into smooth families (i. .5 A look ahead In this heuristic section we interrupt a little bit the flow of arguments to provide the reader a sense of direction.510 CHAPTER 11. g ) one needs several things. Let (V.1. We will not present the details since the applications we have in mind do not require these facts. The direct sum S(V ) = S+ (V ) ⊕ S− (V ) is called the (odd) spinor module. e2 . §11.30. Then S2k+2 becomes naturally a super Cl2k+2 -module − S2k+2 = S+ 2k+2 ⊕ S2k+2 .. . Proof. As in the previous section. (c) A bundle of Clifford modules. . . ı(v )}C = −2g (u. v ∈ Tx M. The above result can be used to describe the complex (super)modules of Cl2k+1 . . and we thus we get the isomorphisms of algebras − Cl(V ) ∼ = Cl(V ⊕ R)even ∼ = Endeven (S2k+2 ) ∼ = End (S+ 2k+2 ) ⊕ End (S2k+2 ). ρ(u† ) = ρ(u)∗ . DIRAC OPERATORS Proposition 11. ∀u ∈ Cl(V ). i. the Clifford multiplication is covariant constant. ∇E (c(α)u) = c(∇M α) + c(α)∇E u. Then e0 canonically determines a section u0 on P ×ρ E which is covariant constant with respect to the induced connection ∇E = ρ∗ (∇).e. . the diagram below is commutative.e. The symmetry group of this principal bundle has to be a Lie group with several additional features which we now proceed to describe. we discuss only the real case. THE STRUCTURE OF DIRAC OPERATORS 511 The above collection of data can be constructed from bundles associated to a common principal bundle. The group Autc V is defined similarly.11. ∀g ∈ G. Any connection ∇ on P induces by association metric connections ∇M on1 T ∗ M and ∇E on the bundle of Clifford modules E = P ×µ E . For brevity. u ∈ C ∞ (E ). such that. This follows from the following elementary invariant theoretic result. We also need a Clifford module c : Cl(V ) → End (E ). i. This leads to significant simplifications in many instances. and ρ : G → Aut (E ) be a linear representation of G. and denote by AutV the group of automorphisms φ of Cl(V ) such that φ(V ) ⊂ V . ∇E u0 = 0.1.2) c(g ·v ) This commutativity can be given an invariant theoretic interpretation as follows. In concrete applications E comes with a metric. Let G be a Lie group. ρ defines a representation ρ : G → GL(V ).1. We need a Lie group G which admits a smooth morphism ρ : G → AutV . Prove the above lemma. Let (V. Exercise 11. and any g ∈ G. and the above commutativity simply means that c is invariant under this action. View the Clifford multiplication c : V → End (E ) as an element c ∈ V ∗ ⊗ E ∗ ⊗ E .. Tautologically.32. Lemma 11. E g c(v ) E K ME Kg ME (11. 1 In practice one requires a little more namely that ∇M is precisely the Levi-Civita connection on T ∗ M .31. using the complexified algebra Cl(V ) instead of Cl(V ). To produce all the data (a)-(d) all we now need is a principal G-bundle P → M such that the associated bundle P ×ρ V is isomorphic with T ∗ M . and an arbitrary connection ∇ on P . g ) denote the standard fiber of T ∗ M .1. The group G acts on this tensor product. i.1.. (This may not be always feasible due to possible topological obstructions). and a representation µ : G → GL(E ). which we assume is orthogonal. Consider an arbitrary principal G-bundle P → X . Assume that there exists e0 ∈ E such that ρ(g )e0 = e0 . for every v ∈ V . ∀α ∈ Ω1 (M ). With respect to these connections. and we also need to require that µ is an orthogonal/unitary representation. DIRAC OPERATORS Apparently. happen in many geometrically interesting situations.1.e. Let (V. ∀g ∈ SO(V ). the map ρx Cl(V ) u −→ α(x)ux−1 ∈ Cl(V ) . g ) be a finite dimensional Euclidean space. Spin. These have the form ϕx : Cl(V ) → Cl(V ) u → ϕx (u) = x · u · x−1 . we can now build our bundle of Clifford modules starting from the principal SO bundle of its oriented orthonormal coframes. where Cl(V ) denotes the group of invertible elements. Using the universality property of Clifford algebras we deduce that each g ∈ SO(V ) induces an automorphism of Cl(V ) preserving V → Cl(V ). g ) be an oriented Euclidean space.2).6 Spin Let (V.33. The corresponding Dirac operator is the Hodge-DeRham operator. where α : Cl(V ) → Cl(V ) denotes the involutive automorphism of Cl(V ) defining its Z2 -grading. Example 11. ∀u ∈ Cl(V ).1. where x is some invertible element in Cl(V ). In general. g ) is an oriented Riemann manifold. such that c(g · v )(ω ) = g · (c(v )(g −1 · ω )). These are the spin groups Spin(n) and its “complexification” Spinc (n). and Spinc . it defines an orthogonal representation on the canonical Clifford module c : Cl(V ) → End (Λ• V ). The very pleasant surprise is that all these. It turns out that all the groups one needs to build Dirac operators are these three classes: SO.1. §11. It is thus natural to determine the subgroup x ∈ Cl(V ) . We will instead try to understand the Clifford group Γ(V ) = x ∈ Cl (V ) α(x) · V · x−1 ⊂ V .512 CHAPTER 11. As connections we can now pick the Levi-Civita connection and its associates. The next two sections discuss two important examples of Lie groups with the above properties. The candidates for the Lie groups with the properties outlined in the previous subsection will be sought for amongst subgroups of interior automorphisms. The group of automorphisms of the Clifford algebra Cl(V ) contains a very rich subgroup consisting of the interior ones. and even more.. If (M. Moreover. v ∈ V. i. the chances that a Lie group G with the above properties exists are very slim. ω ∈ Λ• V. SO(V ) satisfies the equivariance property (11. x · V · x−1 ⊂ V . 