Lectures on the Geometry of Manifolds (Third Edition) [3 ed.] 9811214816, 9789811214813

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Lectures on the

Geometry of Manifolds Third Edition

Other World Scientific Titles by the Author

Introduction to Real Analysis ISBN: 978-981-121-038-9 ISBN: 978-981-121-075-4 (pack)

Lectures on the

Geometry of Manifolds Third Edition

Liviu I Nicolaescu University of Notre Dame, USA

NEW JERSEY

. LONDON . SINGAPORE

World Scientific . BEIJING . SHANGHAI . HONG KONG . TAIPEI . CHENNAI . TOKYO

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Library of Congress Cataloging-in-Publication Data Names: Nicolaescu, Liviu I., author. Title: Lectures on the geometry of manifolds / Liviu I. Nicolaescu, University of Notre Dame, USA. Description: Third edition. | New Jersey : World Scientific Publishing Co., [2021] | Includes bibliographical references and index. Identifiers: LCCN 2020037906 | ISBN 9789811214813 (hardcover) | ISBN 9789811215957 (paperback) | ISBN 9789811214820 (ebook) I ISBN 9789811214837 ( b o o k other) Subjects: LCSH: Geometry, Differential. | Manifolds (Mathematics) Classification: LCC QA649 .N53 2021 | DDC 516.3/62--dc23 LC record available at https://lccn.loc.gov/2020037906

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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Preface

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 wonted 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 vii

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of Manzfolds

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 "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 Caftan's moving frame technique which is extremely useful in concrete computations. As an application of this technique we prove the celebrated Theorem Egregium of Gauss. This section concludes with the first global result of the book, namely the GaussBonnet theorem. We present a proof inspired from [38] 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

Preface

ix

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 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 [22], 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 Poincaré 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 topological type of a manifold. We prove Elie Cartan's old result that the cosmology 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 cosmology of complex Grassmannians relying on Weyl's integration formula and Schur polynomials. The chapter ends with a fifth section containing a concentrated description of tech 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 lin the current edition, this is Chapter 10.

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Lectures on the Geometry of Manzfolds

application of these results we consider a Kazhdan-Warner type equation which recently found applications in solving the Seiberg-Witten equations on a Kohler 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 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 [14]. We then isolate a special subclass we called geometric Dirac operators. Associated to each such operator is a very concrete WeitzenbOck 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 the chapter on elliptic equations 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

Preface

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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 Ionesez 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 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 (Chapter 9) on classical integral geometry. This is a subject that captured my imagination over the last few years, and since the first edition 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 the classical results 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. 2He passed away while I was preparing the second edition. He was the ultimate poet of mathematics.

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Lectures on the Geometry of Manifolds

Preface to the third edition I started writing the first edition 25 years ago as a fresh PhD and I was warned by many "adults" that this was a wrong career move. I embarked in this project with all the energy, enthusiasm, inexperience and confidence of young age. It was meant to be an honest presentation of the basic elements of differential geometry used in global analysis. By "honest" presentation I understood that I should include clear and detailed explanations for many of the folklore results, examples and points of view that are harder to trace in the literature and had helped in my research. It was an exciting experience writing the first edition and I was rewarded for my effort it in many ways. There was an immediate reward for the time spent immersed in the minutia of many examples and proofs. The detailed understanding I achieved allowed me to push my own research to new directions and in such a depth that I did not believe I was capable of at that age. There was a long term reward since it turns out that the useful facts and examples and computations that were harder to trace are still useful and harder to trace, but now I can always turn to this book for details. Personal bias aside, I keep a copy of my book on my desk since I frequently need to look up something in it. There is a personal reward, as an author. I have been receiving input and acknowledgments from many readers from all the corners of the world. I have implemented all the corrections and suggestions I have received. A book lives through its readers and apparently the present one is still alive. So what is new in the third edition? There are some obvious additions reflecting my current research interests. There is the new Chapter 11 on spectral geometry leading to the original results in Subsection 11.4.5. This is the first place where they appear in published form. Describing new results in a monograph rather than in a research journal has allowed me to go to a level of detail and provide perspective that is not possible in a journal. In particular, I have added two new Subsections 4.2.5 and 5.2.3 on Riemannian geometry containing facts that, surprisingly, are not familiar to many geometers. These play an important role in Subsection 11.4.5. There are less conspicuous changes. I have been using various parts of the book for graduate courses I taught over the years. The new edition contains the useful feedback I have received from my students. I have enhanced and cleaned many proofs and I have added new examples. Of course I have corrected the typos pointed out to me by readers. It is likely I have introduced new ones. Notre Dame, 2019.

Contents

vii

Preface

1.

1.1

1.2

2

•

1

Manifolds

1 1 3 6 6 9 10

Preliminaries 1.1.1 Space and Coordinatization 1.1.2 The implicit function theorem 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?

20 23

Natural Constructions on Manifolds 2.1

The tangent bundle Tangent spaces. • . . . . 2.1.1 2.1.2 The tangent bundle . . . . . . . 2.1.3 Transversality 2.1.4 Vector bundles 2.1.5 Some examples of vector bundles A linear algebra interlude 2.2.1 Tensor products 2.2.2 Symmetric and skew-symmetric tensors The "super" slang 2.2.3 2.2.4 Duality................ 2.2.5 Some complex linear algebra . . . . . Tensor fields 2.3.1 Operations with vector bundles 2.3.2 Tensor fields 2.3.3 Fiber bundles . . . . . . . . . . o

2.2

2.3

xiii

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23 23 27 29 33 37 41 41 46 53 56 64 67 67 69 73

Lectures on the Geometry of Manifolds

xiv

3.

Calculus on Manifolds

3.1

3.2

3.3

3.4

4.

79 The Lie d e r i v a t i v e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1.1 Flows on manifolds 81 3.1.2 The Lie d e r i v a t i v e . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.1.3 Examples . 88 Derivations of Q' (M) 3.2.1 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . 88 3.2.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . . 94 3.3.1 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . 94 100 3.3.2 Parallel t r a n s p o r t . . . . . . . . . . . . . . . . . . . 101 3.3.3 The curvature of a connection . . . . . . . . . . . . . 105 3.3.4 Holonomy....................... 108 3.3.5 The Bianchi identities . . . . . . . . . . . . . . . . . 109 3.3.6 Connections on tangent bundles . . . . . . . . . . . . 111 Integration on manifolds . . . . . . . . . . . . . . . . . . . . 111 3.4.1 Integration of 1-densities . . . . . . . . . . . . . . . 115 3.4.2 Orientability and integration of differential forms . . . 3.4.3 Stokes'formula 123 3.4.4 Representations and characters of compact Lie groups . . . . . . 127 3.4.5 Fiberedca1culus..........................133

Riemannian Geometry

139

4.1

139 139 143 147 150 155 165 165 169 172 175 182 192

4.2

5.

79

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. l .4 The length minimizing property of geodesics 4.1.5 Calculus on Riemann manifolds . . . . . . The Riemann curvature 4.2.1 Definitions and properties . . . . . . . . . . . . 4.2.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . Ca1"tan's moving frame method . . . . . . . . . 4.2.3 4.2.4 The geometry of submanifolds . . . . . . . . . Correlators and their geometry 4.2.5 4.2.6 The Gauss-Bonnet theorem for oriented surfaces

Elements of the Calculus of Variations

201

5.1

201 201 207 210

5.2

The least action principle 5.1.1 The 1-dimensional Euler-Lagrange equations 5.1.2 Noether's conservation principle The variational theory of geodesics . . .

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Contents

Variational formulae Jacobi fields The Hamilton-Jacobi equations

211 214

The Fundamental Group and Covering Spaces

227

5.2.1 5.2.2 5.2.3 6.

6.1

6.2

7.

The fundamental group . . 6.1.1 Basic n o t i o n s . . . . . . . . . . . . . . 6.1.2 Of categories and functors . . . Covering Spaces . . . . . . . . . . . . . 6.2.1 Definitions and examples . . . . . . . 6.2.2 Unique lifting property . . . . . . . . 6.2.3 Homotopy lifting property . . . 6.2.4 On the existence of lifts 6.2.5 The universal cover and the fundamental group

220

..228 ..228 .232

... .. .. .. o

t

o

..

o

n

233

..233 ..235 . . .236 . . . . 237 ..239

Cohomology

241

7.1

241 241 245 247 254 257 259

7.2

7.3

7.4

7.5

DeRham c o h o m o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Speculations around the Poincaré lemma . . . . . . .

7.1.2 (tech 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 Kenneth formula . . . . . . . . . . The Poincaré duality . . . . . . . . . . . . . . • . 7.2.1 Cohomology with compact supports 7.2.2 The Poincaré duality . . . . . . Intersection theory . . . . . . . . . . . . . . . 7.3.1 Cycles and their duals . . . . . 7.3.2 Intersection theory . . . . . . . . . . . 7.3.3 The topological degree . . . . . . . . . 7.3.4 The Thom isomorphism theorem 7.3.5 Gauss-Bonnet revisited . . . . Symmetry and topology . . 7.4.1 Symmetric spaces . . . . . . . . . . . 7.4.2 Symmetry and cosmology . . . . . . . . . . . . . . 7.4.3 The cosmology of compact Lie groups 7.4.4 Invariant forms on Grassmannians and Weyl's integral formula 7.4.5 The Poincaré polynomial of a complex Grassmannian éechcohomology................ 7.5.1 Sheaves and presheaves . . . . 7.5.2 tech cosmology . . . . . . .

. . . 262 . 265 . 265 •

•

•

•

•

•

•

•

268 272 272 277 282 284 287 291 291 294 298 299 306 312 313 317

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8.

Lectures on the Geometry

Characteristic Classes

329

8.1

329 329 335 336 339 343 343 343 346 348 351 357 358 363

8.2

8.3

9.

of Manzfolds

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 Important examples 8.2.1 The invariants of the torus T" 8.2.2 Chern classes . 8.2.3 Pontryagin classes 8.2.4 The Euler class . 8.2.5 Universal classes . Computing characteristic classes 8.3.1 Reductions 8.3.2 The Gauss-Bonnet-Chern theorem

373

Classical Integral Geometry

9.1

The integral geometry of real Grassmannians . .. .373 9.1.1 C o - a r e a f o r m u l a e . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9. 1.2 Invariant measures on linear Grassmannians . . . . . . . 386 9. 1.3 Affine Grassmannians . 395 G a u s s - B o n n e t a g a i n ? ! ' ? . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 9.2.1 The shape operator and the second fundamental form . . . . . . 398 9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 405 9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space . 409 Curvature m e a s u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Tame geometry . . . . . . . . . 409 9.3.2 Invariants of the orthogonal group . . . . . . . . . . . . . . . . 414 9.3.3 The tube formula and curvature measures . . . . . . . . . . . . 418 9.3.4 Tube formula => Gauss-Bonnet formula for arbitrary submanifolds of an Euclidean space . . . . . . . . . . . . . . . 429 431 9.3.5 Curvature measures of domains in an Euclidean space 433 9.3.6 Crofton formulae for domains of an Euclidean space . 443 Crofton formulae for submanifolds of an Euclidean space 9.3.7 n

o

n

o

o

o

9.2

9.3

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 .

f

n

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O

o

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451 451 451 457 459 464

xvii

Contents

10.2.1 10.2.2 10.2.3 10.2.4 10.3 Elliptic 10.3.1 10.3.2 10.3.3 10.4 Elliptic 10.4.1 10.4.2 10.4.3

Sobolev spaces in RN . . . . . . . . . . . . . Embedding theorems: integrability properties. . . . Embedding theorems: differentiability properties . Functional spaces on manifolds . . . . . . partial differential operators: analytic aspects Elliptic estimates in RN Elliptic regularity An application: prescribing the curvature of surfaces operators on compact manifolds Fredholm theory . Spectral t h e o r y . . . . . . . . . . . . . . . . . . . . Hodge theory .

464 471 476 480 484 485 489 494 504 504 513 518

11. Spectral Geometry 11.1 Generalized functions and currents . . . . . . . . . . . . 11.1.1 Generalized functions and operations withy them

523

.. . ..

o

f

0

.

11.1.2 C u r r e n t s . . . . . . . . . . . . . . . . . . . . . 11.1.3 Temperate distributions and the Fourier transform . . 11.1.4 Linear differential equations with distributional data 11.2 Important families of generalized functions 11.2.1 Some classical generalized functions on the real axis . 11.2.2 Homogeneous generalized functions . . . . . . . . . 11.3 The wave equation 11.3.1 Fundamental solutions of the wave 11.3.2 The wave family 11.3.3 Local parametrices for the wave equation with variable coefficients....................... I I .4 Spectral geometry l 1.4.1 The spectral function of the Laplacian on a compact manifold 11.4.2 Short time asymptotic for the wave kernel 11.4.3 Spectral function asymptotics . . . . . . . . . . . . . . . . . 11.4.4 Spectral estimates of smoothing operators 11.4.5 Spectral perestroika . . . . . . . . . . . . . . . . . . . . . . 12. Dirac Operators 12.1 The structure of Dirac operators . . . . 12.1.1 Basic definitions and examples 12.1.2 Clifford algebras 12.1.3 Clifford modules: the even case 12.1.4 Clifford modules: the odd case 12.1.5 A look ahead . 12.1.6 The spin group .

