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Lectures on the Geometry of Manifolds Liviu I. Nicolaescu September 9, 2018 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 upwiththefirstpurecreation. Relyingon thecommon experience, hecreated 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”. Therewas 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, theLie groupsareacademically very friendly. They provideamarvelous testing ground for abstract results. We have consistently taken advantage of this feature through- out 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. Eachpointhasatangentspaceattachedtoitsothatweobtaina“bunchoftangentspaces” 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. ThestudyofShapebeginsinearnestinChapter4whichdealswithRiemannmanifolds. We approach these objects gradually. The firstsection 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 completemeasureofhowfarisaRiemannmanifoldfrombeingflat. Thisisaveryinvolved 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. Thissection concludes with the firstglobal resultof the book, namely the Gauss-Bonnet theorem. We present a proof inspired from [26] 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 cur- rently 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 coho- mology 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 iso- morphism 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 topological type of a manifold. We prove E´lie Cartan’s old result that the cohomology of a symmetric space is given by the linear space of its bi-invariant forms. We use this technique to computethe lower degree cohomology of compact semisimpleLie 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 containing a concentrated description of Cˇech 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 im- portant 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 con- nections 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 ex- perience with connections on vector bundles which we hope will attenuate the formalism shock. In proving thevarious identities involving characteristic classes we adopt an invari- ant 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 Ka¨hler manifold. We adopt a variational approach. The uniformization theorem for compactRiemannsurfacesisthenanicebonus. Thismaynotbethemostdirectproofbut 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 operatorsoncompactmanifoldsareFredholmandestablishthehomotopyinvarianceofthe index. TheseareverygeneralHodgetypetheorems. Theclassical onefollows 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 mostdramaticadvancesofmoderngeometry. Webeginbyfirstdescribingageneralnotion 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 Weitzenbo¨ck formula which can be viewed as a bridge between geometryandanalysis,andwhichisoftenthesourceofmanyinterestingapplications. 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 secondhalfofthebookisanextendedversion ofagraduatecourseindifferential 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 knowl- edge 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 en- thusiastic 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 funda- mentals ofmoderngeometry. Heplayed akeyroleinshapingthemannerinwhichIregard 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 hopethefilter of my mindcaptured thefullrichness of theideas they sogenerously shared with me. My friends Louis Funar and Gheorghe Ionesei1 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 1He 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. Thefirstedition was generously sprinkledwith 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. Inpreparingthenewedition,Ihavebeenengagedonamassivetypohunting,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. Theonlymajoradditiontothecontents isanewchapteronclassicalintegralgeometry. 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. Onenovel featurein our presentation of integral geometry is theuseof tame geometry. Thisisarecent extension of thebetter knowarea ofreal algebraic geometry whichallowed us to avoid many heavy analytical arguments, and present the geometric ideas in as clear a light as possible. Notre Dame, 2007. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 1 Manifolds 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Space and Coordinatization . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 The implicit function theorem . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 How many manifolds are there? . . . . . . . . . . . . . . . . . . . . . 20 2 Natural Constructions on Manifolds 23 2.1 The tangent bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.2 The tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.1.3 Sard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.4 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.5 Some examples of vector bundles . . . . . . . . . . . . . . . . . . . . 37 2.2 A linear algebra interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.2 Symmetric and skew-symmetric tensors . . . . . . . . . . . . . . . . 46 2.2.3 The “super” slang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.5 Some complex linear algebra . . . . . . . . . . . . . . . . . . . . . . 64 2.3 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3.1 Operations with vector bundles . . . . . . . . . . . . . . . . . . . . . 69 2.3.2 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3.3 Fiber bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3 Calculus on Manifolds 79 3.1 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1.1 Flows on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1.2 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2 Derivations of Ω (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 • ii CONTENTS iii 3.2.1 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3 Connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3.1 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3.2 Parallel transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.3.3 The curvature of a connection . . . . . . . . . . . . . . . . . . . . . . 101 3.3.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.3.5 The Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.3.6 Connections on tangent bundles . . . . . . . . . . . . . . . . . . . . 109 3.4 Integration on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.1 Integration of 1-densities . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.4.2 Orientability and integration of differential forms . . . . . . . . . . . 115 3.4.3 Stokes’ formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4.4 Representations and characters of compact Lie groups . . . . . . . . 126 3.4.5 Fibered calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4 Riemannian Geometry 138 4.1 Metric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.1.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . 138 4.1.2 The Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . 141 4.1.3 The exponential map and normal coordinates . . . . . . . . . . . . . 147 4.1.4 The length minimizing property of geodesics . . . . . . . . . . . . . 149 4.1.5 Calculus on Riemann manifolds . . . . . . . . . . . . . . . . . . . . . 155 4.2 The Riemann curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.2.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . 164 4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.2.3 Cartan’s moving frame method . . . . . . . . . . . . . . . . . . . . . 170 4.2.4 The geometry of submanifolds . . . . . . . . . . . . . . . . . . . . . 173 4.2.5 The Gauss-Bonnet theorem for oriented surfaces . . . . . . . . . . . 179 5 Elements of the Calculus of Variations 188 5.1 The least action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.1.1 The 1-dimensional Euler-Lagrange equations . . . . . . . . . . . . . 188 5.1.2 Noether’s conservation principle . . . . . . . . . . . . . . . . . . . . 194 5.2 The variational theory of geodesics . . . . . . . . . . . . . . . . . . . . . . . 197 5.2.1 Variational formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.2.2 Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6 The Fundamental group and Covering Spaces 208 6.1 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.1.2 Of categories and functors . . . . . . . . . . . . . . . . . . . . . . . . 213 6.2 Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . 214 6.2.2 Unique lifting property . . . . . . . . . . . . . . . . . . . . . . . . . 216 iv CONTENTS 6.2.3 Homotopy lifting property . . . . . . . . . . . . . . . . . . . . . . . . 217 6.2.4 On the existence of lifts . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.2.5 The universal cover and the fundamental group . . . . . . . . . . . . 220 7 Cohomology 222 7.1 DeRham cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.1.1 Speculations around the Poincar´e lemma . . . . . . . . . . . . . . . 222 7.1.2 Cˇech vs. DeRham . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.1.3 Very little homological algebra . . . . . . . . . . . . . . . . . . . . . 228 7.1.4 Functorial properties of the DeRham cohomology . . . . . . . . . . . 235 7.1.5 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.1.6 The Mayer-Vietoris principle . . . . . . . . . . . . . . . . . . . . . . 240 7.1.7 The Ku¨nneth formula . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.2 The Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.2.1 Cohomology with compact supports . . . . . . . . . . . . . . . . . . 246 7.2.2 The Poincar´e duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.3 Intersection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.3.1 Cycles and their duals . . . . . . . . . . . . . . . . . . . . . . . . . . 254 7.3.2 Intersection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.3.3 The topological degree . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.3.4 Thom isomorphism theorem . . . . . . . . . . . . . . . . . . . . . . . 266 7.3.5 Gauss-Bonnet revisited . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.4 Symmetry and topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.4.1 Symmetric spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 7.4.2 Symmetry and cohomology . . . . . . . . . . . . . . . . . . . . . . . 276 7.4.3 The cohomology of compact Lie groups . . . . . . . . . . . . . . . . 279 7.4.4 Invariant forms on Grassmannians and Weyl’s integral formula . . . 280 7.4.5 The Poincar´e polynomial of a complex Grassmannian . . . . . . . . 288 7.5 Cˇech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.5.1 Sheaves and presheaves . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.5.2 Cˇech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 8 Characteristic classes 309 8.1 Chern-Weil theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.1.1 Connections in principal G-bundles . . . . . . . . . . . . . . . . . . . 309 8.1.2 G-vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 8.1.3 Invariant polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 316 8.1.4 The Chern-Weil Theory . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.2 Important examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.2.1 The invariants of the torus Tn . . . . . . . . . . . . . . . . . . . . . 324 8.2.2 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.2.3 Pontryagin classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.2.4 The Euler class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.5 Universal classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.3 Computing characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . 338 CONTENTS v 8.3.1 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 8.3.2 The Gauss-Bonnet-Chern theorem . . . . . . . . . . . . . . . . . . . 344 9 Classical Integral Geometry 353 9.1 The integral geometry of real Grassmannians . . . . . . . . . . . . . . . . . 353 9.1.1 Co-area formulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 9.1.2 Invariant measures on linear Grassmannians . . . . . . . . . . . . . . 364 9.1.3 Affine Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . 374 9.2 Gauss-Bonnet again?!? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 9.2.1 The shape operator and the second fundamental form of a submanifold in Rn377 9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space. 379 9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space . . . . . . 384 9.3 Curvature measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 9.3.1 Tame geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 9.3.2 Invariants of the orthogonal group . . . . . . . . . . . . . . . . . . . 394 9.3.3 The tube formula and curvature measures . . . . . . . . . . . . . . . 398 9.3.4 Tube formula = Gauss-Bonnet formula for arbitrary submanifolds 408 ⇒ 9.3.5 Curvature measures of domains in an Euclidean space . . . . . . . . 410 9.3.6 Crofton Formulæ for domains of an Euclidean space . . . . . . . . . 413 9.3.7 Crofton formulæ for submanifolds of an Euclidean space . . . . . . . 423 10 Elliptic Equations on Manifolds 430 10.1 Partial differential operators: algebraic aspects . . . . . . . . . . . . . . . . 430 10.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 10.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 10.1.3 Formal adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 10.2 Functional framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.1 Sobolev spaces in RN . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.2.2 Embedding theorems: integrability properties . . . . . . . . . . . . . 451 10.2.3 Embedding theorems: differentiability properties . . . . . . . . . . . 456 10.2.4 Functional spaces on manifolds . . . . . . . . . . . . . . . . . . . . . 460 10.3 Elliptic partial differential operators: analytic aspects . . . . . . . . . . . . 464 10.3.1 Elliptic estimates in RN . . . . . . . . . . . . . . . . . . . . . . . . . 465 10.3.2 Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 10.3.3 An application: prescribing the curvature of surfaces . . . . . . . . . 474 10.4 Elliptic operators on compact manifolds . . . . . . . . . . . . . . . . . . . . 484 10.4.1 The Fredholm theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 484 10.4.2 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 10.4.3 Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 11 Dirac Operators 502 11.1 The structure of Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . 502 11.1.1 Basic definitions and examples . . . . . . . . . . . . . . . . . . . . . 502 11.1.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 11.1.3 Clifford modules: the even case . . . . . . . . . . . . . . . . . . . . . 509

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