1. The element x decomposes into even/odd components x = x0 + x1 .36. and this concludes the proof of the proposition. To establish the opposite inclusion choose an orthonormal basis (ei ) of V . ·) ⊂ Cl(V ) . Its impact is mainly on the æsthetics of the presentation which we borrowed from the elegant paper [7]. We let the reader check the following elementary fact. Proposition 11. Since [·.35. the Clifford group Γ(V ) comes equipped with a tautological representation ρ : Γ(V ) → GL(V ) ρ(x) : v → α(x) · v · x−1 .1. Since this should happen for every i this means x1 = 0. if x ∈ Γ(V ). and the condition α(x)ei x−1 = ei translates into (x0 − x1 )ei = ei (x0 + x1 ). THE STRUCTURE OF DIRAC OPERATORS 513 is not an automorphism of algebras. the even part x0 lies in the center of Cl(V ). ∀i. In particular. In terms of the s-commutator. then x1 should be a linear combination of elementary monomials ej1 · · · ejs none of which containing ei as a factor. (a) N (Γ(V )) ⊂ R∗ . Note that since {x1 . Proposition 11. ei } = 0.1. ∀i. x]s = 0. Clearly R∗ ⊂ ker ρ. ei } = 0. the above two equalities can be written as one [ei . but as we will see by the end of this subsection. ker ρ = (R∗ .34. x] is a superderivation of Cl(V ) we conclude that [y. The spinorial norm is the map N : Cl(V ) → Cl(V ) N (x) = x x. Proof. Definition 11. then ρx = ±ϕx . and hence a posteriori this alteration has no impact. and let x ∈ ker ρ. x]s = 0 ∀i.11. . (b) The map N : Γ(V ) → R∗ is a group morphism. The center of the Clifford algebra is the field of scalars R ⊂ Cl(V ). By construction. Lemma 11. ei ] = x0 ei − ei x0 = x1 ei + ei x1 = {x1 . This is equivalent with the following two conditions [x0 .1.1.37. ∀y ∈ Cl(V ). we deduce that the only possible choice of signs in the above equality is +. ∀v ∈ V . we deduce that α({α(x) · v · x−1 } ) = −α(x) · v · x−1 ∈ V. y ∈ Γ(V ). we have N (v ) = −|v |2 g . Hence x† ∈ Γ(V ). Proof. In particular. We first prove that x ∈ Γ(V ). Hence. y = α(x) · v · x−1 is an element in V . we deduce N (Γ(V )) ⊂ Γ(V ).514 CHAPTER 11.38. the transformation ρ(x) of V is orthogonal. (b) & (c) We only need to show ρ(Γ(V )) = O(V ). For every x ∈ Γ(V ) we get N (ρ(x)(v )) = N (α(x)vx−1 ) = N (α(x))N (v )N (x−1 ) = N (α(x))N (x)−1 N (v ). ρ . since α(Γ(V )) ⊂ Γ(V ). Since both N (v ) and N (ρx (v )) are negative numbers. Theorem 11. Let x ∈ Γ(V ).e. x2 = α(x2 ) = α(x)2 so we conclude that N (x)2 = N (α(x))2 . we deduce α( (x )−1 · v · α(x ) ) ∈ V. which implies y † = α(y ) = −y . every element of Γ(V ) is Z2 -homogeneous.1. ∀v ∈ V . (x )−1 ∈ Γ(V ). (a) Note that. (c) Every x ∈ Γ(V ) can be written (in a non-unique way) as a product x = v1 · · · vk . DIRAC OPERATORS Proof. Using the fact that x → x is an anti-automorphism. or purely odd. N (ρ(x)(v )) = ±N (v ). (b) If x. ∀x ∈ Γ(V ). that is. it is either purely even. (a) For every x ∈ Γ(V ). For any v ∈ V we have α(N (x)) · v · (N (x))−1 = α(x x) · v · (x x)−1 = α(x ) · {α(x) · v · x−1 } · (x )−1 . vj ∈ V . (b) There exists a short exact sequence of groups 1 → R∗ → Γ(V ) → O(V ) → 1. On the other hand. This shows that ρ(x) is an orthogonal transformation. For x ∈ V with |x|g = 1 we have α(x) = −x = x−1 . then N (x · y ) = (xy ) (xy ) = y x xy = y N (x)y = N (x)y y = N (x)N (y ). In particular. so that. α((x )−1 ) · v · x ∈ V. i. Since α(x)vx−1 ∈ V . Hence α(N (x)) · v · (N (x))−1 = −α(x ) · y † · (x )−1 = −α(x ) · α(x )−1 · v † · x · (x )−1 = −v † = v. This means N (x) ∈ ker ρ = R∗ . On the other hand.. . . . ρ T ∈ O(V ). then we deduce ρ(x)v = −λx + u. and simply connected if dim V ≥ 3. Proposition 11. the group Spin(V ) is connected if dim V ≥ 2. Incidentally. this also establishes (c). In particular.39.1. ρ(x) is the orthogonal reflection in the hyperplane through origin which is perpendicular to x. THE STRUCTURE OF DIRAC OPERATORS If we decompose v ∈ V as λx + u.29 and [44. Observe that Spin(V ) is a closed subgroup of the Lie group GL( Cl(V ) ). Set Pin(V ) := and Spin(V ) := x ∈ Γ(V ). Proposition 11. There exist short exact sequences 1 → Z2 → Pin(V ) → O(V ) → 1 1 → Z2 → Spin(V ) → SO(V ) → 1. Suppose V is a finite dimensional Euclidean space.1. Exercise 11. we have a short exact sequence 1 → R∗ → Γ0 (V ) → SO(V ) → 1. vk ∈ V such that T = ρ(v1 ) · · · ρ(vk ). N (x) = 1 = Pin(V ) ∩ Γ0 (V ). we deduce that for each T ∈ O(V ) we can find v1 .1.2.41.40. 515 In other words. when dim V ≥ 3. . Set Γ0 (V ) := Γ(V ) ∩ Cleven .11.1. Definition 11. In particular. this shows that Spin(V ) is a compact topological group. 95]) that Spin(V ) is in fact a Lie group. The morphism ρ : Spin(V ) → SO(V ) is a covering map.1. vi ∈ V. . det T = 1 . Prove that any orthogonal operator T ∈ O(V ) is a product of at most dim V reflections. where λ ∈ R and u ⊥ x. and the map Spin(V ) → Cl(V ) is a smooth embedding. 2k }. k ≥ 0. Since any orthogonal transformation of V is a composition of such reflections (Exercise ). . ∀i = 1. |N (x)| = 1 . The proof of the following result is left to the reader. |vi | = 1. Moreover. The results we proved so far show that Spin(V ) can be alternatively described by the following “friendlier” equality Spin(V ) = {v1 · · · v2k . Spin(V ) is the universal cover of SO(V ).42. . Note that ρ(Γ0 (V )) ⊂ SO(V ) = Hence. This implies (see Remark 1. . x ∈ Γ(V ). and moreover γ (0) = −1. 1] → Spin(V ). ˆ(t) The correspondence t → v (t) is a smooth. v (t)}. DIRAC OPERATORS Proof. then the path γ (t) = (u cos t + v sin t)(u cos t − v sin t). The fact that ρ : Spin(V ) → SO(V ) is a covering map is an elementary consequence of the following simple observations. closed path u ˆ : [0. γ (π/2) = 1. 0 ≤ t ≤ π/2 lies inside Spin(V ). the fact that Spin(V ) is connected would follow if we showed that any points in the same fiber of ρ can be connected by arcs. u ˆ(0) = u ˆ(1) = 1. set u(t) := ρu ∈ SO ( V ). because |u cos t ± v sin t| = 1. (iii) The subgroup ker ρ is discrete. Since SO(V ) is connected if dim V ≥ 2. then Spin(V ) is isomorphic to the group of unit quaternions (see Example 11. and U is a subspace. we argue by induction on dim V . (i) The map ρ is a group morphism . |f0 (t)| . v ∈ V are such that |u| = |v | = 1. f0 (t) := v (t) − (v (t) • x)x. and define v (t) := u(t)e ∈ V . f (t) := 1 f0 (t). In other words. Spin(U ) → Spin(V ). Note that if V is an Euclidean space.516 CHAPTER 11.1. (ii) The map ρ is continuous. Using Sard’s theorem we deduce that there exists a vector x ∈ S dim V −1 . such that the diagrams below are commutative Spin(V ) K ρ i Spin(U ) O ρ M SO(V ) O . it suffices to show that if dim V > 3. such that v (t) = ±x. closed path on the unit sphere S dim V −1 ⊂ V . Denote by f (t) the vector obtained by applying the Gramm-Schmidt orthonormalization process to the ordered linearly independent set {x. Fix a unit vector e ∈ V . It suffices to look at the fiber ρ−1 (1) = {−1. the vectors v (t) and x are linearly independent. then for every smooth. and proper. 1]. If u. it is homeomorphic to the sphere S 3 which is simply connected. In particular. the natural inclusion U → V induces morphisms Cl( U ) → Cl(V ). If dim V = 3. and in particular. 1}. M SO(U ) K Spin(V ) K O M Cl(V ) O O M Cl(U ) Spin(U ) K Hence. there exists a codimension one subspace U → V such that u ˆ is homotopic to a loop in Spin(U ).55). ∀t ∈ [0. u ⊥ v . SO(U ) → SO(V ). To prove that Spin(V ) is simply connected. for any t. σ (t)−1 (t) = cos θ − xf sin θ. ρσ(t) x = v (t). 