. 523 . . 523 528 529 531 536 536 542 548 548 550 567 574 574 580 588 593 598

611 611 611 614 617 622 623 625

xviii

Lectures on the Geometry

12.1.7 The complex spin group 12.1.8 Low dimensional examples . . 12.1.9 Dirac bundles 12.2 Fundamental examples 12.2.1 The Hodge-DeRham operator . 12.2.2 The Hodge-Dolbeault operator 12.2.3 The spin Dirac operator . . . . 12.2.4 The spine Dirac operator

of Manifolds 633 635 640 644 644 648 654 660

Bibliog raphy

669

Index

675

Chapter 1

Manifolds

1.1 Preliminaries 1.1.1 Space and Coordinatization Mathematics is a natural science with a special modus operandi. It replaces concrete natural objects with mental abstractions which serve as intermediaries. One studies the properties of these abstractions in the hope they reflect facts of life. So far, this approach proved to be very productive. The most visible natural object is the Space, the place where all things happen. The first and most important mathematical abstraction is the notion of number. Loosely speaking, the aim of this book is to illustrate how these two concepts, Space and Number, fit together. It is safe to say that geometry as a rigorous science is a creation of ancient Greeks. Euclid proposed a method of research that was later adopted by the entire mathematics. We refer of course to the axiomatic method. He viewed the Space as a collection of points, and he distinguished some basic objects in the space such as lines, planes etc. He then postulated certain (natural) relations between them. All the other properties were derived from these simple axioms. Euclid's work is a masterpiece of mathematics, and it has produced many interesting results, but it has its own limitations. For example, the most complicated shapes one could reasonably study using this method are the conics and/or quadrics, and the Greeks certainly did this. A major breakthrough in geometry was the discovery of coordinates by René Descartes in the 17th century. Numbers were put to work in the study of Space. Descartes' idea of producing what is now commonly referred to as Cartesian coordinates is familiar to any undergraduate. These coordinates are obtained using a very special method (in this case using three concurrent, pairwise perpendicular lines, each one endowed with an orientation and a unit length standard. What is important here is that they produced a one-to-one mapping

Euclidian Space -> R3

7

P

I

> (;v(P)?y(Pl»2(P))-

We call such a process coordinatization. The corresponding map is called (in this case) Cartesian system of coordinates. A line or a plane becomes via coordinatization an algebraic object, more precisely, an equation. 1

2

Lectures on the Geometry of Manzfolds

In general, any coordinatization replaces geometry by algebra and we get a two-way correspondence

> Study of Equations.

Study of Space
(1~(p),9(p))@(0,oo)

X

( or, fl-

This choice is related to the Cartesian choice by the well known formulae at

rosH

y=rsin9.

(1.1.1)

A remarkable feature of (l.1.l) is that ;L°(Pl and y(P) depend smoothly upon r(Pl and @(p). As science progressed, so did the notion of Space. One can think of Space as a configuration set, i.e., the collection of all possible states of a certain phenomenon. For example,

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 space associated with this problem consists of all pairs (position, velocity) a particle can possibly have. We can coordinatize this space using six

3

Manifolds

functions: three of them will describe the position, and the other three of them will describe the velocity. We say the configuration space is 6-dimensional. We cannot visualize this space, but it helps to think of it as an Euclidian space, only "roomier". There are many ways to coordinatize the configuration space of a motion of a particle, and for each choice of coordinates we get a different description of the motion. Clearly, all these descriptions must "agree" in some sense, since they all reflect the same phenomenon. In other words, these descriptions should be independent of coordinates. Differential geometry studies the objects which are independent of coordinates. The coordinatization process had been used by people centuries before mathematicians accepted it as a method. For example, sailors used it to travel from one point to another on Earth. Each point has a latitude and a longitude that completely determines its position on Earth. This coordinatization is not a global one. There exist four domains delimited by the Equator and the Greenwich meridian, and each of them is then naturally coordinatized. Note that the points on the Equator or the Greenwich meridian admit two different coordinatizations which are smoothly related. The manifolds are precisely those spaces which can be piecewise coordinatized, with smooth correspondence on overlaps, and the intention of this book is to introduce the reader to the problems and the methods which arise in the study of manifolds. The next section is a technical interlude. We will review the implicit function theorem which will be one of the basic tools for detecting manifolds.

1.1.2 The implicit function theorem We gather here, with only sketchy proofs, a collection of classical analytical facts. For more details one can consult [34]. Let X and Y be two Banach spaces and denote by L ( X , Y) the space of bounded linear operators X -> Y. For example, if X = R", Y = RT", then L ( X , Y) can be identified with the space of m X n matrices with real entries. For any set S we will denote by k g the identity map S -> S.

Definition 1.1.1. Let F U C X -> Y be a continuous function (U is an open subset of X). The map F is said to be (Fréchet) differentiable at u 6 U if there exists T e L ( X , Y ) such that

lIF0

l

IIHIIx num0

h)

'

F(Uol - Telly = 0.

Loosely speaking, a continuous function is differentiable at a point if, near that point, it admits a "best approximation" by a linear map. When F is differentiable at U0 6 U, the operator T in the above definition is uniquely determined by . 1 d Th F(u0 -|- t h ) nm - (F(u0 -|- th) - F(u0l) t->0 t dt i

4

Lectures on the Geometry of Manzfolds

We will use the notation T = DU0 F and we will call T the Fréchet derivative of F at U0 . Assume that the map F : U -> Y is differentiable at each point u 6 U. Then F is said to be of class C1, i f the map u +-> D'LLF 6 L ( X , Y) is continuous. F is said to be of class C2 if u +-> D'LLF is of class C1. One can define inductively Ck and coo (or smooth) maps. Example 1.1.2. Consider F : U C R" ->

Run.

QUO

.CE

'LL

'LL

1

7 .

Using Cartesian coordinates

.. ,$ n

in R" and 1 7 °

.

. ,U

m

in Run, we can think of F as a collection of m functions on U 'LL

1

1 l 'u,(;13,...,a3"),

,um

U'771(;U17°°'7a:n)'

The map F is differentiable at a point p = (p1, . . . ,1>") 6 U if and only if the functions u'll are differentiable at p in the usual sense of calculus. The Fréchet derivative of F at p is the linear operator DpF : R" -> RM given by the Jacobian matrix

(al 3(7u,1 3(a21 7 7

D,,F

um 7

in

31/,2

3:151

(p)

31/,1

5':1:2

nu2 31/32

(al (al

(p) 51/,2

3515"

(p)

7

(p) The map F is smooth if and only if the functions

sum 31/32

(p)

sum 3113"

(p)

_

) are smooth.

Exercise 1.1.3. (a) Let 'LL C LGR" ,WI denote the set of invertible n X n matrices. Show that U is an open subset of L(]R"', Rn). (b) Let F : 'LL -> U be defined as A -> A-1_ Show that DAF(H) = -A-1HA-1 for any n X n matrix H . (c) Show that the Fréchet derivative of the map det : L(]Rn*,R") -> R, A »-> det A, at A = Ilia" 6 L(Rn*, Rn) is given by tr H, i.e., d

L

det(J1Rn + t H ) = t H , oH € L(]R",]R"). 0

Theorem 1.1.4 (Inverse function theorem). Let X , Y be two Banach spaces, U C X open and F : U C X -> Y a smooth map. If at a point U 0 6 U the derivative DU0 F E L(X,Y) is invertible, then there exits an open neighborhood U1 of U 0 in U such that F(U1) is an open neighborhood of Uo = F(u0) in Y and F : U1 -> F(U1) is bijective, with smooth inverse. The spirit of the theorem is very clear: the convertibility of the derivative DU0 F "propagates" locally to F because DU0 F is a very good local approximation for F.

5

Manifolds

More formally, if we set T

DuoF, then

F(u0 + h) = F(u0) + Th

+ ¢~(h),

where r(h) = o(llhll) as h -> 0. The theorem states that, for every 'u s u f i ciently close to U0, the equation F('u.) = 'U has a unique solution 'LL = up -|- h, with h very small. To prove the theorem one has to show that, for llv - 'U0 fly sufficiently small, the equation below t)0+Th+v°(h)

='u

has a unique solution. We can rewrite the above equation as Th = 'U - U0 - r(h) or, equivalently, as h = T"1('v - U0 - r(h)). This last equation is a fixed point problem that can be approached successfully via the Banach fixed point theorem.

Let X, Y , Z be Banach spaces, W C X ,

Theorem 1.1.5 (Implicit function theorem). "//c Y open sets and F :W

X

"If -> Z

a smooth map. Let (330, ygl 6 W X "//, and Z0 := F(:z;0,

y0l. Set

$2 : "// -> Z, F2(Y) = FllimYlAssume that D90 F2 G L(Y, Z ) is invertible. Then there exist open neighborhoods U C W

of £170

in X , V C 7/ of Yo in Y , and o smooth map G : U -> V such that the set S of solution (as, y ) of the equation F(:c, y ) = Z0 which lie inside U X V can be identified with

the graph of G, i.e.,

{(a:,y)€UxV;

F(w,y)=20}={(w,G(w)l

G U x V ; a2€U}.

In pre-Bourbaki times, the classics regarded the coordinate y as a function

of :u defined

implicitly by the equality F(a2, yl = Z0»

Proof. Consider the map

H:X

X

Y->X

X

Z,

The map H is a smooth map, and at £0 = has the block decomposition

§ = ( a : , y ) »-> (:c,F(;z3,y)). (1U073/0)

its derivative D€01-I : X

X

Y -> X

X

Z

D H= so -D€oF1 D€0 F2 Above, DF1 (respectively DF2) denotes the derivative of ac »-> F(:L°, 3/ol (respectively the derivative of y »-> F(x0, y)). The linear operator Do0 H is invertible, and its inverse has the block decomposition

Hx

0

(D50H)

(D1oe)-1 o (D§0F1) (D 0F2)* Thus, by the inverse function theorem, the equation (as, F(;u, y)) = (in, zo) has a unique solution (M) = H-1(x, Zeal in a neighborhood of (5130, yo). It obviously satisfies 5: = as and F(i;,g) = 20. Hence, the set {(x,y); F(ac,y) = z0} is locally the graph of ac +-> H-1 (ac, 20).

6

Lectures on the Geometry of Manzfolds

1.2 Smooth manifolds 1.2.1 Basic definitions We now introduce the object which will be the main focus of this book, namely, the concept of (smooth) manifold. It formalizes the general principles outlined in Subsection 1.1.1.

Definition 1.2.1. A smooth manifold of dimension m is a locally compact, second countable' Hausdorff space M together with the following collection of data (henceforth called atlas or smooth structure) consisting of the following. (a) An open cover {U'i}i@I of M, (b) A collection of homeomorphisms

\Pi

u

-> \p(u-> C Run; i 6

I}

(called charts or local coordinates) such that, 'Pt (Up) is open in RM and, if UZ-O(/'j ¢ (Z), then the transition map \IJjo

'Pi

1

\P¢(U¢ M

Url C RM

\Ifjlu¢

O Url C RM

is smooth. (We say that the various Cha s are smoothly compatible, see Figure 1.2).

J

Ui

J

W.

'Vi

J

"Ii 'Vi-1

/

Run

Fig. 1.2

Transition maps.

l A second countable space is a topological space that admits a countable basis of open sets.

7

Manifolds

Remark 1.2.2. (a) Each chart 'Pi : Uz- -> RM can be viewed as a collection of m functions ( X 1 7 ° ° ° 7 a 3 M 7 on Up,

l

1-1 1,2

(p) (p)

1

We (p)

(p) Similarly, we can view another chart 'PJ as another collection of functions (y1, - - JW)The transition map 'Pa o 'Pi l can then be interpreted as a collection of maps at771

•

(x1,...,;u"") +-> l3/1(a31,...,a:""),...,yM(ac1,...,a:M)l.

(b) Since a manifold is a second countable space we can always work with atlases that are at most countable.

The first and the most important example of manifold is R" itself. The natural smooth structure consists of an atlas with a single chart, Ilia" : R" -> R". To construct more examples we will use the implicit function theorem.

Definition 1.2.3. (a) Let M, N be two smooth manifolds of dimensions m and respectively f : M -> N is said to be smooth if, for any local charts QS on M and on N, the composition o f o (V1 (whenever this makes sense) is a smooth map Run -> Rn_ (b) A smooth map f : M -> N is called a dyffeomorphism if it is invertible and its inverse is also a smooth map.

n. A continuous map

w

w

Example 1.2.4. The map t »-> et is a diffeomorphism ( -oo, oo) -> (0,oo). The map t +-> t3 is a homeomorphism R -> R, but it is not a diffeomorphism! If M is a smooth m-dimensional manifold, we will denote by cOO (M) the linear space of all smooth functions f : M -> R. Let us point out a simple procedure that we will use frequently in the sequel. Suppose that f : M -> R is a smooth function. If (U, \If) is a local chart on M so \P(Ul is an open subset of Run, then, by definition, the composition f o \I/-1 : \P(Ul -> R is a smooth function on the open set \If(U) C R". If we denote by x1, . . . ,513771 the canonical Euclidean coordinates on Run, then o \I/-1 is a function depending on the m variables .CU1 . . m and we will use the notation f lx, . . . ,LU m when referring to this function.

w

7 '

Remark 1.2.5. Let U be an open subset of the smooth manifold M (dim M

m ) and

\If:U-HRM

a smooth, one-to-one map with open image and smooth inverse. Then \If defines local coordinates over U compatible with the existing atlas of M. Thus (U, \I/) can be added to the original atlas and the new smooth structure is diffeomorphic with the initial one. Using Zermelo's Axiom we can produce a maximal atlas (no more compatible local chart can be added to it).

8

Lectures on the Geometry of Manzfolds

Our next result is a general recipe for producing manifolds. Historically, this is how manifolds entered mathematics. Proposition 1.2.6. Let M be a smooth manzfold of dimension m and f l , cOO

7

fk 6

( M ) . Define

Z

Z»(f1,

7

Assume that the functions

fl,

fol

MP)

{ppM;

=

=f1=(p) =0}-

. . . , fk are functionally independent along Z, i.e., for each

I

p 6 Z, there exist local coordinates (331, . . .

dejined in a neighborhood

of p

in M

such that so (p) = 0, i = 1, . . . ,m, and the matrix

3f1

6f1 $f1

3x1 3x2

hmm

ask 6 f k 3x1 3x2

hmm

. a

95; p

ask

:n 1

:1;'"=0

has rank k. Then Z has a natural structure of smooth manifold of dimension m - k.

Proof. Step 1: Constructing the charts. Let Po E Z., and denote by ( X 1 7 ' ° ° 7 X M 7 local coordinates near Po such that x'll (pg) = 0. One of the A: X k minors of the matrix fifl hmm

..

3:5

3fk

p 31/31

hmm

31/32

:nm 0

11:1

is nonzero. Assume this minor is determined by the last k columns (and all the k lines). We can think of the functions fl, . . . , fk as defined on an open subset U of Run. Split RM as R m - k X Rk, and set al

1 -k (:z3,...,a:M ),

ml/

("-'°+1, . . . , mM).