517 x cos θ θ − f sin 2 2 −x cos θ θ − f sin . The next result offers a more precise description.1.11. We can find a smooth map θ : [0. Set σ (t) := cos θ + xf sin θ ∈ Cl(V ). The Lie bracket is given by the commutator in Cl(V ). 1] define σs (t) := u ˆs (t) = cos sθ + xf sin sθ. v (t) = x cos 2θ(t) + f (t) sin 2θ(t). 1]. . On the other hand. and as such we can identify its Lie algebra spin(V ) with a linear subspace of Cl(V ). such that. Proposition 11. θ(0) = θ(1) = 0. ∀t ∈ [0. 2 2 Observe that for s = 0 we have u ˆ0 (t) = u ˆ(t). 2π ). and σ (t)xσ (t)−1 = (x cos θ + f sin θ)(cos θ − xf sin θ) = x cos 2θ + f sin 2θ = e0 . If we denote by U the orthogonal complement of x in V . We can view Spin(V ) as a submanifold of Cl(V ). For every s ∈ [0. and 1 us (t) := ρ− σs (t) u(t)ρσs (t) ∈ SO (V ). Consider the quantization map q : Λ∗ V → Cl(V ). 1] → [0. On the other hand.43. Note that σ (t) = so that σ (t) ∈ Spin(V ). In other words. for s = 1 we have 1 u1 (t)x = ρ− σs (t) u(t)ρσs (t) x = ρσs (t)−1 v (t) = x. u ˆs (t) := σs (t)−1 u ˆ(t)σs (t). THE STRUCTURE OF DIRAC OPERATORS where we denoted the inner product in V by •. we deduce that u1 (t) ∈ Spin(U ). Then spin(V ) = q(Λ2 V ).1. DIRAC OPERATORS Proof. where x0 ∈ Cl(V )even . The Lie bracket of x ˙ (0) and y ˙ (0) is then found (using the Exercise 3. y : (−ε. en . . Since dim spin(V ) = dim so(V ) = dim Λ2 V we conclude that the above inclusion is in fact an equality of vector spaces. For every x ∈ spin(V ). where the above bracket is the commutator of x ˙ (0) and y ˙ (0) viewed as elements in the associative algebra Cl(V ). we deduce 1 1 e1 x1 = [e1 . To get a more explicit picture of the induced morphism of Lie algebras ρ∗ : spin(V ) → so(V ). Since [x0 .21) from the equality x(t)y (t)x(t)−1 y (t)−1 = 1 + [x ˙ (0). . . en } of V . and {x1 . xv − vx ∈ V. Since the elements of Γ(V ) are either purely even. . y ˙ (0)]t2 + O(t3 ) (as t → 0).1. Now consider two smooth paths x. 2 2 In particular this means x1 ∈ R ⊕ V ∈ Cl(V ). . We can decompose x as x = x0 + e1 x1 . we deduce that the tangent space at 1 ∈ Γ(V ) is a subspace of E= x ∈ Cleven (V ). .518 CHAPTER 11. and let e1 . and x1 ∈ Cl(V )odd are linear combinations of monomials involving only the vectors e2 . . e1 } = 0. T1 Γ0 (V ) ⊂ R ⊕ q(Λ2 V ). we fix an orientation on V . The tangent space to Spin(V ) satisfies a further restriction obtained by differentiating the condition N (x) = 1. . or purely odd. This gives spin(V ) ⊂ {x ∈ R ⊕ q(Λ2 V ) . ∀v ∈ V . x + x = 0} = q(Λ2 V ). Fix x ∈ E . (n = dim V ). ε) → Spin(V ) such that x(0) = y (0) = 1. and then choose a positively oriented orthonormal basis {e1 . e1 ] = 0. . Thus. . en be an orthonormal basis of V . The group Γ(V ) is a Lie group as a closed subgroup of the group of linear transformations of Cl(V ). x1 ] = [e1 . Repeating the same argument with every vector ei we deduce that E ⊂ R ⊕ span{ei · ej . . x] ∈ V. If x= i<j xij ei ej . the element ρ∗ (x) ∈ spin(V ) acts on V according to ρ∗ (x)v = x · v − v · x. . . 1 ≤ i < j ≤ dim V } = R ⊕ q(Λ2 V ). . this shows ρ∗ is an isomorphism.11. where σ : Cl(V ) → Λ• V is the symbol map. THE STRUCTURE OF DIRAC OPERATORS then ρ∗ (x)ej = −2 i 519 xij ei . ei ej → ei ∧ ej . (xij = −xji ). In particular. ρ∗ (x)ej )ei ∧ ej = 2 i<j xij ei ∧ ej = 2σ (x) ∈ Λ2 V. If we identify as usual so(V ) ∼ = Λ2 V by so(n) A → ωA = i<j g (Aei . Note the following often confusing fact. ej )ei ∧ ej = − i<j g (ei . Aej )ei ∧ ej .1. and that the map ρ : Spin(V ) → SO(V ) is also a submersion. then the Lie algebra morphism ρ∗ takes the form ωρ∗ (x) = − i<j g (ei . (n = dim V ). Example 11. and in many concrete problems it makes a world of difference.1. A word of warning. They are called the positive/negative complex spin representations. If A ∈ so(V ) has the matrix description Aej = i aij ei with respect to an oriented orthonormal basis {e1 . (The complex spinor representations). en }. . . Assume next that dim V is odd. g ). The orientation on V induces a Z2 -grading on the spinor module S = S+ ⊕ S− . nonisomorphic complex Spin(V )-modules. Assume first that dim V is even. Any Clifford module φ : Cl(V ) → EndK (E ) defines by restriction a representation φ : Spin(V ) → GLK (E ).j (11.1. i. . The (complex) representation theory described in the previous sections can be used to determine the representations of Spin(V ). .44. then the 2-form associated to A has the form ωA = − aij ei ∧ ej . Since Spin(V ) ⊂ Cleven (V ) we deduce that each of the spinor spaces S± is a representation space for Spin(V ). Consider a finite dimensional oriented Euclidean space (V. i<j so that 1 ρ− ∗ (A) = − 1 2 aij ei ej = − i<j 1 4 aij ei ej .3) The above negative sign is essential. The spinor module S(V ) S(V ) = S+ (V ) ⊕ S− (V ). . They are in fact irreducible. 5. For each positive integer n we will denote by Sn the Spin(n)-module defined by S2k ∼ = S+ (R2k ) ⊕ S− (R2k ) if n = 2k. and N (x) = x · x.0 V . and S2k+1 ∼ = S+ (R2k+1 ) ∼ =Spin(2k+1) S− (R2k+1 ) if n = 2k + 1.0 (V ). Prove the above proposition.0 V ). α(x) · v · x−1 ∈ V ∀v ∈ V }. v = Jv. The complex Clifford group Γc (V ) is defined by Γc (V ) = {x ∈ Cl(V ) . Hence ∆(q ) = ∆+ (q ) ⊕ ∆− (q ) where ∆+/− (q ) acts on Λeven/odd. e2n+1 of V then the Clifford multiplication by ω = e1 · · · e2n+1 intertwines the ± components.520 CHAPTER 11. Let (V. The module Sn will be called the fundamental complex spinor module of Spin(n).46.0 V are non-isomorphic Spin(V )-modules. The canonical involutive automorphism α : Cln → Cln extends by complex linearity to an automorphism of Cln .48.1.1. Exercise 11.47. This is a Spin(V ) isomorphism since ω lies in the center of Cl(V ). . DIRAC OPERATORS is not irreducible as a Cl(V ) modules. Let φ : Cl(V ) → End (E ) be a selfadjoint Clifford module. g ) be an Euclidean space.  