..

We are now in the setting of the implicit function theorem with

x=Rm-'" and F X

X

7

y=1Re.'"

z=R'*

7

7

Y - > Z given by

f1l£ul 6 R/*

an->

fk(x)) OF In this case, DF2 = ( 5933" ) : Re' -> Rk is invertible since its determinant corresponds to our nonzero minor. Thus, in a product neighborhood UP0 = UPo X U" of Po, the set Z, is Po the graph of some function

g

_

. UPo)

C

RM- k

>U/I CRk*, Po

i.e.,

Z O UPo

(;u/,g(xI) ) 6 Rm-k

X

Re. co' 6 7

UPo) 7

at / small

.

9

Manifolds

We now define MY I Z FW

UP()

-> ]RM"* by (22',

9(11') )

+¢P0>

513I

E R""-k.

The map 1% is a local chart of Z near Po.

Step 2. The transition maps for the charts constructed above are smooth. The details are left to the reader. Exercise 1.2.7. Complete Step 2 in the proof of Proposition 1.2.6. Definition 1.2.8. Let M be a m-dimensional manifold. A codimension k submamfold of M is a subset S C M locally defined as the common zero locus of k functionally independent functions fl, . . . , fk 6 COO(M). More precisely, this means that, for any Po 6 S, there exists an open neighborhood U of Po e M and k functionally independent smooth functions f17...7fk¢2U-

such that p 6 U m S if and only if fl (p)

R

fA¢(Pl=0-

Proposition 1.2.6 shows that any submanifold N C M has a natural smooth structure so it becomes a manifold per se. Moreover, the inclusion map z' : N M is smooth.

Exercise 1.2.9. Suppose that M is a smooth m-dimensional manifold. Prive that S C M is a codimension k-submanifold of M if and only if, for any Po 6 M, there exists a coordinate chart (U, \I/) with local coordinates (x1, . . . ,arM) such that Po 6 U and

urns 1.2.2

{ p a U ; 1v1(p)

1vI"(p)

0 }~

Partitions of unity

This is a very brief technical subsection describing a trick we will extensively use in this book. Recall that manifolds are locally compact, second countable topological spaces. As such, they are paracompact so they admit continuous partitions of unity, see [30, §3.7]. A much more precise result is in fact true.

Definition 1.2.10. Let M be a smooth manifold and (UJQQA an open cover of M. A (smooth) partition of unity subordinated to this cover is a family (fB)Be8 C COOIMI satisfying the following conditions. (i) 0 S f B S 1. (ii) EIQS : 8 -> A such that supp fa C U¢(/3)(iii) The family (supp f ii) is locally finite, i.e., any point as 6 M admits an open neighborhood intersecting only finitely many of the supports supp fa . (iv) EB f g l i ) = 1 for allay 6 M .

10

Lectures on the Geometry of Manifolds

We include here for the reader's convenience the basic existence result concerning partitions of unity. For a proof we refer to [128].

Proposition 1.2.11. (a) For any open cover U = ( U ) o € A of a smooth manifold M there exists at least one smooth partition of unity IJBIBQ8 subordinated to U such that supp f B is compact for any B. (b) If we do not require compact supports, then we can _lind a partition of unity in which 'B = A and QUO = HA. Exercise 1.2.12. Let M be a smooth manifold and S C M a closed submanifold, i.e., S is a closed subset of M. Prove that the restriction map

7:Croo(M)->CYOO(S) f *->fls is surjective. Deduce that for any finite set X C M and any function g : F -> R there exists a smooth, compactly supported function f : M -> R such that f(:z:) = glatl, Van 6 X .

1.2.3 Examples Manifolds are everywhere, and in fact, to many physical phenomena which can be modelled mathematically one can naturally associate a manifold. On the other hand, many problems in mathematics find their most natural presentation using the language of manifolds. To give the reader an idea of the scope and extent of modern geometry, we present here a she list of examples of manifolds. This list will be enlarged as we enter deeper into the study of manifolds.

Example 1.2.13. (The n-dimensional sphere) This is the codimension l submanifold of Rn+l given by the equation n

I I :u

2

(W 'i

T'

2

,

at

0

.CU

7

7

so'n) 6 R'n+l

0

One checks that, along the sphere, the differential of laval2 is nowhere zero, so by Proposition 1.2.6, S" is indeed a smooth manifold. In this case one can explicitly construct an atlas (consisting of two charts) which is useful in many applications. The construction relies on stereographic projections. Let N and S denote the North and resp. South pole of S" (N = (0, . . . ,0, l) 6 lR"+1, S = (0, . . . ,0, -1) 6 R"+1). Consider the open sets Un = S"\{N} and Us = S"\{S}. They form an open cover of S". The stereographic projection from the North pole is the map uN : Un -> R" such that, for any P 6 Un, the point oN (P) is the intersection of the line NP with the hyperplane {;u" = 0} 2 R" . The stereographic projection from the South pole is defined similarly.

..

For P Q U n we denote by lY1lPl) For Q GUs we denote by (zwQ),

7 7

y"(P) ) the coordinates of uN (P). z"(Q) ) the coordinates of us (Q).

11

Manifolds

A simple argument shows the map

(y1(p),...,y"(p)) »-> (21(P)7...72"(Pl)7 P G Un VW Us, is smooth (see the exercise below). Hence {(UN, oN), (US, 0g)} defines a smooth structune on S". Exercise 1.2.14. Show that the functions y Zi constructed in the above example satisfy ni Zi

vi = 1,

y

,n.

lE]=1lY]l2 Example 1.2.15. (The n-dimensional torus) This is the codimension n submanifold of the vector space R211 with Cartesian coordinates (al , y1, , can, yn), defined by the equalities 2

2

£I31+?/1

Mn

2

2 -l-yn

= 1.

Note that T1 is diffeomorphic with the 1-dimensional sphere Sl (unit circle). As a set T" is a direct product of n circles (see Figure 1.3) T"={=v21+yI=1}

X

I

Fig. 1.3

{:u2

2

Yn

1} S1 X

X

51.

I

The 2-dimensional torus.

The above example suggests the following general construction.

Example 1.2.16. Let M and N be smooth manifolds of dimension m and respectively Then their topological direct product has a natural structure of smooth manifold of dimension m - i - n .

n.

Example 1.2.17. (The connected sum of two manifolds) Let M1 and M2 be two manifolds of the same dimension m. Pick Pi 6 Mt (i = 1, 2), choose small open neighborhoods Up of Pt in M and then local charts QM identifying each of these neighborhoods with 8220), the ball of radius 2 in RM. Let Vi C Up correspond (via ) to the annulus {1/2 < lac°| < 2} C Run. Consider

go: {1/2 Projk(vl, L

I-)

PL

with inverse P *-> Range (P). The set Proj k l ) is a subset of the vector space of symmetric endomorphisms End+(v)

== {A E EndlVl,

A*

= A }.

The space End+ (V) is equipped with a natural inner product ( B ) :=

We denote by

5l tr(AB),

vA, B E End-*(V).

(1.2.2)

II II the norm on End+ (V) induced by this inner product, l

11A112 = 5 ttlA2l. Note that the subset Proj J V ) C End+ (V) can alternatively be described by the equalities

P2 = p, t r P = l . This proves that Projm / ) is a closed subset of End+ (V). From the equality 1 k - t I P 2 = § t r P = §» 'v'P G Projm / ) 2

lIPII 2

we deduce that Projm / ) is also bounded subset of End+ (V). The bijection

P : Gr k l ) -> Projk(V), L »-> PL induces a topology on Grk (V), and with this topology Grk (V) is a compact metric space. We want to show that Grk,(V) has a natural structure of smooth manifold compatible with this topology. To see this, we define for every L C Gr k(V) the set Gr

Lemma 1.2.23. (a) Let L U

F)

: = { U € G r MV); U M L J ' = 0 } . G

Gr /K7(V)- Then

L; = 0 ~< >» II - PL -|- PU : V -> V is an isomorphism.

(b) The set Grk(V, L ) is an open subset ofGrk(V).

(1.2.3)

15

Manifolds

Proof. (a) Note first that a dimension count implies that

UmL_L

0

i

`\I

\, P

U+Li

Let us show that U O L_L = 0 implies that show that

]1

Ker(]l - PL

V

i

- PL

-|-

`\I

-|-

\, P

U re L

0.

PL is an isomorphism. It suffices to

Pu) = 0.

Suppose 'u E ker(]1 - PL -|- PU). Then

0=pL(11-pL 'l'PU)'U=PLPU'U=0--)>PU'U

Q Urwke1~pL

Hence PU?) = 0, so thats E U I . From the equality (11 - pa (II - PL)V = 0 so thats E L. Hence uQU1'-LM

L

=Ural = 0 . 0 we also deduce

- Pu)V

0.

Conversely, we will show that if II - PL + PU = PL_L + PU is onto, then U -|- LJIndeed, let 'u E V. Then there exists :IJ E V such that

= V.

FU=PLX+Pua;eLl+U. (b) We have to show that, for every K 6 Grk(V, L), there exists satisfying

lIP

6

> 0 such that any U

- PK I < e

intersects LJ- trivially. Since K e Gr k(V, L) we deduce from (a) that the map It - PL PK : V -> V is an isomorphism. Note that

I (11

PK) - (11 - PL -

PL

PUlll = lInK - PU||-

The space of isomorphisms of V is an open subset of End(Vl. Hence there exists 6 > 0 such that, for any subspace U satisfying I PU - PK I < 5, the endomorphism (it - PL - PU ) is an isomorphism. We now conclude using pa (a).

Since L 6 Gr k l , L l , v L @ Gr h e ) , the collection Gr k(V§Ll; L

Q

Gr I Gr k(V, L), that associates to each linear map S : L -> L-L its graph (see Figure 1.5)

Fs =

{;u-i-S566

L + Ll

V; $ 6 L}.

The map (1.2.4) is obviously injective. We claim that it is in fact a bijection. Indeed, if U E Gr k:(V, L), then the restriction of PL to U is injective since Unke1~pL = U n L l = 0 .

(1 .2.4)

16

Lectures on the Geometry of Manifolds

LJ_

v 1

vL L Sx

\ "

I

v

L

Fig. 1.5

rS

\

I I I

L

x

Subspaces as graphs o f linear operators.

Thus, the linear map PLlu : U -> L, U 9 Up »-> PLU is a linear isomorphism because dim U dimL k. Denote by HU its inverse, HU : L -> U. It is easy to see that U is the graph of the linear map i

Z

5 : L -> L l , Sac =

PLiHUX.

We will show that the bijection (1.2.4) is a homeomorphism. We first prove that it is continuous by providing an explicit description of the orthogonal projection PpS Observe first that the orthogonal complement of FS is the graph of -S* : L-L -> L.

More precisely, 1"SJ.=1*_S*={y-s*yeL_L+L=v; y Q L - L } .

Let ' U = P L ' U + P L J . ' U = ' U L - l - U L + E V (see Figure 1.5). Then P1*S'U=LU+SCC, :UE /

\.

\

I

§1xEL,

L

\

'U

(;z:+Sa9l€II;9§ at

Z / € L J - such that

+ S*y

UL

Sac y 'UL_L L_L Consider the operator S : L ® LJ- -> L ® which has the block decomposition

8 Then the above linear system can be rewritten as 8. Now observe that S2

H :u

'UL

y

'UL-L

]lL+S*S 0

] 0

HL +SS*

Hence S is invertible, and S-12

(]lL-I-S*S) 0

0

l:l]_L_l_ -l-SS*)'1

S

17

Manifolds

(HL + S*S)'1S* -(EL -|- SS*l-1

-I-S*S)'1

l]l L

u

+ss*)-1s

We deduce l]lL + S*S)-1'UL

QUO

-|-

(HL

-|-

S*Sl-1S*'ULJ_

and,

Pos?)

QUO

[ ] S33

Hence PpS has the block decomposition IlL

Pos

S

]

l]l L

so

[lm + s*s)-1

(aL + s * 5 > - l s *

-I-S*Sl-1

(aL -|- S*S)'1S*

+ 5*s)

S(J1L +S *s) -15 *

(l.2.5)

This proves that the map

Hom(L, Ll) 9 S »-> Pos E Grklv, L) is continuous. Note that if U 6 Grk(V, L), then with respect to the decomposition V L + L-L the projector PU has the block form

A C

PU

PLPuIL

B D

PLPUIL

]

7

PLJ.PUIL PLLPUILJ. where for every subspace K V we denoted by IK : K -> V the canonical inclusion, then U = Fs. If U = Fs, then (l.2.5) implies 0(U) = S(11L + s*5> -1 7 so S = C A 1 Since A and C depend continuously on PU we deduce shows that the inverse of the graph map

PLPuIL

A(U) = (HL +5* 5>- 1

7

PLPUILJ.

H o m ( L , L l 9 S »-> FS 6 Gr 1mlL1,Ll), i

= 0, 1,

and the correspondence So -> $1 is smooth. This shows that Grk(V) has a natural structure of smooth manifold of dimension

dim Grklvl = dim Hom(L, Ll)

k ( n - k).

~

Gr k(@") is defined as the space of complex k-dimensional subspaces of Q". It can be structured as above as a smooth manifold of dimension 2k(n - k). Note that Gp1 (RTLI = RP7n-1, and Gp1 (ml = 9-1V1

is smooth. These structures provide an excellent way to formalize the notion of symmetry. (a) (Rn, -I-) is a commutative Lie group. (b) The unit circle S1 can be alternatively described as the set of complex numbers of norm one and the complex multiplication defines a Lie group structure on it. This is a commutative group. More generally, the torus T" is a Lie group as a direct product of n circles.2 (c) The general linear group GL(n, K) defined as the group of invertible n X n matrices with entries in the field K = R, (C is a Lie group. Indeed, GL(n, K) is an open subset (see Exercise l.1.3) in the linear space of n X n matrices with entries in K. It has dimension den2, where do is the dimension of K as a linear space over R. We will often use the

alternate notation GL(K'") when referring to GL(n, K). (d) The orthogonal group O(n) is the group of real n

T.Tt

X

n matrices satisfying

Jl.

m,,(1R()#

o f

To describe its smooth structure we will use the Cayley transform trick as in [113] (see also the classical [133]). Set

{T

E MAR); det(l +Tl ¢ 0

}.

The matrices in m,,(1R()# are called non exceptional. Clearly II E O(n)# = O(n) VI Mn (lR)# so that O(n)# is a nonempty open subset of O(nl. The Cayley transform is the map # : My(Rl# -> m(02a) defined by

A »-> A#

O(n)

-

S »-> LT(S) = T S.

20ne can show that any connected commutative Lie group has the from Tn X R "

19

Manifolds

We obtain an open cover of O(n): O(n) =

U

T - O(n)#.

T€O(n)

Define QT : T - O ( n ) #

-)

Q(n) by S »-> (T-1 - S)#. One can show that the collection

IT . o#,@T )

T€0(n)

defines a smooth structure on O(n). In particular, we deduce dim O(n)

n(n - 1)/2.

Inside O(n) lies a normal subgroup (the special orthogonal group)

SO(n)

= {T

6 O(n)

det T = 1}.

The group SO(n) is a Lie group as well and dim SO(n)

= dim O(n).

(e) The unitary group U (in) is defined as

U(n) = {T 6 GL (n,(Cl; T - T * = ] l } , where T* denotes the conjugate transpose (adjoint) of T. To prove that U (n) is a manifold one uses again the Cayley transform trick. This time, we coordinatize the group using the space u(n) of skew-adjoint (skew-Hermitian) n X n complex matrices (A = -A*). Thus U (n) is a smooth manifold of dimension

dim U(n)

= dim_u(n)

n2

Inside U(n) sits the normal subgroup SU(n), the kernel of the group homomorphism det : U(n) -> S1. SU(nl is also called the special unitary group. This a smooth manifold of dimension 112 - 1. In fact the Cayley transform trick allows one to coordinatize SU(n) using the space

W)

{A E _u(nl , teA

0}.

Exercise 1.2.26. (a) Prove the properties (i)-(iii) of the Cayley transform, and then show that (T 0(n)#, \ P T ) T € 0 ( n ) defines a smooth structure on O(nl. (b) Prove that U(n) and SU(n) are manifolds. (c) Show that O(n), SO(n), U(n), SU(n) are compact spaces. (d) Prove that SU(2) is diffeomorphic with S3. (Hint: think of S3 as the group of unit

-

quaternions.)

Exercise 1.2.27. Let SL(n; K) denote the group of n X n matrices of determinant 1 with entries in the field K = K, C Use the implicit function theorem to show that SL(n; K) is a smooth manifold of dimension dK(77,2 - 1), where did = dim K.

20

Lectures on the Geometry of Manifolds

Exercise 1.2.28. (Quillen). Suppose Vo, V1 are two real, finite dimensional Euclidean space, and T : V0 -> V1 is a linear map. We denote by T* is adjoint, T* : V1 -> Vo, and by FT the graph of T,

FT =

{ ( U 0 7 U1l

E V0 ® V13

'U1

T'U0)}.

I

We form the skew-symmetric operator

XzV0QEV1->V0@BV1, X

[ ] [ 00

U1

0 T* T 0

N ] U0

U1

We denote by CT the Cayley transform of X , C'T=(]1-Xl(]l-i-X)

7

and by R0:V0€BV1->V0€BV1 the reflection

Ro Show that RT = CTRL is an orthogonal involution, i.e.,

R§~=1, R T = R T , and ker(]1 - RT)

FT. In other words, RT is the orthogonal reflection in the subspace

FT,

RT = 2PPT

11,

where PPT denotes the orthogonal projection onto FT.

Exercise 1.2.29. Suppose G is a Lie group, and H is an abstract subgroup of G. Prove that the closure of H is also a subgroup of G. Exercise 1.2.30. (a) Let G be a connected Lie group and denote by U a neighborhood of 1 6 G. If H is the subgroup algebraically generated by U show that H is dense in G. (b) Let G be a compact Lie group and g 6 G. Show that l 6 G lies in the closure of {9" ; n G Z

\ {0}}-

Remark 1.2.31. If G is a Lie group, and H is a closed subgroup of G, then H is in fact a smooth submanifold of G, and with respect to this smooth structure H is a Lie group. For a proof we refer to [61, 128]. In view of Exercise 1.2.29, this fact allows us to produce many examples of Lie groups.

1.2.4 How many manifolds are there? The list of examples in the previous subsection can go on for ever, so one may ask whether there is any coherent way to organize the collection of all possible manifolds. This is too general a question to expect a clear cut answer. We have to be more specific. For example, we can ask

Manifolds

21

Question 1: Which are the compact, connected manifolds of a given dimension d? For d = 1 the answer is very simple: the only compact connected 1-dimensional manifold is the circle S1. (Can you prove this?) We can raise the stakes and try the same problem for d = 2. Already the situation is more elaborate. We know at least two surfaces: the sphere S2 and the torus T2. They clearly look different but we have not yet proved rigorously that they are indeed not diffeomorphic. This is not the end of the story. We can connect sum two tori, three tori or any number g of tori. We obtain doughnut-shaped surface as in Figure 1.6

Fig. 1.6 Connected sum o f 3 tori.

Again we face the same question: do we get non-diffeomorphic surfaces for different choices of g? Figure 1.6 suggests that this may be the case but this is no rigorous argument. We know another example of compact surface, the projective plane RIP2, and we naturally ask whether it looks like one of the surfaces constructed above. Unfortunately, we cannot visualize the real projective plane (one can prove rigorously it does not have enough room to exist inside our 3-dimensional Universe). We have to decide this question using a little more than the raw geometric intuition provided by a picture. To kill the suspense, we mention that RIP2 does not belong to the family of donuts. One reason is that, for example, a torus has two faces: an inside face and an outside face (think of a car rubber tube). RP2 has a weird behavior: it has "no inside" and "no outside". It has only one side! One says the torus is orientable while the projective plane is not. We can now connect sum any numbers of lRlPD2's to any donut an thus obtain more and more surfaces, which we cannot visualize and we have yet no idea if they are pairwise distinct. A classical result in topology says that all compact surfaces can be obtained in this way (see [90]), but in the above list some manifolds are diffeomorphic, and we have to describe which. In dimension 3 things are not yet settled3 and, to make things look hopeless, in dimension > _ 4 Question 1 is algorithmically undecidable . We can reconsider our goals, and look for all the manifolds with a given property X . In many instances one can give fairly accurate answers. Property X may refer to more than the (differential) topology of a manifold. Real life situations suggest the study of manifolds with additional structure. The following problem may give the reader a taste of the types of problems we will be concerned with in this book. 3 Things are still not settled in 2007, but there has been considerable progress due to G. Perelman's proof of the Poincaré's and Thursron's conjectures.

22

Lectures on the Geometry of Manifolds

Question 2: Can we wrap a planar piece of canvas around a metal sphere in a one-to-one fashion? (The canvas is flexible but not elastic). A simple do-it-yourself experiment is enough to convince anyone that this is not possible. Naturally, one asks for a rigorous explanation of what goes wrong. The best explanation of this phenomenon is contained in the celebrated Theorem Egregium (Golden Theorem) of Gauss. Canvas surfaces have additional structure (they are made of a special material), and for such objects there is a rigorous way to measure "how curved" are they. One then realizes that the problem in Question 2 is impossible, since a (canvas) sphere is curved in a different way than a plane canvas. There are many other structures Nature forced us into studying them, but they may not be so easily described in elementary terms. A word to the reader. The next two chapters are probably the most arid in geometry but, keep in mind that, behind each construction lies a natural motivation and, even if we do not always have the time to show it to the reader, it is there, and it may take a while to reveal itself. Most of the constructions the reader will have to "endure" in the next two chapters constitute not just some difficult to "swallow" formalism, but the basic language of geometry. 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. "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, nevertheless, it cannot be avoided. Just as anyone who wishes to give expressions to his thoughts with ease must spend laborious hours learning language and writing, so here too the only way we can lessen the burden of formulae 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, our object being to get an insight into the nature of space [...]. Whoever sets out in quest of these goals must possess a perfect mathematical equipment from the outset." H. Weyl: Space, Time, Mattel:

Chapter 2

Natural Constructions on Manifolds

The goal of this chapter is to introduce the basic terminology used in differential geometry. The key concept is that of tangent space at a point which is a first order approximation of the manifold near that point. We will be able to transport many notions in linear analysis to manifolds via the tangent space.

2.1 The tangent bundle 2.1.1 Tangent spaces We begin with a simple example which will serve as a motivation for the abstract definidons Example 2.1.1. Consider the sphere

S2) 2 +

(

2

Z 2 I +

l

in RB.

We want to find the plane passing through the North pole N (0, 0, 1) that is "closest" to the sphere. The classics would refer to such a plane as an oscillator plane. The natural candidate for this oscillator plane would be a plane given by a linear equation that best approximates the defining equation 3132 +3/2 -l- Z2 = 1 in a neighborhood of the North pole. The linear approximation of 232 -1- y2 -l- Z2 near N seems like the best candidate. We have 1v2+y2+22

l

2(z

1) + O(2),

where 0(2) denotes a quadratic error. Hence, the oscillator plane is Z = 1. Geometrically, it is the horizontal amine plane through the North pole. The linear subspace {z = 0} C $3 is called the tangent space to S2 at N. The above construction has one deficiency: it is not intrinsic, i.e., it relies on objects "outside" the manifold S2. There is one natural way to fix this problem. Look at a smooth path 'y(t) on S2 passing through N at t = 0. Hence, t »-> 'y(t) 6 R3, and

Iv(t)l 2 = 1.

(2.1.1)

If we differentiate (2.1.1) at t = 0 we get 07(0), 'y(0)) = 0, i.e., (0) J_ 'y(0), so that (0) lies in the linear subspace Z = 0. We deduce that the tangent space consists of the tangents to the curves on S2 passing through N. 23

24

Lectures on the Geometry of Manifolds

This is apparently no major conceptual gain since we still regard the tangent space as a subspace of R3, and this is still an extrinsic description. However, if we use the stereographic projection from the South pole we get local coordinates (u, Fu) near N, and any curve 'y(t) as above can be viewed as a curve t »-> (u(t), v(t)) in the (u, UI plane. If 'u,(t),y(t)), then (WI is another curve through N given in local coordinates by t »-> (_ (0l

(/5(0)

(U°(0l»'ul(0) )

\, P

`\I

The right-hand side of the above equality defines an equivalence relation on the set of smooth curves passing trough (0, 0). Thus, there is a bijective correspondence between the tangents to the curves through N, and the equivalence classes of "~". This equivalence relation is now intrinsic modulo one problem: "~" may depend on the choice of the local coordinates. Fortunately, as we are going to see, this is a non-issue.

Definition 2.1.2. Let M771 be a smooth manifold and Po a point in M. Two smooth paths a, B : (-e, 5 -> M such that a(0) = B(0) = Po are said to have a ]?rst order contact at Po if there exist local coordinates (x) = (xl, . . . ,:z:m*) near Po such that

w'o(0)

£i35(0l,

i

where

a(t) = ( m )

(al (t),

it) = ( t ) =

(we), . . »w2"(t))-

•

•

•

,x;"(t) )

7

and •

We write this O d ~ 1

Lemma 2.1.3.

~1 is an equivalence relation.

Sketch of proof. The binary relation -1 is obviously reflexive and symmetric, so we only have to check the transitivity. Let a -1 B and B -1 fy. Thus there exist local coordinates (w)=(x% ,xm) and (y) = Ly1, . 7 ym) near Po such that

(mum) = (am) and (y°@(0) )

IzM0II-

The transitivity follows from the equality 9(0)

Z

v€TpM.

ppM

Any local coordinate system .cc = ( x ) defined over an open set U C M produces a natural basis l88,(p)l of TpM, for any p 6 U. Thus, an element U 6 TU LIpaU TpM is completely determined if we know T

..

which tangent space does it belong to, i.e., we know the point p the coordinates of 'U in the basis (6,,,, (p) ),

7r(0) E M ,

II

D

§jx(v)6(p). i

We thus have a bijection m,,,.TU->U°">M)

,a:'"),...,y"(:c1, . . . ,a:'")).

»-> (y1(:1:1,

A basis in TpM is given by { 3 . while a basis of TqN is given by { by, If a, 5 : (-6, 5) -> M are two smooth paths such that a(0) = 5(0) = p, then in local coordinates they have the description

a(t) do)

1

(5/31

(al,

(131 (go,

Z

B(t)=(H1E(t)»

»H1Q"(t))»

7

HvZ"(t))»

7

ia""(0))» 5(0)=(i:}2(0)»

7

=i:Z'(0) l.

Then F(@(¢)) = (y1(t),

,y;:(t))

F(@()) = (1/é»(t)»

»yE(t)) = (3/1(=@;()),

Flag)

DF( co) )

-|-

aw)

d dt

llllllllll

Z

(y1(w;lt)

(1/1(=@;()),

I

-|-

H1200 ),

'rL

F(@(t))

( 2- 1- 3) j=1

3/3(0)3yi ..

7

y"(:v2,(t) )),

, y " ( ; ( t ) ) ),

,y"(£uZ,(t)+w2(t)))-

iii

691

j= 1

i=1

33%

I

=i;;(0) E9yj

7

29

Natural Constructions on Manifolds

DF(@(0)

-|-

n

d dt

DF(B(0) )

F(@(t)) (a, b, c). Describe explicitly the discriminant set

of Tr.