Convention. Then the induced representation of Spin(V ) is orthogonal (unitary). As in the real case set x† := α(x) . while the anti-automorphism : Cln → Cln extends to Cln according to the rule (v ⊗ z ) = v ⊗ z. · · · . Note that q ∈ Spin(V ) so that ∆(q ) preserves the parities when acting on Λ∗.7 Spinc The considerations in the previous case have a natural extension to the complexified Clifford algebra Cln .1.45.1. Proposition 11. Compute tr (∆+/− (q )) and then conclude that Λeven. Choose u ∈ V such that |u|g = 1 and then for each t ∈ R set q = q (t) = cos t + u · v sin t. §11. Let (V. Exercise 11.1./odd. Prove that the group Spin(V ) satisfies all the conditions discussed in Subsection 10. If we pick an oriented orthonormal basis e1 . Each of the modules S± (V ) is a representation space for Spin(V ). Thus we have an explicit isomorphism ∆ : Cl(V ) → End (Λ•. g ) be a 2k -dimensional Euclidean space and J : V → V a complex structure compatible with the metric g . Exercise 11. They are irreducible but also isomorphic as Spin(V )-modules.1. Define Pinc (V ) = {x ∈ Γc (V ) . any any isomorphism mCl(V ) ∼ = EndC (S(V )) induces a complex unitary representation Spinc (V ) → Aut (S(V )) called the complex spinorial representation of Spinc . Then. As in the real case.11. |N (x)| = 1}. THE STRUCTURE OF DIRAC OPERATORS 521 We denote by ρc the tautological representation ρc : Γc (V ) → GL(V.50. Proposition 11. The inclusions Pin(V ) ⊂ Cl(V ).51. It is not irreducible since. −z ) ∀(x.1. Proof. z ) ∈ Pin(V ) × S 1 . As in the real case one can check that ρc (Γc (V )) = O(V ). Exercise 11.5.1. S(V ) splits into a direct sum of irreducible representations S± (V ).1. Arguing as in the above corollary we deduce Spinc (V ) ∼ = (Spin(V ) × S 1 )/ ∼∼ = (Spin(V ) × S 1 )/Z2 . by definition. the space End (S(V )) has a natural superstructure and. z ) ∼ (−x. The image of this morphism lies obviously in Γc (V ) ∩ {|N | = 1} so that (Pin(V ) × S 1 )/ ∼ can be viewed as a subgroup of Pinc (V ). ker ρc = C∗ . once we fix an orientation on V .1. S 1 ⊂ C induce an inclusion (Pin(V ) × S 1 )/ ∼→ Cl(V ). R). Prove Spinc (V ) satisfies all the conditions outlined in Subsection 10. . Assume now dim V is even. Fix a complex structure J on V . Corollary 11. where “∼” is the equivalence relation (x. The sought for isomorphism now follows from the exact sequence 1 → S 1 → (Pin(V ) × S 1 )/ ∼→ O(V ) → 1. The spinorial norm N (x) determines a group morphism N : Γc (V ) → C∗ . There exists a natural isomorphism Pinc (V ) ∼ = (Pin(V ) × S 1 )/ ∼. Spinc acts through even automorphism. This complex structure determines two things. We define Spinc (V ) as the inverse image of SO(V ) via the morphism ρc : Pinc (V ) → O(V ). There exists a short exact sequence 1 → S 1 → Pinc (V ) → O(V ) → 1. We let the reader check the following result.49.1. (ii) A natural subgroup CHAPTER 11.52. = ±. Let ω ∈ U (V ). J ) −→ Spinc (V ) such that the diagram below is commutative.2. We have to check two things.35 of Subsection 6. 1] → Spin(V ) such that γ ˜ (0) = . (ii) The map σ is a smooth group morphism. [T. The map det : U (V ) → S 1 induces an isomorphism between the fundamental groups (see Exercise 6. J ) = {T ∈ SO(V ) . and λ : [0. 1] → U (V ) is a different path connecting  to ω . ν (t)ei = ei ∀i ≥ 2. There exists a natural group morphism U (V. Such a loop will be homotopic to γ ∗ η −1 in U (V ). ρ We can identify as the holonomy of the covering Spin(V ) → SO(V ) along the loop γ ∗ η − which goes from  to ω along γ . δ (1)) in Spinc (V ). in describing the holonomy it suffices to replace the loop γ ∗ η − ⊂ U (V ) by any loop ν (t) such that det ν (t) = det(γ ∗ η − ) = ∆(t) ∈ S 1 .2. J ) ı M Spin (V )   Kρ ξJ c J c SO(V ) Proof. . z → z 2 . 1] → S 1 is such that λ(0) = 1 and λ(t) = det η (t)2 . Denote by ıJ : U (V. J ) → SO(V ) the inclusion map.1. To prove (i). we can find a path t → δ (t) ∈ S 1 such that δ (0) = 1 and δ 2 (t) = det γ (t). Define ξ (ω ) to be the image of (˜ γ (1). (i) The map ξ is well defined. then (˜ η (1). DIRAC OPERATORS U (V.522 (i) A canonical orientation on V . J Proposition 11. δ (1)) in Spinc (V ). J ] = 0} ⊂ SO(V ). we need to show that if η : [0. Hence γ ˜ (1) = η ˜(1). Hence. ξ U (V. 1] → U (V ) connecting  to ω . Select ν (t) of the form ν (t)e1 = ∆(t)e1 . λ(1)) = (˜ γ (1). and then back to  along η − (t) = η (1 − t). Using the double cover S 1 → S 1 . The elements γ ˜ (1) and η ˜(1) lie in the same fiber of the covering Spin(V ) → SO(V ) so that they differ by an element in ker ρ. Via the inclusion U (V ) → SO(V ) we may regard γ as a path in SO(V ). and thus in SO(V ) as well. As such. it admits a unique lift γ ˜ : [0. and consider a path γ : [0.5). . Under this identification the operation x → x coincides with the conjugation in H x = a + bi + cj + dk → x = a − bi − cj − dk.  where θ : [0. e2 } in R2 . The Clifford algebra Cl3 is isomorphic. We leave the reader to check that ξ is indeed a smooth morphism of groups. This proves ξ is well defined.1. e2 → j . . More relevant is the isomorphism Cleven = 3 ∼ Cl2 = H given by 1 → 1.8 Low dimensional examples In low dimensions the objects discussed in the previous subsections can be given more suggestive interpretations. .1. e2 . a2 + b2 = 1} ∼ = S1. e1 → i. where i. Set fi = Jei . j .1. J ). The Z2 -grading is Re C ⊕ Im C. . . (The case n = 2). e3 e1 → k. The lift of ν (t) to Spin(V ) has the form ν ˜(t) = cos θ(t) θ(t) − e1 f1 sin . and k are the imaginary units in H. ) the operator ν (t) (viewed as an element of SO(V )) has the matrix description   cos θ(t) − sin θ(t) · · ·  sin θ(t) cos θ(t) · · ·   . (The case n = 3). The isomorphism is given by 1 → 1. This can be seen by choosing an orthonormal basis {e1 . b ∈ R. z2 .11. orthonormal basis of (V. . f1 . 1] → R is a continuous map such that ∆(t) = eiθ(t) . The group Spin(1) is isomorphic with Z2 . The Clifford algebra Cl2 is isomorphic with the algebra of quaternions H. Example 11. §11.1. 2 2 We see that the holonomy defined by ν ˜(t) is nontrivial if and only if the holonomy of the loop t → δ (t) in the double cover S 1 → S 1 is nontrivial. With respect to the real basis (e1 . f2 . Note that Spin(2) = {a + bk . The natural map Spin(1) → SO(2) ∼ = S 1 takes the form eiθ → e2iθ . e3 } is an orthonormal basis in R3 . e1 e2 → i. The Clifford algebra Cl1 is isomorphic with the field of complex numbers C. THE STRUCTURE OF DIRAC OPERATORS 523 where (ei ) is a complex. This means that δ (1) and λ(1) differ by the same element of Z2 as γ ˜ (1) and η ˜(1). . Example 11.1. a. as an ∼ ungraded algebra. e2 . where {e1 .54.55. . e2 e3 → j . . Example 11. In this subsection we will describe some of these interpretations. e1 e2 → k.53. to the direct sum H ⊕ H. (The case n = 1). and the diagram below is commutative. The isomorphism Spin(3) ∼ = SU (2) can be visualized by writing each q = a + bi + cj + dk as q = u + j v.. Moreover. the correspondences e1 e2 → Si ⊕ Si ∈ End (C2 ) ⊕ End (C2 ) . a simple