30

Lectures on the Geometry of Manifolds

Definition 2.1.16. Suppose U, V are finite dimensional real Euclidean vector spaces, (9 C U is an open subset, and F : (9 -> V is a smooth map. Then a point u 6 (9 is a critical point of F if and only if

rank (D'UF : U -> V) < m i n ( dim U, d i v ) . Exercise 2.1.17. Show that if F : M -> N is a smooth map and dim N < _ dim M, hen for every q 6 N \ As the fiber F-1(q) is either empty, or a submanifold of M of codimension dim N. Theorem 2.1.18 (Sard). Suppose U, V are ]?nite dimensional real Euclidean vector spaces, (9 C U is an open subset, and F : C) -> V is a smooth map. Assume dim V < dim U . Then the discriminant set As is negligible. Proof. We follow the elegant approach of J. Milnor [97] and L. Pontryagin [111]. Set n = dim U, and m = dim V. We will argue inductively on the dimension n. For every positive integer k we denote by Cfrkp C CTF the set of points 'LL G (9 such that all the partial derivatives of F up to order A: vanish at u. We obtain a decreasing filtration of closed sets CTF

nor; nor; D

.

The case n = 0 is trivially true so we may assume n > 0, and the statement is true for any n' < n, and any m < n'. The inductive step is divided into three intermediary steps. Step 1. The set F ( C T p \ C r ) is negligible. Step 2. The set F(CT'F \ CT"'F+1) is negligible for all k > l. Step 3. The set F(CT'F) is negligible for some s u f i ciently large ac. Step 1. Set CTF := CTF u p . We will show that there exists a countable open cover of CTF such that F ( 0 j VI CTFI is negligible for all j > 1. Since CT'F is separable, it suffices to prove that every point u 6 CTF admits an open neighborhood N such that F ( N VI CTF ) is negligible. Suppose up 6 C T F . Assume first that there exist local coordinates (5/31, . . .,;1:") defined in a neighborhood N of up, and local coordinates Ly1 , . . . , ym*) near 100 = F(u0) such

{(97}j>1

that, @i('U,0)

W = 1 MY to N is described by functions ye yi(:1:1,...,a:M)

=0,

W

1,

3/(v0) = 0 ,

Na

7

and the restriction of F x1. For every t E R we set

al

3\ft:= {(:1:1,...,:1:")€3\f, : c 1 = t }

such that

7

and we define

m) Gt I Nt -> R m - 1 7 ( . . . a Observe that

l->

2 2 (3/ (t : c , . . . , §UTL)a°°°7YMlt7X27°°°aIn)

F(NN C T F I = U { t } t

X

Gt(C'rGt ).

I.

31

Natural Constructions on Manifolds

The inductive assumption implies that the sets Gt (Cfrgt) have trivial (m - l)-dimensional Lebesgue measure. Using Cavalieri's principle or Fubini's theorem we deduce that F(N O CTF) has trivial m-dimensional Lebesgue measure. To conclude Step l is suffices to prove that the above simplifying assumption concerning the existence of nice coordinates is always fulfilled. To see this, choose local coordinates (so, . . . , s") near U0 and coordinates l y , . . . ,yM) near U0 such that s£('u,0)=0,

'v'i=1,...nI y7(v0)=0,

'v'j=1,...m,

The map F is then locally described by a collection of functions yj(s17 7 sn), j l , . . . , n . Since u 6 C T F/ , we can assume, after an eventual re-labelling of coordinates, that 31/1 Now define 381 y 1 (s 1 , . . . , s n ), a:¢ -_ s~ ' , ' v-' z_ - 2 , . . . , n .

a: l

The implicit function theorem shows that the collection of functions (1131 ,...,.cc") defines a coordinate system in a neighborhood of U0. We regard by as functions of so. From the definition we deduce y1 = acl . k CTF ( )

.

k

:= CHI; Crl+1. Since U0 6 C r i ) , we can find local coordinates (s1, . . .,s") near U0 and coordinates (y1, . . . , ym) near "U0 such that

Step 2. Set

•

smog

Z

0,

Vi

I

31y1

6(81)j

l,...n, (Uu)

ye I v ) = 0,

'v'j

Z

l,...m,

0, v j = 1 , . . . , k ,

and 3/c-I-lyl

3(81)k+1 (Uo) ¢ 0. Define

alkyl

2(8)

6(51)k

7

and set 1:i := s', V11 2, . . . , n. Then the collection ( x ) defines smooth local coordinates on an open neighborhood N o f U Q , and Cfrkp HN is contained in the hyperplane {a:1 = 0}. Define

G : NVW{:I:1 = 0 } -> V ; G(:z:2,...,:1:M) = F(0,:1:2,..

7

so"

.

Then

co;my = Org,

F(C1°kF

mol =

G(c¢~'g),

and the induction assumption implies that G(C'r'é) is negligible. By covering CTF( k ) with a countably many open neighborhood {Ne}e>1 such that F(C'r'§" Vulg) is negligible we conclude that F(C'rF( k ) ) is negligible.

Step 3. Suppose k > Q . We will prove that F(C'rk'F) is negligible. More precisely, we will show that, for every compact subset S C (9, the set F ( S o C7r°k'F) is negligible.

32

Lectures on the Geometry of Manifolds

From the Taylor expansion around points in C'rak'F OS we deduce that there exist numbers To 6 (0, l ) and A0 > 0, depending only on S, such that, if C is a cube with edge 1° < to which intersects CW; OS, then diamF(C) < A0rk

7

where for every set A C V we define

diam(A) := sup{

la1

G12l5

611102

pA}.

In particular, if um denotes the m-dimensional Lebesgue measure on V, and ,un denotes the n-dimensional Lebesgue measure on U, we deduce that there exists a constant A1 > 0 such that

urn(F(cll
Y

A x X 9 (A,:1:) »-> FIZUI 6 Y. Suppose S is a submanifold of Y such that F is transversal to S. Define

ZS Z = F-1lgl

C

A

X

X.

Prove that if A0 6 A is a regular value of the natural map

ZsC A xX->A, then the map F o : X -> Y is transversal to S.

2.1.4 Vector bundles The tangent bundle TM of a manifold M has some special features which makes it a very particular type of manifold. We list now the special ingredients which enter into this special structure of TM since they will occur in many instances. Set for brevity E := TM, and F := RM (m = dim M). We denote by Aut(F) the Lie group GL(n, R) of linear automorphisms of F. Then (a) E is a smooth manifold, and there exists a surjective submersion 7r E -> M. For every U C M we set E IN:= 7r"1(U). (b) From (2.l.4) we deduce that there exists a trivializing cover, i.e., an open cover U of M , and for every U G 'LL a diffeomorphism

\ItuZElu->UXF,

U

»-> (p =

v), @g(@)).

(bl) (Dp is a diffeomorphism Ep -> F for any p € U. (b2) If U, V G 'LL are two trivializing neighborhoods with non empty overlap U n V pV o (8) then, for any p Q U Fl V, the map VU(p) : F -> F is a linear isomorphism, and moreover, the map

p I-> @VU(P) G Aut(F)

is smooth. In our special case, the map QvUIP) is explicitly defined by the matrix (2. 1.4) I

App)

i

(p) 1E.

The smoothness of the scalar multiplication means that it is smooth map

RxE->E. There is an equivalent way of defining vector bundles. To describe it, let us introduce a notation. For any vector space F over the field K = R, (C we denote by GLK(F), (or simply GL(F) if there is no ambiguity concerning the field of scalars K) the Lie group of linear automorphisms F -> F. According to Definition 2.1.23, we can find an open cover (Ua) of M such that each of the restrictions Ea = E INa is isomorphic to a product \110, : Ea = F X UQ- Moreover, on the overlaps UQ VW UP, the transition maps .905 = \1/o\1/0 1 can be viewed as smooth maps

~

.QoB

: Uo, VI UB -> GL(F).

They satisfy the cocycle condition (a) gas I HF (b) QQBQBWQwQ

up over Ua O UB H U p

Conversely, given an open cover (Ua) of M, and a collection of smooth maps £106 : Uo, O UB -> GL(F) satisfying the cocycle condition, we can reconstruct a vector bundle by gluing the product bundles Ea = F X UCI on the overlaps Uo, n UB according to the gluing rules the point (v,.5c) 6 Ea is identified with the point

(9@(:@)w)

6 EB Va: 6 Uo, re UB-

The details are can*ied out in the exercise below. We will say that the map 96a is the transition from the a-trivialization to the Btrivialization, and we will refer to the collection of maps (950) satisfying the cocycle

35

Natural Constructions on Manifolds

condition as a gluing cocycle. We will refer to the cover (Ua) as above as a trivializing cover. Exercise 2.1.24. Consider a smooth manifold M, a vector space V, an open cover (UQ), and smooth maps Q0B : Uo, re UB -> GLIV)

satisfying the cocycle condition. Set

X:=

UV x U O x { a } . Of

We apologize X as the disjoint union of the topological spaces U a X V , and we define a relation ~ C X X X by V X U a x { a } 9 ('u,,:1:,a)

i v

(v, '

a

B) Q V

X

UB

def a

X

{/5'} -:

zu,

'U =gg0,l£l3)'LL.

equipped with the quotient (a) Show that is an equivalence relation, and E = X/ topology has a natural stnicture of smooth manifold. (b) Show that the projection 77' : X -> M , (7u,,a:,a) »-> as descends to a submersion I'\J

E -> M. (c) Prove that (E, 7r7 M, V) is naturally a smooth vector bundle.

Definition 2.1.25. A description of a vector bundle in terms of a trivializing cover, and a gluing cocycle is called a gluing cocycle description of that vector bundle. Exercise 2.1.26. Find a gluing cocycle description of the tangent bundle of the 2-sphere. In the sequel, we will prefer to think of vector bundles in terms of gluing cocycles. Definition 2.1.27. (a) A section in a vector bundle E l> M defined over the open subset U C M is a smooth map S : U -> E such that s(p) E Ep

'/r"1(p), Vp E U

4--> f r o s = ]1U.

The space of smooth sections of E over U will be denoted by I`(U, E) or COO(U, E) . Note that F(U, E) is naturally a vector space. We will use the simpler notation C°°(E) when referring to the space of sections of E over M. (b) A section of the tangent bundle of a smooth manifold is called a vector field on that manifold. The space of vector fields over on open subset U of a smooth manifold is denoted by Vect (U). Proposition 2.1.28. Suppose E -> M is a smooth vector bundle with standard fiber F, defined by an open cover lUa)a6AJ and gluing cocycle

960 : Uog -> GL(F). Then there exists a natural bueetion between the vector space of smooth sections of E, and the set of families of smooth maps { sa : Uo, -> F; o f 6 A }, satisfying the following gluing condition on the overlaps sO(:1:) = go5(:1:)s5(:1:), Va: 6 Uo,

UB-

36

Lectures on the Geometry of Manifolds

Exercise 2.1.29. Prove the above proposition. Definition 2.1.30. (a) Let Ei 1 Mt be two smooth vector bundles. A vector bundle map consists of a pair of smooth maps f : M1 -> M2 and F : E1 -> E2 satisfying the following properties.

.

The map F covers f , i.e., F(E,} ) diagram below is commutative

Equivalently, this means that the F

E1

>

'/T2

W1

.

ET

f

M1

)

M2

The induced map F .• E p1 - > E f l2p l is linear.

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. (b) If E and F are two vector bundles over the same manifold, then we denote by Hom(F, F) the space of bundle maps E -> F which cover the identity ]lM. Such bundle maps are called bundle morphisms.

For example, the differential Df : TM -> TN covering f .

Df

of a smooth map

f

M - > N is a bundle map

Definition 2.1.31. Let E l> M be a smooth vector bundle. A bundle endomorphism of E is a bundle morphism F : E -> E. An automorphism (or gauge transformation) is an invertible endomorphism. Example 2.1.32. Consider the trivial vector bundle M -> M over the smooth manifold M. A section of this vector bundle is a smooth map u : M -> R". We can think of u as a smooth family of vectors (141:) G RnlrrélVfAn endomorphism of this vector bundle is a smooth map A : M -> EI1dRlRnl. We can think of A as a smooth family of n X n matrices

Ax

GW)

0é(:11)

021(H3)

a )

(.l a2 (.1)

al

7

a;(a3) 6 C°°(M).

61/i'(H13 @§(f1>) 0,26/1) The map A is a gauge transformation if and only if det A ; é 0 , V : 1 : € M .

Exercise 2.1.33. Suppose El, E2 -> M are two smooth vector bundles over the smooth manifold with standard fibers F1, and respectively F2. Assume that both bundles are defined by a common trivializing cover (U0,)0,6A and gluing cocycles .QBQ : Uog -> GL(F1)7 h e : Uog -> GL(F2)-

37

Natural Constructions on Manifolds

Prove that there exists a bijection between the vector space of bundle morphisms Hom(E, F), and the set of families of smooth maps

{TO, : Uo -> Hom(F1,F2);

a 6 A },

satisfying the gluing conditions Tglilt)

= h 0 0 ( ) T O ( ) ( x ) 9 0 1,

V113

6 Ugh.

um

Exercise 2.1.34. Let V be a vector space, M a smooth manifold, an open cover of M , and gag, hog : Uo, O UB -> GL(V) two collections of smooth maps satisfying the cocycle conditions. Prove the two collections define isomorphic vector bundles if and only they are cohomologous, i.e., there exist smooth maps (pa : Uo, -> GL(V) such that -l

h

2.1.5

¢a9QB¢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.

Example 2.1.35. (The tautological line bundle over RIP" and (CIP'"). First, let us recall that a rank one vector bundle is usually called a line bundle. We consider only the complex case. The total space o f the tautological or universal line bundle over CIP" is the space

un =

UC =

{ (z,L)

G

(c"+1

x @1p"§

Z

belongs to the line L C

(STL-l-1

Let 77` : UC -> CIP" denote the projection onto the second component. Note that for every L Q CIP", the fiber through 7r-1 (L) = 'LLC L coincides with the one-dimensional subspace in c"+1 defined by L. Example 2.1.36. (The tautological vector bundle over a Grassmannian). We consider here for brevity only complex Grassmannian Gr klCCnl . The real case is completely similar. The total space of this bundle is Uk,n =

up" =

(z, L) E (Zn

X

Gr I¢(@N) ;

Z

belongs to the subspace L C Q"

}.

If or denotes the natural projection 'fr : U p T L -> Gr IO, u0=s2\{5}, U1

52\{N}.

In this case, we have only a single nontrivial overlap, Un O Us- Identify U0 with the complex line (C, so that the North pole becomes the origin Z = 0. For every n 6 Z we obtain a complex line bundle Ln->S2 defined by the open cover {U0, U1} and gluing cocycle 9

n

910(Z)

Z

7

UP \

'v'z€(C*

w

A smooth section of this line bundle is described by a pair of smooth functions u0:U0-HC, u1:U1-MC,

which along the overlap U0

n U1 satisfy the equality u1(z)

Z

'"'u,0(z). For example, if

n > 0, the pair of functions u0(z) = z"', u1(p) = l, Vp E U1

defines a smooth section of Ln.

Exercise 2.1.39. We know that (CIP1 is diffeomorphic to S2_ Prove that the universal line bundle Un -> (III1 is isomorphic with the line bundle L-1 constructed in the above example. Exercise 2.1.40. Consider the incidence set j

{(flaL) E ( CIP". This manifold is called the complex blowup of lcn-1-1 at the origin. The family of vector bundles is very large. The following construction provides a very powerful method of producing vector bundles.

Definition 2.1.41. Let f : X -> M be a smooth map, and E a vector bundle over M defined by an open cover (UQ) and gluing cocycle (gag). The pullback of E by f is the vector bundle f*E over X defined by the open cover f-1(UO,), and the gluing cocycle (QQ5 o

f)-

39

Natural Constructions on Manycolds

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. The pullback operation defines a linear map between the space of sections of E and the space of sections of f*E. More precisely, if s 6 I`(E) is defined by the open cover (UQ), and the collection of smooth maps (so, then the pullback f*s is defined by the open cover f-1(U0,), and the smooth maps (sOf o f ). Again, there is no difficulty to check the above definition is independent of the various choices. 7

Exercise 2.1.42. For every positive integer lc consider the map Pa : QIP1 -> QIP1,

Show that paLn Example 2.1.38.

f\J

Lkna

P1 M of rank T' as a collection of 7° sections up, . . . , u which are pointwise linearly independent. One can naively ask the following question. Is every vector bundle trivial? We can even limit our search to tangent bundles. Thus we ask the following question. 15 it true that for every smooth manzfold M the tangent bundle T M i5 trivial (as a vector bundle)? Let us look at some positive examples. Example 2.1.47. TS1 'E KS.. Let 9 denote the angular coordinate on the circle. Then 36 is a globally defined, nowhere vanishing vector field on S1. We thus get a map

£51 -> TS1 7 (s, 9) 1-> (S3979) E $951 which is easily seen to be a bundle isomorphism. Let us carefully analyze this example. Think of S1 as a Lie group (the group of complex numbers of norm l). The tangent space at Z = l, i.e., 9 = 0, coincides with the subspace 8 Re Z = 0, and 69 11 is the unit vertical vector j. Denote by Re the counterclockwise rotation by an angle 9. Clearly RE is a diffeomorphism, and for each 9 we have a linear isomorphism De

l0=0

RE : T151 -> T9s1.

Moreover,

69 = DO l9=0 Red The existence of the trivializing vector field 36 is due to to our ability to "move freely and coherently" inside S1. One has a similar freedom inside a Lie group as we are going to see in the next example.

Example 2.1.48. For any Lie group G the tangent bundle TG is trivial. To see this let n = dim G, and consider e l , . . . , en a basis of the tangent space at the origin, To G. We denote by Rg the right translation (by g) in the group defined by R g : :co-> : I : - g , VJJGG.

Rg is a diffeomorphism with inverse R9-1 so that the differential DRg defines a linear isomorphism DRg : TIG -> TgG. Set ,n. Eilis ) = DRgl€¢) 6 TAG, i = 1, Since the multiplication G X G -> G, (g, h) »-> g h is a smooth map we deduce that the vectors Et (g) define smooth vector fields over G. Moreover, for every g Q G, the collection

-

{E1 (g), . . . , En (g)} is a basis of TgG so we can define without ambiguity a map =@G -> TG,

I

(9;X1 ,. . . ,XTL) »-> 9;

XE¢l9)

).

One checks immediately that is a vector bundle isomorphism and this proves the claim. In particular TS3 is trivial since the sphere S3 is a Lie group (unit quaternions). (Using the Cayley numbers one can show that TS7 is also trivial, see [112] for details).

41

Natural Constructions on Manifolds

We see that the tangent bundle TM of a manifold M is trivial if and only if there exist vector fields Xl, . . . ,Xm (m = d i M ) such that for each p 6 M, X1(p), . . . , X m ( p ) span T p M . This suggests the following more refined question.

Problem Given a manifold M , compute v ( M ) , the maximum number ofpointwise linearly independent vector fields over M . Obviously 0 < v ( M ) < dim M and T M i5 trivial if and only if v ( M ) = dim M . A special instance of this problem is the celebrated vector field problem: compute vlgnl for any n > l. We have seen that 'u(S") = n for n = 1, 3 and 7. Amazingly, these are the only cases when the above equality holds. This is a highly nontrivial result, first proved by J.F. Adams in [3] using very sophisticated algebraic tools. This fact is related to many other natural questions in algebra. For a nice presentation we refer to [91]. The methods we will develop in this book will not suffice to compute v(S'") for any n, but we will be able to solve "half" of this problem. More precisely we will show that v(S") = 0 if and only if n is an even number. In particular, this shows that TS211 is not trivial. In odd dimensions the situation is far more elaborate (a complete answer can be found in [3]).

Exercise 2.1.49. vls2k 1) > l for any k

l.

The quantity v ( M ) can be viewed as a measure of nontriviality of a tangent bundle. Unfortunately, its computation is highly nontrivial. In the second part of this book we will describe more efficient ways of measuring the extent of nontriviality of a vector bundle. 2.2 A linear algebra interlude We collect in this section some classical notions of linear algebra. Most of them might be familiar to the reader, but we will present them in a form suitable for applications in differential geometry. This is perhaps the least glamorous part of geometry, and unfortunately cannot be avoided.

"Convention. All the vector spaces in this section will tacitly be assumed finite dimensignal, unless otherwise stated. 2.2.1 Tensor produets Let E, F be two vector spaces over the field K (K sum 'J'(E, F)

lm). Consider the (infinite) direct

(e,f)€EXF

K.

Equivalently, the vector space 'J'(E, F) can be identified with the space of functions C : E X F -> K with finite support. The space 'J'(E, F) has a natural basis consisting of "Dirac functions" 5ef z E X F - > ] K ,

(.ac,y)»->

l if (:13,y) (6,f) 0 if (Hay) vi (6,/') Z

42

Lectures on the Geometry of Mamfolds

In particular, we have an injection 1

E

X

F

Q

m

,

h

Inside 'J'(E, F) sits the linear subspace R(E, F) spanned by

*6e,f

-

5»e,f» A5e,f

(5e+e',f

- 5e,Af»

-

5e,f

-

5e',f» 5e,f+f' - 5e,f - 5e,f'

7

where 6,6I 6 E , f , f ' 6 F,andA 6 K. Now define

E ®11< F : = or(E,F)/R(E,F), and denote by 71' the canonical projection or : 'J'(E, F) -> E

f

€

a I

(X)

F. Set

7f((5e,f)~

1

We get a natural map

E

F->E

F,eX

(X)

f »->e®f.

Obviously L is bilinear. The vector space E ®1K F is called thetensor product of E and F over K. Often, when the field of scalars is clear from the context, we will use the simpler notation E (X) F. The tensor product has the following universality property. Proposition 2.2.1. For any bilinear map QUO : E X F -> G there exists a unique linear map CD : E ( X ) F -> G such that the diagram below is commutative.

E

X

F

I,

>

E

(X)

F

:l

I G

d>

The proof of this result is left to the reader as an exercise. Note that if (80 is a basis of E, and (fy) is a basis of F, then (6z ® fy) is a basis of E ® F, and therefore dirge E ®K F =

ldiII1K

El - ldimg

F) .

~

Exercise 2.2.2. Using the universality property of the tensor product prove that there exists a natural isomorphism E (X) F = F (X) E uniquely defined by € f »-> f (X) e. The above construction can be iterated. Given three vector spaces $1 , Et, E3 over the same field of scalars K we can construct two triple tensor products: (El

(X)

EQ)

(X)

E3 and E1

(X)

(Ez

(X)

E3).

Exercise 2.2.3. Prove there exists a natural isomorphism of K-vector spaces (El lA word

of cuufiwl;

6 is not linear!

~

(X)

EQ) (X) E3 = E1 ® (E2 ® E3).

43

Natural Constructions on Manifolds

The above exercise implies that there exists a unique (up to isomorphism) triple tensor product which we denote by E1 (X) E2 (X) E3. Clearly, we can now define multiple tensor products: E1 (X) (X) En .

---

Definition 2.2.4. (a) For any two vector spaces U, V over the Held K we denote by Hom(U, V), or HOMK(U, V) the space of IN-linear maps U -> V. (b) The dual of a K-linear space E is the linear space E* deaned as the space HOMK(E, K) of K-linear maps E -> K. For any e* 6 E* and e 6 E we set (e*,e) := e*(e).

The above constructions are factorial. More precisely, we have the following result.

Proposition 2.2.5. Suppose Ei, Ft, Gt, i = 1,2 are K-vector spaces. Let To 6 Hom (Ei, Fi), Si 6 Hom(Fi, Gi), i = 1, 2, be two linear operators. Then they naturally induce a linear operator T = T 1 ® T 2 I E 1 ® E 2 - F 1 ® F 2 » S1®S2-F1®F2

uniquely deaned by

T1 ® T2(€1 ® 821 =

lT181l

®

lT2€2l,

\V/€¢ G

E

and satisfying

(51 ® SQ) O (Tl

(X)

TQ) = (51 O To) (X) (52 O TQ).

(b) Any linear operator A : E -> F induces a linear operator AT : F* -> E* uniquely defined by

= (f*,Ae)> ve E E , f* E F*. is called the transpose or adjoint of A. Moreover (A*f*,@)

The operator At

(A o

BV = BT o AT,

VA E Hom(F,G),

B E Hom(E,F).

Exercise 2.2.6. Prove the above proposition. Remark 2.2.7. Any basis (€zl1 V which maps the vector 'U = up et to the vector LAU = ajfuf et.

46

Lectures on the Geometry of Mamfolds

On the tensor algebra there is a natural contraction (or trace) operation tr :

-1

T; ->

_1

uniquely defined by tr

'U1

®-~-®v~®

'LL

1

1

®---®uS`

(Fu,

7

U1),U2

®""U»,~®U2®°_®

S

'LL

7

'v'"7€,u7€V*

In the coordinates determined by a basis 060 of V, the contraction can be described as ¢2...¢,.

(tr T ) . ] 2 . . . _ ] 5

)

(Tub2

7

where again we use Einstein's convention. In particular, we see that the contraction coincides with the usual trace on by; (V) 2 End (V). 2.2.2

Symmetric and skew-symmetric tensors

Let V be a vector space over K = R, (C. We set W W ) := :kg (V), and we denote by S the group of permutations of T' objects. When T' = 0 we set 80 := {1}. Every permutation O' 6 S determines a linear map 'J'""(V) -> 'J"l(V), uniquely determined by the correspondences 01

®v1~»->00(1)®---®00(¢),

(X)

'v'v1,...,'v',°€V

We denote this action of O' 6 S on an arbitrary element t 6 U""(V) by it. In this subsection we will describe two subspaces invariant under this action. These are special instances of the so called Schur functors. (We refer to [46] for more general constructions.) Define S T' 'J"l(V) -> 'T"(V), S

t~(t):

1 al

07t,

u'S

and T'

'T"lvl -> 'T"(V)7

A t)

s 0

A

L

EGGS, 6(c7)(7t

0

dim V dim V

if if

Above, we denoted by 6(0) the signature of the permutation U. Note that

A0

HK.

SO

The following results are immediate. Their proofs are left to the reader as exercises.

Lemma 2.2.13. The Op€l'C1tOl"S A 7° and S are projectors of 'T"° SO

S fr 7 A 2or

i.e.,

A or.

Moreover, 0S,~(t) = S,~(ot)

S ,~(t), 0A,~(t)

A 1-(0¢) = 6(0) A ,~(t), Vt

Q

gmv).

47

Natural Constructions on Manifolds

Definition 2.2.14. A tensor T 6 'T"(V) is called symmetric (respectively skew-symmetric) if

=T

S MT)

(respectively A ,,(T)

T).

The nonnegative integer 1° is called the degree of the (skew -)symmetric tensor. The space of symmetric tensors (respectively skew-symmetric ones) of degree r will be denoted by S V (and respectively A"°V). Set

S'V

Ql§s'°v,

. o

andA'V

'r'>0

@A s 'r">0

Definition 2.2.15. The exterior product is the bilinear map

A I A V x ASV -> A'°+sv, deaned by w"A17S

(r + s)! A v~+sl°u (X) to), Vi res!

G AWE Us G A.SV.

Proposition 2.2.16. The exterior product has the following properties. (a) (Associativity)

(04A/3)A7=@/\(/5'/W)\ \1a, l5'v € A°v. In particular; U1 A

- - - /\ up =

k!Ak(v1 ®

(X)

up) =

6(07)U0(1) (X)

®UU(k)7

\V'Ui G

V.

o'€Sk

(b) (Super-commutativily) wAsS

I-l)'r'g17S /\ w ,

Vi

Q AV,

wS E ASV.

Proof. We first define a new product "Al" by CU/\17]S

A 1°-i-slW (X) UI)

which will turn out to be associative and will force A to be associative as well. To prove the associativity of /\1 consider the quotient algebra Q* = T* IN*, where 'T* is the associative algebra (@ ,~>0T'°(V), -|-, ®), and al* is the bilateral ideal generated by the set of squares { v ® v / U 6 V } . Denote the (obviously associative) multiplication in Q* by U. The natural projection 'fl' : 'T* -> Q* induces a linear map 7'l` : A' V -> Q*. We will complete the proof of the proposition in two steps.

Step 1. We will prove that the map Tl' : A' V -> Q* is a linear isomorphism, and moreover 7r(aA1B) = tr(a) U

In particular, /\1 is an associative product.

B).

(2.2.1)

48

Lectures on the Geometry of Mamfolds

The crucial observation is

frlvl.