computation shows that xR3 x−1 ⊂ R3 . the matrix Sq is in SU (2). Define T : H → C2 by q = u + jv → T q = u v .1. M C2 L K T S K2 MC H H T q q In other words. On the other hand. any x ∈ Cleven \{0} is invertible.2.524 CHAPTER 11. Thus.e. the representation Spin(3) q → Lq ∈ GLC (H) of Spin(3) is isomorphic with the tautological representation of SU (2) on C2 . DIRAC OPERATORS In particular. so that 3 Γ0 (R3 ) ∼ = H \ {0}. ∀q ∈ Spin(3) ∼ = S 3 . and 3 x−1 = 1 x. Hence Spin(3) ∼ = { x ∈ H. For each quaternion q ∈ H we denote by Lq (respectively Rq ) the left (respectively right) multiplication.2. To a quaternion q = u + j v one associates the 2 × 2 complex matrix Sq = u −v ¯ v u ¯ ∈ SU (2). The right multiplication by i defines a complex structure on H. the spinorial norm coincides with the usual norm on H N (a + bi + cj + dk) = a2 + b2 + c2 + d2 . ∗ Note that Sq ¯ = Sq . ∀x ∈ Cleven \{0}. N (x) Moreover. v = (c − di) ∈ C. i. A simple computation shows that T (Ri q ) = iT q . The natural map Spin(3) → SO(3) is precisely the map described in the Exercise 6.8 of Subsection 6. T is a complex linear map. ∀q ∈ H. |x| = 1 } ∼ = SU (2). u = a + bi. The Clifford algebra Cl4 can be realized as the algebra of 2 × 2 matrices with entries in H. THE STRUCTURE OF DIRAC OPERATORS e2 e3 → Sj ⊕ Sj ∈ End (C2 ) ⊕ End (C2 ) e3 e1 → Sk ⊕ Sk ∈ End (C2 ) ⊕ End (C2 ) e1 e2 e3 → R = C2 ⊕ (−C2 ) ∈ End (C2 ) ⊕ End (C2 ) 525 extend to an isomorphism of algebras Cl3 → End (C2 ) ⊕ End (C2 ).58. Exercise 11. The left multiplication by i introduces a different complex structure on H. This plays an important part in the formulation of the recently introduced Seiberg-Witten equations (see [102]). and (ii) it is isomorphic with complex spinorial representation described by the left multiplication. Ri } = 0. A dimension count concludes it must also be surjective. We let the reader check this morphism is also injective.1. we deduce that Rj defines an isomorphism of Spin(3)-modules Rj : S3 → S3 . Prove the representation Spin(3) q → (Rq : H → H) is (i) complex with respect to the above introduced complex structure on H. This implies there exists a Spin(3)-invariant bilinear map β : S3 × S3 → C. (The case n = 4). This proves that the tautological representation of SU (2) is precisely the complex spinorial representation S3 .57. Example 11. Lq ] = {Rj . Spin(4) ∼ = SU (2) × SU (2). Proposition 11.1.1.11. .1. and this correspondence extends to a bona-fide morphism of algebras Cl4 → M2 (H).56. From the equalities [Rj . To describe this isomorphism we have to start from a natural embedding R4 → M2 (H) given by the correspondence H∼ = R4 x→ 0 −x x 0 A simple computation shows that the conditions in the universality property of a Clifford algebra are satisfied. q2 : H → H. |p| = |q | = 1 ⊂ M2 (H). e3 . T defines an (obviously smooth) group morphism T : SU (2) × SU (2) → SO(4).3). Thus each pair (q1 . so that each Tq1 . so that its range must contain an entire neighborhood of  ∈ SO(4). e4 } be an oriented orthonormal basis of R4 . Using the identification Cl4 = to the subgroup diag (p. ∗2 = . Let ∗ denote the Hodge operator defined by the canonical metric and the above chosen orientation.q2 x = q1 xq 2 . all the operators Tq1 . η3 . q ∈ H. Since the range of T is closed (verify this!) we conclude that T must be onto because the closure of the subgroup (algebraically) generated by an open set in a connected Lie group coincides with the group itself (see Subsection 1. q ). where 1 1 ± ± = √ (e1 ∧ e3 ± e4 ∧ e2 ). so that T is 2 : 1.q2 belong to the component of O(4) containing . Exercise 11. The above result shows that so(4) ∼ = spin(4) ∼ = su(2) ⊕ su(2) ∼ = so(3) ⊕ so(3).526 CHAPTER 11. Let {e1 . Again we think of SU (2) as the group of unit quaternions. In order to prove T is a double cover it suffices to show it is onto. η1 = √ (e1 ∧ e2 ± e3 ∧ e4 ). 2 (b) Show that the above splitting of Λ2 R4 corresponds to the splitting so(4) = so(3) ⊕ so(3) under the natural identification Λ2 R4 ∼ = so(4). Note that ∗ : Λ 2 R4 → Λ 2 R4 is involutive. DIRAC OPERATORS Proof. We will use the description of Spin(4) as the universal (double-cover) of SO(4) so we will explicitly produce a smooth 2 : 1 group morphism SU (2) × SU (2) → SO(4). x → Tq1 . Since SU (2) × SU (2) is connected. This follows easily by noticing T is a submersion (verify this! ). η2 2 2 1 ± η3 = √ (e1 ∧ e4 ± e2 ∧ e3 ).1.1.59. η2 . q2 ) ∈ SU (2) × SU (2) defines a real linear map Tq1 .2. p. Note that ker T = {1.60. so that we can split Λ2 into the ±1 eigenspaces of ∗ 4 2 4 Λ2 R4 = Λ2 + R ⊕ Λ− R .e. −1}.q2 is an orthogonal transformation of H. i. Clearly |x| = |q1 | · |x| · |q2 | = |Tq1 . e2 . ∼ M2 (H) show that Spin(4) corresponds Exercise 11..q2 x| ∀x ∈ H. . (a) Show that ± ± ± Λ2 ± = spanR η1 . Identify C2 in the obvious way with Λ1 C2 Λodd C2 . Show that for any η ∈ Λ2 + (V ). Exercise 11. We start from the morphism of R-algebras u −v ¯ H x = u + j v → Sx = v u This extends by complexification to an isomorphism of C-algebras H ⊗R C ∼ = M2 (C).1.61. We deduce the S± 4 representations of Spin(4) = SU (2) × SU (2) are given by S+ 4 : SU (2) × SU (2) S− 4 : SU (2) × SU (2) (p. (11. Exercise 11. C). oriented Euclidean space. and with Λeven C2 via the map e1 → 1 ∈ Λ0 C2 . C).1.4). e2 → e1 ∧ e2 .4) Note that the chirality operator Γ = −e1 e2 e3 e4 is represented by the canonical involution Γ → C2 ⊕ (−C2 ).11. Denote by q : Λ• V → Cl(V ) the quantization map. The space R4 ∼ = H has a canonical complex structure defined by Ri which defines (following the prescriptions in §10.1. q ) → q ∈ GL(2. q ) → p ∈ GL(2. EndH (H ⊕ H) ⊗R C ∼ = M2 (H) ⊗R C ∼ = EndC (C2 ⊕ C2 ). Show that under these identifications we have c(x) = Tx ∀x ∈ R4 ∼ = (H.1.62.3) an isomorphism c : Cl4 → End (Λ• C2 ). (Verify this!) We now use this isomorphism to achieve the identification.1. Exactly as in the case of Spin(3). The embedding R4 → Cl4 now takes the form H∼ = R4 x → Tx = 0 −Sx ¯ Sx 0 ∈ End (C2 ⊕ C2 ). where Tx is the odd endomorphism of C2 ⊕ C2 defined in (11. the image ∆ ◦ q(η ) ∈ End (S(V )) is an endomorphism + of the form T ⊕ 0 ∈ End (S (V )) ⊕ End (S− (V )). (p. and fix an isomorphism ∆ : Cl(V ) → End (S(V )) of Z2 -graded algebras. Let V be a 4-dimensional.1. . Ri ) ∼ = C2 . THE STRUCTURE OF DIRAC OPERATORS 527 To obtain an explicit realization of the complex spinorial representations S± 4 we need to describe a concrete realization of the complexification Cl4 . these representations can be given quaternionic descriptions. X ∈ Vect (M ).e. Note that if (hi . Denote by D the (possible empty) family of Dirac structure on E . (a2) For any α ∈ Ω1 (M ). and a Clifford connection. g ). such that h = h0 ⊕ h1 . i. (This condition means that the Clifford multiplication is covariant constant). (a) A Dirac structure on E is a pair (h. and f ∈ C ∞ (M ). Then there exist Dirac structures on E . Denote by V a 4-dimensional oriented Euclidean space.1. + (a) Show that the representation S+ 4 ⊗ S4 of Spin(4) descends to a representation of SO (4) and moreover S+ (V ) ⊗C S+ (V ) ∼ = (Λ0 (V ) ⊕ Λ2 + (V )) ⊗R C as SO(4) representations.65.. f ∇1 + (1 − f )∇2 ) ∈ D. ∇i ) ∈ D. We denote by c : Ω1 (M ) → End(E ) the Clifford multiplication. g ). . The special characteristics of the most important concrete examples are studied in some detail in the following section. DIRAC OPERATORS Exercise 11. i = 1. The next result addresses the fundamental consistency question: do there exist Dirac bundles? Proposition 11. We will touch only the general aspects. where hi (respectively ∇i ) is a metric (respectively a metric connection) on Ei .528 CHAPTER 11. Proof. Show that via the correspondence at (a) the isomorphism φ spans Λ0 (V ).1. (a1) For any α ∈ Ω1 (M ). a connection ∇ on E compatible with h and satisfying the following conditions. Definition 11. ∇) consisting of a Hermitian metric h on E .1. §11. then (f h1 + (1 − f )h2 . In the sequel all Clifford bundles will be assumed to be complex. u ∈ C ∞ (E ) ∇X (c(α)u) = c(∇M X α)u + c(α)(∇X u). Let E → M be a Clifford bundle over the oriented Riemann manifold (M.64. ¯+ (V ) as Spin(4) modules.1. h. Dirac structure) will be called a Dirac bundle. where ∇M denotes the Levi-Civita connection on T ∗ M . the Clifford multiplication by α is a skew-Hermitian endomorphism of E .9 Dirac bundles In this subsection we discuss a distinguished class of Clifford bundles which is both frequently encountered in applications and it is rich in geometric information. A pair (Clifford bundle. ∇) is a Z2 grading of the underlying Clifford structure E = E0 ⊕ E1 .63. (b) A Z2 -grading on a Dirac bundle (E. 2. and ∇ = ∇0 ⊕ ∇1 . (b) Show that S+ (V ) ∼ =S (c)The above isomorphism defines an element φ ∈ S+ (V ) ⊗C S+ (V ). Let E → M be a Clifford bundle over the oriented Riemann manifold (M. We will continue to denote by c the restriction of the Clifford multiplication to Spin(n) → Cln . hE . The straightforward details are left to the reader. The representation theory of the Clifford algebra Cln with n even shows that any Clifford bundle over M must be a twisting SM ⊗ W of the spinor bundle SM . ∇E ) over M . it suffices to consider only the case when M is an open subset of Rn . . ∇ej = ωej = i ωij ⊗ ei . Step 1. n = dim M is even. hW ) equipped with a Hermitian connection ∇W we can construct the tensor product E ⊗ W equipped with the metric hE ⊗ hW and the product connection ∇E ⊗W .e. Constructing general Dirac bundles. c(u† ) = c(u)∗ . ∀u ∈ C ∞ (SM ).. We will distinguish two cases.. The odd case is dealt with similarly using the different representation theory of the Cl2k+1 . but the proof is conceptually identical.1.11. ∀u ∈ Cln . Denote by ω = (ωij ) the connection 1-form of the Levi-Civita connection on T ∗ M with respect to this moving frame. i. and E is a trivial vector bundle. Using the canonical isomorphism ρ∗ : spin(n) → so(n) we define 1 ω ˜ := ρ− ∗ (ω ) = − 1 2 ω ij ei · ej ∈ Ω1 (M ) ⊗ spin(n).1. Fix a Dirac bundle (E. Now instead of one generating model S there are two. For any Hermitian vector bundle (W. These two data define a Dirac structure on E ⊗ W (Exercise 11. Thus. i<j This defines a connection ∇S on the trivial vector bundle SM = Sn × M given by ∇S u = du − 1 2 ωij ⊗ c(ei ) · c(ej )u. i<j The considerations in §10. THE STRUCTURE OF DIRAC OPERATORS 529 This elementary fact shows that the existence of Dirac structures is essentially a local issue: local Dirac structures can be patched-up via partitions of unity. we cannot assume that the metric g is also trivial (Euclidean) since the local obstructions given by the Riemann curvature cannot be removed.5 show that ∇S is indeed a Clifford connection so that SM is a Dirac bundle. On the other hand. The proof will be completed in three steps. A special example. This completes the proof of the proposition when dim M is even. ω ∈ Ω1 (M ) ⊗ so(n). Consider the (complex) spinor module (representation) c : Cln → End (Sn ).e. A.1. Fix a global. We can assume that c is selfadjoint. orthonormal frame (ei ) of T M .68 at the end of this subsection). Step 2. Step 3 Conclusion. i. oriented. and denote by (ej ) its dual coframe. then at x0 the associated Dirac operator can be described as D= c(ei )∇i (∇i = ∇ei ). h. Using the quantization map q : Λ• T ∗ M → Cl(T ∗ M ) we get a section D2 q(F ) ∈ Cl(T ∗ M ) ⊗ End (E ). i i i On the other hand. is a generalized Laplacian. and according to Proposition 10. h. On the other hand.1. Let (E. g ). ∇ a very explicit description with remarkable geometric consequences. Let D be a geometric operator associated to the Dirac bundle (E. and R is the Weitzenb¨ where ∇ ock remainder which is an endo˜ = ∇. For geometric Dirac operators. Denote by (ei ) the dual coframe of (ei ). . Since the connection ∇ is compatible with h and div g (ei ) |x0 = 0. If (h. the Clifford multiplication is covariant constant. c(ei )] |x0 = 0 ∀i. To formulate it we need a little bit of foundational work. By definition. g ). ∇).34 we must have an equality of the form ˜ ∗∇ ˜ + R. and (∇M i e ) |x0 = 0 so we conclude [∇i . Set ei = ∂x . D2 = ∇ ˜ is a connection on E .66. The curvature F (∇) of ∇ is a section of Λ2 T ∗ M ⊗ End (E ). h. and this remainder can be given morphism of E .530 CHAPTER 11. ∇) is a Dirac struci ture on the Clifford bundle E . i ∇ c We deduce D∗ = (∇i )∗ c(ei )∗ . Proposition 11. Since the Clifford multiplication is skew-Hermitian. we deduce that at x0 we have D∗ = ∇i ◦ c(ei ) = [∇i . the Clifford multiplication c : Cl(T ∗ M ) → End (E ) defines a linear map Cl(T ∗ M ) ⊗ End (E ) → End(E ).1. c(ei )∗ = −c(ei ). we deduce (∇i )∗ = −∇i . ∇) be a Dirac bundle over the oriented Riemann manifold (M. ω ⊗ T → c(ω ) ◦ T. The assertion in the above proposition is local so we can work with local orthonormal moving frames. c(ei )] + D. A Dirac operator associated to a Dirac structure is