7r(T) = or( A ,-(T)), Vt 6 It suffices to check (2.2.2) on monomials T

; € 1 ® . . . (X)

(2.2.2)

e , et 6 V. Since

@+v2e,wvev u(mod U*). Hence, for any 0 QS 7r(€1

(X)

- --

e)

(X)

= e(o)tr(eu(1l

(X)

(X)

When we sum over O' 6 S in (2.2.3) we obtain (2.2.2). To prove the injectivity of 71' note First that A* w*) As V, then w 6 kerw = U* re A' V so that w

A *(w)

(2.2.3)

€o('r)l°

0. If tr(w)

0 for some Ld 6

0.

The surjectivity of 7/ follows immediately from (2.2.2). Indeed, any 7r(T) can be alternatively described as tr(w) for some w 6 A' V. It suffices to take w = A* (T). To prove (2.2.1) it suffices to consider only the special cases when a and B are monomia1s:

A 7 ° l € 1 ® ' ° ' ® € r l 7 /3

a

As(f1 ( X )

(X)

fs)-

mfr) ( X ) A 5(f1 (X)

(X)

full

We have (X)

(X)

®e,~)® A 5l.t1®

®e,~))U

(X)

of( A 5(f1

fs)

(X)

(X)

) f5)) = we)

LJ

w).

Thus /\1 is associative.

Step 2. The product A is associative. Consider a 6 AT'V, have (aAB)A'y

f(

+

)! QA1/5

(r

-|- S

+ t)!

r!s!t!

As

(@A15 lA17

(r+sl!

B6

ASV and

iT+s+tl!

v

6

Ate.

We

A17

Hs! (7° -|- s)!t! (@A15) (T + S -|- t)! r!8!t! &A1(5Aw) = a /\ (0 /\ vl-

The associativity of A is proved. The computation above shows that e1A---Aek=k!Ak(e1®-~-®ek).

(b) The supercommutativity of A follows from the supercommutativity of /\1 (or U). To prove the latter one uses (2.2.2). he details are left to the reader.

Exercise 2.2.17. Finish the proof of part (b) in the above proposition.

49

Natural Constructions on Manifolds

The space A' V is called the exterior algebra of V. The operation A is called the exterior product. The exterior algebra is a Z-graded algebra, i.e.,

( A V /\ ASK pA*l+SV, 'v'1°,s. Note that AT'V

0 for 1°> dim V (pigeonhole principle).

Definition 2.2.18. Let V be an n-dimensional K-vector space. The one dimensional vector space A'"V is called the determinant line of V, and it is denoted by det V. There exists a natural injection

LV

:VA' V, 1,V(0)

LV('u) /\ Le (Fu)

U,

such that

0, Vu € V

This map enters crucially into the formulation of the following universality property.

Proposition 2.2.19. Let V be a vector space over K. For any K-algebra A, and any linear map QUO : V -> A such that Tb(;z:)2 = 0, there exists an unique morphism of K-algebras ® : A' V -> A such that the diagram below is commutative

V `

r

LV

>

A. V I I

I I I

7

A i.e., ® O LV

¢,

Exercise 2.2.20. Prove Proposition 2.2.19. The space of symmetric tensors S' V can be similarly given a structure of associative algebra with respect to the product S ~+5(a®B), v a e s ' ° v , @ @ s 8v

s

•

a-B

The symmetric product "-" is also commutative. Exercise 2.2.21. Formulate and prove the analogue of Proposition 2.2.19 for the algebra s' V.

617

It is often convenient to represent (skew-)symmetric tensors in coordinates. , en is a basis of the vector space V then, for any l < 7° < n, the family 6111 /\

/\€,/ 1 Gil

is a basis for AT'V so that any degree sented as

7°

If

s ' w uniquely dejined by their actions on monomials

A. Ll'U1 /\

--

A U ) = (Lil) /\

- - /\ ILUI,

and

S.

Moreover; If U

A)

V

B

>W

Ll'U1,

. . . ,u,~) =

(Lil) - - - ( L ) .

are Mo linear maps, then

ATI(BAl = (ATB)(A'°A)7 S"(BA) = (S"B)(STA). Exercise 2.2.23. Prove the above proposition. In particular, if n = dirn]K V, then any linear endomorphism L : V -> V defines an endomorphism

A"L : derV = A"V -> derV. Since the vector space det V is 1-dimensional, the endomorphism A"L can be identified with a scalar det L, the determinant of the endomorphism L. Definition 2.2.24. Suppose V is a finite dimensional vector space and A : V -> V is an endomorphism of V. For every positive integer T' we denote by 07,~(A) the trace of the induced endomorphism

AT'A A*V -> A V , and by ,MAl the trace of the endomorphism AT` : V -> V. We define

A) = 1,

A) = d i v .

Exercise 2.2.25. Suppose V is a complex n-dimensional vector space, and A is an endomorphism of V . (a) Prove that if A is diagonalizable and its eigenvalues are CL17 7 A N a then Ur(/4)

(Lil ° ' ° & ¢ _ ,

Z

¢,.(A)

ICL1

-I-

+ Ag.

1$1;1!

0.

U", n-1

C-) dt

Proof. From (2.2.4a) and (2.2.4b) we deduce using Exercise 2.2.27 that for any vector spaces V and W we have

PA- (ve9w) (to = PA-V(t) • PA- w('5l and Ps- (vebw) (to = P5»V(t) ' P$»W(t). In particular, if V has dimension n, then V 2 K" so that PA~ vltl (PA~ K(t))" and Ps' vltl (P5'° . The proposition follows using the equalities 1 PA HA,

defined on homogeneous elements wi E A , Do E As by

lwnvjls == w t v j

- (-1)"17i(»Jj

An s-algebra is called .9-commutative, if the suppercommutator is trivial, [o 7 .]

S

E

0.

Example 2.2.32. Let E = E0 ® E1 be a 5-space. Any linear endomorphism T 6 End ( E ) has a block decomposition II

7

where Tri Q End (18 Et ). We can use this block decomposition to describe a structure of s-algebra on End (E). The even endomorphisms have the form

7

while the odd endomorphisms have the form

54

Lectures on the Geometry of Mamfolds

Example 2.2.33. Let V be a finite dimensional space. The exterior algebra A'V is naturally a s-algebra. The even elements are gathered in Aeven

V

AT'V , T'

even

T'

odd

while the odd elements are gathered in Aoddv

=

As.

The .9-algebra A'V is .9-commutative.

Definition 2.2.34. Let A = A0 ® A1 be a s-algebra. An s-derivation on A is a linear operator on D 6 End (A) such that, for any an 6 A,

[D, Lx]End(Al

Z

(2.2.5)

I/Da: a

where [ , SEnd(.A) denotes the supercommutator in End (A) (with the s-structure deaned in Example 2.2.32), while for any Z 6 A we denoted by LZ the left multiplication operator a »-> Z a. An .9-derivation is called even (respectively odd), if it is even (respectively odd) as an element of the .9-algebra End (A). a

Remark 2.2.35. The relation (2.2.5) is a super version of the usual Leibniz formula. Indeed, assuming D is homogeneous (as an element of the s-algebra End (All then equality (2.2.5) becomes D

)

= (Dlv)y + (-1)l""Dlw(Dy),

for any homogeneous elements Xv by 6 A. Example 2.2.36. Let V be a vector space. Any u* 6 V* defines an odd 5-derivation of A'V denoted by in* uniquely determined by its action on monomials. T'

in* l'U0 /\

U1 /\

(-1Y [0, ac]. A is called an s-Lie algebra if it is s-anticommutative, i.e., [as, y] -|- (-l)|""||Y|[y, at] = 0, for all homogeneous elements x, y e A, and Van 6 A, Ras is a s-derivation. When A is purely even, i.e., A1 = {0}, then A is called simply a Lie algebra. The multiplication in a (s-) Lie algebra is called the (s-)bracket.

55

Natural Constructions on Manifolds

The above definition is highly condensed. In down-to-earth terms, the fact that s-derivation for all as 6 A is equivalent with the super Jacobi identity

Hy, 111, z] + l-1ll33ll1/l[y

Hy, 2], x]

[z, SDH 7

Ras

is a

(2.2.6)

for all homogeneous elements cc, y, Z Q A. When A is a purely even K-algebra, then A is a Lie algebra over K if [ , ] is anticommutative and satisfies (2.2.6), which in this case is equivalent with the classical Jacobi identity, Hw,y], z] + He, z],;u] -I-

[[w1]»y]

0> Van°,y,z€A.

(2.2.7)

End (El is a Lie Example 2.2.39. Let E be a vector space (purely even). Then A algebra with bracket given by the usual commutator: [a, 6] = ab - be. Proposition 2.2.40. Let A = A0 ® A1 be a 5-algebra, and denote by DerS (Al the vector space of s-derivations of A.

(a) For any D 6 DerS(A), its homogeneous components Do, D1 6 End (Al are also s-derivations. (b) For any D, D' 6 DerS(A), the s-cornrnutator [D, D']End(Al is again an 5-derivation. (c) Vi

6 A the bracket B 11: : a »->

[a, buzz]S

is a s-derivation called the bracket derivation

determined by at. Moreover

[B"', By]End(A)

= B[""i"]S,

Vat, y

G

A.

Exercise 2.2.41. Prove Proposition 2.2.40. Definition 2.2.42. Let E = E0 ® E1 and F = F0 ® F1 be two 5-spaces. Their 5-tensor product is the 5-space E (X) F with the Z2-grading, (X)

Fr

E¢®Fi,

..II

(E

6

0, 1.

'i-l-jEe (2)

To emphasize the super-nature of the tensor product we will use the symbol sc®9s instead of the usual "®".

Exercise 2.2.43. Show that there exists a natural isomorphism of .9-spaces

V* A°V (X) uniquely determined by u*

X

f"*J

DerS (A°V),

w »-> DU*®"", where

D'U*®wlul

Do*®w

is s-derivation deaned by

(u*,u)w, 've 6 V

Notice in particular that any s-derivation of A°V is uniquely determined by its action on A1V. (When Ld = 1, D'U* ®1 coincides with the internal derivation discussed in Example 2.2.36.)

56

Lectures on the Geometry of Mamfolds

Let A = A0 ® A1 be an .9-algebra over K map 7' A -> K such that,

R,(c. A supertrace on A is a K-linear

T([22,y]s) 0 Very 6 A. If we denote by 1A»A1s the linear subspace of A spanned by the supercommutators i

{[X7Y]S

3 $79 E A

}»

then we see that the space of s-traces is isomorphic with the dual of the quotient space A/[A,A]S.

Proposition 2.2.44. Let E = E0 ® E1 be ajinite dimensional 5-space, and denote by A the 5-algebra of endomorphisms of E. Then there exists a canonical .9-trace trS on A uniquely dejined by tI's]1E

dim E0 - dim E l .

I

In fact, If T 6 A has the block decomposition II

$4

7

then tr . T = tr T00 - tr T11.

Exercise 2.2.45. Prove the above proposition. 2.2.4

Duality

Duality is a subtle and fundamental concept which permeates all branches of mathematics. This section is devoted to those aspects of the atmosphere called duality which are relevant to differential geometry. In the sequel, all vector spaces will be tacitly assumed finite dimensional, and we will use Einstein's convention without mentioning it. K will denote one of the fields R or C

Definition 2.2.46. A pairing between two K-vector spaces V and W is a bilinear map B V X W->]K. Any pairing B V

X

W-> K defines a linear map ] I 8 . V - > W * , v»->B(u,

.

)kW* 7

called the adjunction morphism associated to the pairing. Conversely, any linear map L : V -> W* defines a pairing

BL V X W -> K , B(v,w) = (L'u)(w), Vu G V, w € W. Observe that EBL = L. A pairing B is called a duality if the adjunction map III is an isomorphisms.

..

Example 2.2.47. The natural pairing ( , ) : V* X V -> K is a duality. One sees that H( 0 , 0 ) = ] L V * Z V * - > V * This pairing is called the natural duality between a vector space and its dual.

57

Natural Constructions on Manifolds

..

Example 2.2.48. Let V be a finite dimensional real vector space. Any symmetric nondegenerate quadratic form ) : V X V -> R defines a (self)duality, and in particular a natural isomorphism 7

..

U( 0 , 0 I Z V - > V *

When , ) is positive definite, then the operator L is called metric duality or loweringthe-indices map. This operator can be nicely described in coordinates as follows. Pick a basis ( I of V, and set gig

Let ( e ) denote the dual basis of V* defined by 'Soj , W, j. (et, € )

The action of L is then Lei

9¢j€J

Example 2.2.49. Consider V a real vector space and Ld : V X V -> R a skew-symmetric bilinear form on V. The form Ld is said to be symplectic if this pairing is a duality. In this case, the induced operator Hw : V -> V* is called symplectic duality.

Exercise 2.2.50. Suppose that V is a real vector space, and w : V x V - > ] R is a symplectic duality. Prove the following. (a) The V has even dimension. OJ(e£ , € j l , then d€t(w¢j)1g¢,jgdimv ¢ 0. (b) If (80 is a basis of V, and we The notion of duality is compatible with the factorial constructions introduced so far.

Proposition 2.2.51. Let B : V X We -> R (i Then there exists a natural duality

B = 81

B2 I (V1 (X) V2)

(X)

= 1, 2) be two pairs of spaces in duality. X

(W1®W2)->]R,

uniquely determined by

HB1®B2 = HB1 (X) HB2

;

\

\

I

8(U1 (X)

'U2,'U)1 (X) 'U1QI

=

B1l'U1,'U)1l ° B 2 ( U 2 , W 2 l -

Exercise 2.2.52. Prove Proposition 2.2.51. Proposition 2.2.51 implies that given two spaces in duality B : V naturally induced duality

X

W -> K there is a

B (Xm : v®'" x W ®"f* ->K. This defines by restriction a pairing AfrB : ATV X A"°W -> K uniquely determined by l'Ul /\

I\'v»,~,w1 /\

Aw

..II

AB

det ( B

Exercise 2.2.53. Prove the above pairing is a duality.