said to be a geometric Dirac operator. There exists a Dirac operator on E canonically associated to this structure given by D = c ◦ ∇ : C ∞ (E ) → C ∞ (T ∗ M ⊗ E ) → C ∞ (E ). g ). Any geometric Dirac operator is formally selfadjoint. DIRAC OPERATORS Denote by (E. and denote by (xi ) a collection of normal coordinates ∂ near x0 . Fix x0 ∈ M . ∇) a Dirac bundle over the oriented Riemann manifold (M. Denote by Cl(T ∗ M M ) → M the bundle of Clifford algebras generated by (T ∗ M. Proof. We denote by DW the corresponding geometric Dirac operator. Such a choice is always possible because the torsion of the Levi-Civita connection is zero. g ) with associated Dirac operator D. q c . g ).1. h ⊗ hW . orthonormal moving frame for T ∗ M . oriented. and c(F ) = i<j c(ei )c(ej )Fij = 1 2 c(ei )c(ej )Fij . h. i. The theorem now follows by observing that (∇∗ ∇) |x = − i ∇2 i |x . Finally. Since [∇i .1.11. THE STRUCTURE OF DIRAC OPERATORS 531 This map associates to the element q(F ) an endomorphism of E which we denote by c(F ). ej ] |x = (∇M ei ) |x = 0. ˆ = ∇ ⊗ 1W + 1E ⊗ ∇W ) defines a Dirac structure on (a) Show that (E ⊗ W. then we can write F (∇) = i<j ei ∧ ej ⊗ Fij . i<j =− i ∇2 i + i<j c(ei )c(ej )[∇i . and then choose an oriented.68. Proof. w ∈ C ∞ (W ). ∇ E ⊗ W in which the Clifford multiplication by α ∈ Ω1 (M ) is defined by c(α)(e ⊗ w) = (c(α)e) ⊗ w. h. where ∇M denotes the Levi-Civita connection. c(ej )] |x = 0 we deduce D2 |x = i. Fix x ∈ M .j c(ei )c(ej )∇i ∇j = − i ∇2 i + i=j c(ei )c(ej )∇i ∇j c(ei )c(ej )Fij (∇). ∇) be a Dirac bundle over the oriented Riemann manifold (M. Consider a Hermitian bundle W → M and a connection ∇W compatible with the Hermitian metric hW . ∇) over the oriented Riemann manifold (M. Then D2 = ∇∗ ∇ + c(F (∇)). Then   D2 |x = i c(ei )∇i  j c(ej )∇j  .1. Let D be the geometric Dirac operator associated to the Dirac bundle (E. ∇j ] = − i ∇2 i + We want to emphasize again the above equalities hold only at x. e ∈ C ∞ (E ). Let (E. (b) Denote by cW (F (∇W )) the endomorphism of E ⊗ W defined by the sequence F (∇W ) ∈ C ∞ (Λ2 T ∗ M ⊗ End (W )) → C ∞ (Cl(T ∗ M ) ⊗ End (W )) → C ∞ (End (E ⊗ W )). local orthonormal moving frame (ei ) of T M near x such that [ei . denote by (ei ) the dual coframe of (ei ). If (ei ) is a local. Exercise 11.j Theorem 11.67 (Bochner-Weitzenb¨ ock)). 2. In fact. · · · . We will denote by ∇ all the Levi-Civita connection on the tensor bundles of M . the Dolbeault operator the spin and spinc Dirac.2 Fundamental examples This section is entirely devoted to the presentation of some fundamental examples of Dirac operators. §11. We will provide more concrete descriptions of the Weitzenb¨ ock remainder presented in Subsection 10. DW =∇ 11.1. where ε denotes the exterior multiplication map. g ) be an oriented Riemann manifold and set ∗ • ∗ Λ• C T M := Λ T M ⊗ C. Proposition 11. then for any ordered multi-index I we have ε∗ (θI ) = j iXj θI .1 The Hodge-DeRham operator Let (M. The pair (g. and θ1 . Thus.2. Thus d + d∗ = ε ◦ ∇ + ∇∗ ◦ ε∗ .9 and show some of its uses in establishing vanishing theorems. We have already seen that the Hodge-DeRham operator d + d∗ : Ω• (M ) → Ω• (M ) is a Dirac operator. More specifically we will discuss the Hodge-DeRham operator. θn is its dual coframe. . DIRAC OPERATORS 2 ˆ ∗∇ ˆ + c(F (∇)) + cW (F (∇W )). In Subsection 4. If X1 . · · · .. Proof.1. we will prove this operator is a geometric Dirac operator.5 we have proved that d can be alternatively described as the composition ∇ ε C ∞ (Λ• T ∗ M ) → C ∞ (T ∗ M ⊗ Λ• T ∗ M ) → C ∞ (Λ• T ∗ M ).g. ∇g ) defines a Dirac structure on the Clifford bundle ΛC T ∗ M and d + d∗ is the associated Dirac operator. When we want to be more specific about which Levi-Civita connection we are using at a given moment we will indicate the bundle it acts on as a superscript. Xn is a local orthonormal frame of T M . for any ω ∈ C ∞ (Λ• T ∗ M ) we have (∇∗ ◦ ε∗ )ω = − k ∇Xk iXk + div g (Xk )iXk ω. E.532 Show that CHAPTER 11. Continue to denote by g the Hermitian metric induced by the metric g on the com∗ g plexification Λ• C T M . ∗ ∇T M is the Levi-Civita connection on T ∗ M .1. For simplicity we continue to denote by Ω• (M ) the space of smooth differential forms on M with complex coefficients. Thus R is a bundle morphism R : C ∞ (T M ) → C ∞ (Λ2 T ∗ M ⊗ T M ).2.2. Y. X.e.. Z ∈ Vect (M ). . and thus d∗ ω |x0 = − k i∂k ∇∂k ω. We have a dual morphism ˜ : C ∞ (T ∗ M ) → C ∞ (Λ2 T ∗ M ⊗ T ∗ M ). we have Xk = ∂ k . i. Lemma 11. With such a choice div g (Xk ) = 0 at x0 . the Clifford multiplication is covariant constant. FUNDAMENTAL EXAMPLES 533 Fortunately. ∂xk where (x ) denotes a collection of normal coordinates near x0 . We want to spend some time elucidating the structure of the Weitzenb¨ ock remainder. Derivating along Y ∈ Vect (M ) we get (∇Y ∇Z α)(X ) = Y · (∇Z α)(X ) − (∇Z α)(∇Y X ) = Y · Z · α(X ) − Y · α(∇Z X ) − Z · α(∇Y X ) − α(∇Z ∇Y X ). ∇ c ˜ is the curvature of the Levi-Civita connection ∇T ∗ M .11. Z ∈ Vect (M ). We leave the reader to verify that the Levi-Civita connection on Λ• T ∗ M is indeed a Clifford connection. where c denotes the usual Clifford multiplication on an exterior algebra. Z )X ) (R ∀α ∈ Ω1 (M ) X. The Levi-Civita connection on T ∗ M is determined by the equalities T (∇Z ∗M M 1 α)(X ) = Z · α(X ) − α(∇T Z X ) ∀α ∈ Ω (M ). i. First of all. Z )α)(X ) = −α(R(Y. at x0 . Fix an arbitrary point x0 ∈ M and choose (Xk ) such that. Passing to the complexification ∗ ∗ Λ• C T M we deduce that d + d is a geometric Dirac operator.. we need a better description of the curvature of ∇g viewed as a connection on Λ• T ∗ M .e. we have the freedom to choose the frame (Xk ) in any manner we find convenient. Denote by R the Riemann curvature tensor. R uniquely determined by the equality ˜ (Y. the curvature of the Levi-Civita connection ∇g : C ∞ (T M ) → C ∞ (T ∗ M ⊗ T M ). R Proof. This shows d + d∗ can be written as the composition C ∞ (Λ• T ∗ M ) → C ∞ (T ∗ M ⊗ Λ• T ∗ M ) → C ∞ (Λ• T ∗ M ).2. e ) ˜ (ek . where c(ej ) = e(ej ) − i(ej ). Set ei = ∂x 0 i At x0 the Riemann curvature tensor R has the form R= k< i e =R where R(ek . for every X ∈ Vect (M ). . e )ej = Rijk ei . .j •T ∗M (ei . ej ) = 1 2 c(ei )c(ej )RΛ i. and any 1-forms α1 . To understand its form. To better understand its action we need to pick a local.2 we get ek ∧ e R(ek . . Z )X ). The Riemann curvature tensor can be expressed as R= ei ∧ ej Rij .j . .2. . . . = (R for any vector fields X. Y )(α1 ∧ · · · ∧ αk ) ˜ (X. pick normal ∂ ∗ M. Since R0 ≡ 0. oriented. αk ∈ Ω1 (M ). Prove the above lemma.Z ] and we get (RT ∗M (Y. ∗ Note that since both ∇∗ ∇. Lemma 11. Using Lemma 11. i. the first interesting case is R1 . and any α1 .3. ej ) : T M → T M . The curvature of the Levi-Civita connection on Λ• T ∗ M is defined by RΛ •T ∗M (X. We denote by (ei ) its dual coframe. Exercise 11. e )ej = Rjk i ijk ei . More precisely. Z )α)(X ) = −α(RT M (Y.534 CHAPTER 11. Y .2. ej ). i<j • ∗ where Rij is the skew-symmetric endomorphism Rij = R(ei .2. k. orthonormal moving frame (ei ) of T M .4. Y )α1 ) ∧ · · · ∧ αk + · · · + α1 ∧ · · · ∧ (R ˜ (X. DIRAC OPERATORS Similar computations give ∇Z ∇Y α and ∇[Y. . R Using this in the expression of R1 at x0 we get R1 ( j α j ej ) = 1 2 αj Rijk c(ek )c(e )ei . The Weitzenb¨ ock remainder of the Dirac operator d + d∗ is c(RΛ T M ). Thus c(RΛ •T ∗M )= i<j c(ei )c(ej )RΛ •T ∗M (ei . αk . Y )αk ). and consequently it must split as c(RΛ •T ∗M ) = ⊕k≥0 Rk . so does the Weitzenb¨ ock remainder. we define g g ∇g X (α1 ∧ · · · ∧ αk ) = (∇X α1 ) ∧ · · · ∧ αk + · · · + α1 ∧ · · · ∧ (∇X αk ). ∗ The Levi-Civita connection ∇g = ∇T M extends as an even derivation to a connection on Λ• T ∗ M . |x0 ∈ Tx0 M and ei = dxi |x0 ∈ Tx coordinates (xi ) near x0 ∈ M . and (d + d∗ )2 preserve the Z-grading of Λ• C T M . 5 (Bochner).j. but it is somewhere strictly positive definite then b1 (M ) = 0. where G is a compact semisimple Lie group. we conclude that Because of the skew-symmetry Rijk = −Rij ijk αj Rijk δk ei = 0. Hence R1 = Ric .11. and c(ek )c(e )ei = ek ∧ e ∧ ei − δi ek − δk ei + δik e . If the uniformity assumption is dropped then the conclusion of the Myers theorem no longer holds (think of the flat torus). We have c(e )ei = e ∧ ei − δi . Under more restrictive assumptions on the Ricci tensor (uniformly positive definite) one deduces a stronger conclusion namely that the fundamental group is finite. This result gives yet another explanation for the equality H 1 (G) = 0. The above result is truly remarkable. Let (M. (b) If Ricci tensor is non-negative definite. the Ricci tensor Ric is regarded (via the metric duality) as a selfadjoint endomorphism of T ∗ M . (11.j. connected. The condition on the Ricci tensor is purely local. We have proved a similar result using geodesics (see Myers Theorem. 6. k Rjik ei ∧ ek ∧ e = 0. FUNDAMENTAL EXAMPLES We need to evaluate the Clifford actions in the above equality. i Rijik αj Rijki ek = i.j.k i. but it has global consequences.k αj Rijik ek = jk αj Ricjk ek .2 . then b1 (M ) ≤ dim M .k. Using the first Bianchi identity we deduce that Rijk ek ∧ e ∧ ei = − i.j.5).k.k. . Subsection 5. oriented Riemann manifold.k.j. ∀j.1) has a beautiful consequence.2. where Ricjk = denotes the Ricci tensor at x0 .2. (a) If the Ricci tensor is non-negative definite.1) In the above equality. (Recall that b1 (M ) denotes the first Betti number of M ). i. Hence R1 ( j 535 α j ej ) = 1 2 αj Rijk (ek ∧ e ∧ ei − δi ek − δk ei + δik e ). The identity (11.2. Theorem 11. = 1 2 αj Riji e − i. i. Recall that in this case the Ricci curvature is 1 4 × {the Killing metric}. g ) be a compact.2.2.2.2. Hence R1 ( j α j ej ) = 1 2 1 2 αj Rijk (δik e − δi ek ) i.4 and 6. 536 Proof. and then integrating by parts. If the Ricci tensor is positive at some x0 ∈ M . η ∈ Ω1 .2) we deduce that any harmonic 1-form η must satisfy (Ric(x)ηx . In fact. we conclude that η ≡ 0.6.2) M M Since Ric is non-negative definite we deduce ∇η = 0.2.1) we deduce that if η ∈ ker ∆1 . to find the first Betti number. Taking the L2 -inner product by η . Since we had almost no contact with this beautiful branch of geometry we will present only those aspects concerning the “Dirac nature” of these operators.2. . then η (x0 ) = 0. Hodge theory asserts that b1 (M ) = dim ker ∆1 so that.2. (a) Let E → M be a smooth real vector bundle over the smooth manifold M . Remark 11. then ∇∗ ∇η + Ric η = 0. Note that any almost complex manifold is necessarily even dimensional and orientable. An almost complex structure on E is an endomorphism J : E → E such that J 2 = −E . Since η is also covariant constant. so that any harmonic 1-form must be covariant constant. since M is connected. so from the start we know that not any manifold admits almost complex structures. ηx )x = 0 ∀x ∈ M. To define this operator we need a little more differential geometric background.2. η )dvg = 0. (a) Let ∆1 denote the metric Laplacian CHAPTER 11.7.2. the number of linearly independent harmonic 1-forms is no greater than the rank of T ∗ M which is dim M . §11. we get |∇η |2 dvg + (Ric η. (b) An almost complex structure on a smooth manifold M is an almost complex structure J on the tangent bundle. (b) Using the equality (11. and M is connected. DIRAC OPERATORS ∆1 = dd∗ + d∗ d : Ω1 (M ) → Ω1 (M ). the existence of such a structure is determined by topological invariants finer than the dimension and orientability. Using the Bochner-Weitzenb¨ ock theorem and the equality (11. on M. For a very nice survey of some beautiful applications of this technique we refer to [10]. Definition 11. In particular.2.2 The Hodge-Dolbeault operator This subsection introduces the reader to the Dolbeault operator which plays a central role in complex geometry. we need to estimate the “number” of solutions of the elliptic equation ∆1 η = 0. An almost complex manifold is a pair (manifold. (11. almost complex structure). Hence.8.1 . and we get a similar decomposition T ∗ M ⊗ C = (T ∗ M )1. v ∈ Tx M .2.q (M ) := C ∞ (Λp. which is the kernel of the differential of the canonical projection π : T M → M . x) ∈ T M . this endomorphism induces the almost complex structure on T M . FUNDAMENTAL EXAMPLES 537 Example 11.2. Indeed. Using the results of Subsection 2. Prove that a smooth manifold M of dimension 2n admits an almost complex structure if an only if there exists a 2-form ω ∈ Ω2 (M ) such that the top exterior power ω n is a volume form on M .1 .0 ⊕ (T ∗ M )0.2. the differential of π induces an isomorphism π∗ : H(v.x) T M is vertical if and only if it is tangent to a smooth path which entirely the fiber Tx M ⊂ T M . Let (M. denote by E the tangent bundle of T M .9.x .q T ∗ M. Note that we have a canonical identification V(v.0 is isomorphic (over C) with (T M.2. Using the direct sum decomposition E = H ⊕ V we define the endomorphism J of E in the block form J= 0