(U p, W y l l l g g r •

58

Lectures on the Geometry of Manifolds

..

In particular, the natural duality ( ,

):V*

K induces a duality

AT'V->]R,

and thus defines a natural isomoiphism

AW*

(A*v>*

2

This shows that we can regard the elements of AfrV* as skew-symmetric 1°-linear forms

V" ->K. A duality B : V

X

W-> K naturally induces a duality B U V * Bt('u*,w*)

X

W * - > K by

=l'u',I[Blw*),

where IIB : V -> W* is the adjunction isomorphism induced by the duality B. Now consider a (real) Euclidean vector space V. Denote its inner product by ( The self-duality defined by l 7 ) induces a self-duality

.. l. . ,

):A"V

X

. .>7

ATIV->]R,

determined by (U1 A - - - A v , ~ , w 1 A - - - A w , - ) ..

The symmetric

7° X T

s

det((v,w))1s

(2.2.8)

matrix

G (v1,...,v,~) ..

(U1,'U1 ('U1»U2l

('U1»Uv~l

(U2»U1l ('U2»U2l

IU?)

u)

( u m ) ('u»,°,v2) is called the Grammian of the vectors "up, . . . , v . It is positive semidefinite and, if the vectors '01, . . . , o are linearly independent, then it is also nonsingular. This shows that l'07f°7'0'f°l

the bilinear form in (2.2.8) is symmetric and positive definite. We have thus proved the following result. Corollary 2.2.54. An inner product on a real vector space V naturally induces an inner

product on the tensor algebra y ( V ) , and in the exterior algebra A°V.

In a Euclidean vector space V the inner product induces the metric duality L : V - > V * . This induces an operator L : T; (V) -> T§°+11(V) defined by L ( v 1 ® . . . ® v , - ® u1

uS)

1

(X) 'u»S)- (2.2.9) ) ® IILu1 (X) u (X) The operation defined in (2.2.9) is classically referred to as lowering the indices. The reason for this nomenclature comes from the coordinate description of this operation. If T 6 'y§(V) is given by

T

(X)

I

...®

;

(112

(X) v

(X)

il-

Tm...j;€i1 (X)

® e , ®ei1 (X)

(X)

eds 7

then LT

¢.2....¢,.

=

..T¢¢2....¢,

( IJJ1---§v~ 91] J 1 . - - a s where gig ( i i 7 o i l . The inverse of the metric duality L-1 operation 7; (V) -> V induces a linear

59

Natural Constructions on Manifolds

Exercise 2.2.55 (Cartan). Let V be a Euclidean vector space. For any 'U Q V denote by /Le* 6v (resp. i'U) the linear endomorphism of A*V defined by e'Uw = 'U A Ld (resp. i'U where iv* denotes the interior derivation defined by u* 6 V*-the metric dual of "u, see Example 2.2.36). Show that for any u, 'U Q V

[e

7'u,]s

I

e v e -|-

iuev

(to, U ) ] l A . V .

I

Definition 2.2.56. Let V be a real vector space. A volume form on V is a nontrivial linear form on the determinant line of V , / u : d e t V - > R Equivalently, a volume form on V is a nontrivial element of det V* (n = dim V). Since det V is 1-dimensional, a choice of a volume form corresponds to a choice of a basis of det V.

Definition 2.2.57. (a) An orientation on a vector space V is a continuous, surjective map or : derV \ {0} -> {::1}. We denote by O'r(V) the set of orientations of V. Observe that O'r(V) consists of precisely two elements. (b) A pair (V, or), where V is a vector space, and or is an orientation on V is called an oriented vector space. (c) Suppose or E Or(V). A basis LU offset V is said to be positively oriented if or(w) > 0. Otherwise, the basis is said to be negatively oriented. There is an equivalent way of looking at orientations. To describe it, note that any nontrivial volume form n on V uniquely specifies an orientation or given by

0ru(W) := sign,u(w), Vi E det V

\ {0}.

We define an equivalence relation on the space of nontrivial volume forms by declaring H1

H2 1 ' u1(w)u2((»J)

> 0, Vi

6 derV

\ {0}.

Then M1

iv

M2

KI

\ ,I

07'/-01

I

OTm2

.

To €V€I'Y orientation 07' WC ChIll ElSSOCi2lt€ 2111 €qI.liV211€I1C€ class [/J]o»,» of volume forms such that 1u(w)o'r(w) > 0, Vi 6 det V \ {0}.

Thus, we can identify the set of orientations with the set of equivalence classes of nontrivial volume forms. Equivalently, to specify an orientation on V it suffices to specify a basis w of det V. The associated orientation orb, is uniquely characterized by the condition

orw(w) = 1.

60

Lectures on the Geometry of Manifolds

---

To any basis {et, ..., en} of V one can associate a basis el A A eTL of det V. Note that a permutation of the indices 1, . . . , n changes the associated basis of det V by a factor equal to the signature of the permutation. Thus, to define an orientation on a vector space, it suffices to specify a total ordering of a given basis of the space. An ordered basis of an oriented vector space (V, or) is said to be positively oriented if so is the associated basis of det V. Definition 2.2.58. Given two orientations 0T1, 0T2 on the vector space V we define 0 r 1 / 0 r 2 Q {d:1} to be 0m/0r2

:= or1(w)or2(w), Vi 6 det V

\ {0}.

We will say that 0T1 /org is the relative signature of the pair of orientations o f , 0T2.

..

Assume now that V is a Euclidean space. Denote the Euclidean inner product by g( 7 The vector space det V has an induced Euclidean structure, and in particular, there exist exactly two length-one-vectors in det V. If we fix one of them, call it w, and we think of it as a basis of det V, then we achieve two things.

. .

First, it determines a volume form #9 defined by #go/\Wl Second, it determines an orientation on V.

Z

A.

Conversely, an orientation or 6 Or(V) uniquely selects a length-one-vector w Wor in det V, which determines a volume form ,ug = 1 g". Thus, we have proved the following result. I

Proposition 2.2.59. An orientation 07° on a Euclidean vector space (V, g ) canonically selects a volume form on V , henceforth denoted by Detg = D e t g . Exercise 2.2.60. Let (V, g) be an n-dimensional Euclidean vector space, and 07° an orientation on V. Show that, for any basis 111, . . . 'un of V, we have

Detg'°(v1 /\

- - - As"

07°

01

A - - - A v n ) v 0 Denote by it the rescaled metric it = t2g. If * is the Hodge operator corresponding to the metric g and orientation or, and *t is the Hodge operator corresponding to the metric it and the same orientation, show that Dettg = t" Detg

7

and *to =

tn-2p

*w

Vi G ApV*

We conclude this subsection with a brief discussion of densities. Definition 2.2.72. Let V be a real vector space. For any T' > 0 we define an 1°-density to be a function f : det V -> R such that f(A~) = Il.°f(u), Vu Q derV

\ {0},

vA ¢ 0.

The linear space of r-densities on V will be denoted by

IAIV.

When

T'

1 we set

IAlv := III,, and we will refer to 1-densities simply as densities. Example 2.2.73. Any Euclidean metric g on V defines a canonical l-density | Detg | E which associated to each w € d e t V its length, Iw I g.

no

Observe that an orientation or to,

: derV* ->

Q

O'r(V) defines a natural linear isomorphism

IAlv

derV* 9 M »->

u E IAIv,

(2.2.13)

where z0,..u,(w)

or(w),u(w), Vi 6 det V

\ {0}.

In particular, an orientation in a Euclidean vector space canonically identifies

IAlv with R

64

Lectures on the Geometry of Manifolds

2.2.5

Some complex linear algebra

It is convenient to have a coordinate description of the abstract objects introduced above. Let V be an n-dimensional complex vector space and h a Hermitian metric on it. Pick an unitary basis 61, ..., iiTL of V, i.e., n = d i n g V, and h(ei, co) = i i . For each j, we denote by f j the vector i e j . Then the collection e l , fl, . . . , en, fn is an R-basis of V. Denote by e1, f1, . . . , e", fn the dual R-basis in V*. Then Re h ( i i » € j l

60

R e d ( f , f j ) and Re h l € z » f j l = 0 ,

Le

Re h

®€'i

-I-f

(X)

fz)-

ii

Also

Lull¢,fjl

I

-lm

h(€i,'i€jl

(So

=

7

'Jlffjl

Wlezaejlz

0 WJ,

which shows that

Imh

CU

et /\

f'L.

Any complex space V can be also thought of as a real vector space. The multiplication by ii = v -1 defines a real linear operator which we denote by J. Obviously J satisfies $2 -iv. Conversely, if V is a real vector space then any real operator J V -> V as above defines a complex structure on V by I

( a + b ' £ ) ' v = a v + b J v , VuGV, a + b i € l C . We will call an operator J as above a complex structure. Let V be a real vector space with a complex structure J on it. The operator J has no eigenvectors on V. The natural extension of J to the complexification of V, Vc = V ® (C, has two eigenvalues ii, and we have a splitting of Vi as a direct sum o f complex vector spaces (eigenspaces)

Va: = her (J - i) ® (her J

-|-

i).

Exercise 2.2.74. Prove that we have the following isomorphisms of complex vector spaces V&"ker(J-fi)

V%lkerJ-I-i).

Set V1,0

Thus V(: 2

V.'1,0

®

:z Ker(J-fi) V0,1

2

V®

V

V.

V0,1

== Ker(J+i)

We obtain an isomorphism of Z-graded complex

vector spaces

A°V(C

V.

f"*J

A.Vl,0 (X)

Ai/0,1

65

Natural Constructions on Manifolds

If we set Aznqy

:I

ApV1 ,0

AqV0 ,1

®(F:

7

then the above isomoiphism can be reformulated as

Am; 2

AMS.

(2.2.14)

p-l-qzk

Note that the complex structure J on V induces by duality a complex structure J* on and we have an isomorphism of complex vector spaces

v;

(v, J);

v;

7

( w ) .

2

We can define similarly AWIV* as the Aw?-construction applied to the real vector space equipped with the complex structure J*. Note that

v:

A1,0V*

~_ (A1,0V)2»

A0,1V*

g

Ap» R, AM* 9 n »->

1

g(v"7D€tg).

lm h is real 2-form associated with the Hermitian metric h, then

Det 9

l Dotg'rc = -w" in!

7

and we conclude

pf

hw

l.

2.3 Tensor Iields 2.3.1

Operations with vector bundles

We now return to geometry, and more specifically, to vector bundles. Let K denote one of the fields R or (C, and let E -> M be a rank T' K-vector bundle over the smooth manifold M . According to the definition of a vector bundle, we can find an open cover (Url of M such that each restriction E I UQ is trivial: E | Up, V X Up, where V is an 1°-dimensional vector space over the field K. The bundle E is obtained by f"*J

68

Lectures on the Geometry of Manifolds

gluing these trivial pieces on the overlaps Ua re UB using a collection of transition maps QQB : UO( O UB -> GL(V) satisfying the cocycle condition. Conversely, a collection of gluing maps as above satisfying the cocycle condition uniquely defines a vector bundle. In the sequel, we will exclusively think of vector bundles in terms of gluing cocycles. Let E, F be two vector bundles over the smooth manifold M with standard fibers VE and respectively Vp, given by a (common) open cover (Ua), and gluing cocycles QQB : Uog ->

GL(VE ), and respectively hog : UQ5 -> GLWFI-

Then the collections QQB ® hog : UQ5 -> GL(VE ® Vp), QQB

(ITB)

:U

-> GLWE

>,

(X)

hog : Uog -> GL(VE (X) Vp),

ATM : u

-> GL(A'°vE),

where T denotes the transpose of a linear map, satisfy the cocycle condition, and therefore define vector bundles which we denote by E ® F, E (X) F, E*, and respectively A"IE. In particular, if T' = ranklE, the bundle AT'E has rank 1. It is called the determinant line bundle of E, and it is denoted by det E. Given the adjunction isomorphism VE (X) VF E HOM(VE, Vp), we set

Hom(E, F) := E* (X) F. We set End(E) := Hom(E,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.l.34). The bundle E* is called the dual of the vector bundle E. The direct sum E ® F is also called the Whitney sum of vector bundles. All the factorial constructions on vector spaces discussed in the previous section have a vector bundle correspondent. (Observe that a vector space can be thought of as a vector bundle over a point.) These above constructions are natural in the following sense. Let E' and F' be vector bundles over the same smooth manifold M/. Any bundle maps S : E -> E' and T : F -> F' , both covering the same dweomorphism go : M -> M', induce bundle morphisms

S ® T : E ® F->E' ® T/ 7 S

(X)

T:E

covering (25, a morphism

s* : (E')* -> E* covering (a

1

etc.

Exercise 2.3.1. Prove the assertion above.

7

(X)

F->E'

(X)

FI

7

Natural Constructions on Manifolds

69

Example 2.3.2. Let E, F, E' and F' be vector bundles over a smooth manifold M. Consider bundle isomorphisms S : E -> E' and T : F -> F' covering the same diffeomorphism of the base, : M -> M. Then (S-1)t : E* -> (EI)* is a bundle isomorphism covering go, so that we get an induced map (5-1)'r (X) T : E* (X) F -> (E/)* (X) F'. Note that we have a natural identification COO(

Hom(E, F ) ) = c°°(E* (X) F) E Hom(E, F),

where we recall that Hom(E, F) denotes the space of smooth bundle moiphisms E ->F.

Definition 2.3.3. Let E -> M be a K-vector bundle over M . A metric on E is a section h OfE* ®1I< E* (F: Eif11< = R) such that, for any m Q M, h(m) defines a metric on Et (Euclidean if K = R or Hermitian if K G). 2.3.2 Tensor jields We now specialize the previous considerations to the special situation when E is the tangent bundle of M , E 5 T M . The cotangent bundle is then

T*M

(TM)*.

= 537 'v'i,j. A basis in 'J'.§(T,,M) is given by

w

(X)

(X)

a,

® do

®

(X)

dxjs, 1 S

7317

7

i , ~ < n , 1

7

j.S S n

Hence, any tensor T 6 7 § ( M ) has a local description .

T

Tel---j; ex11 (X)

(X)

33,,,. (X) data

(X)

(X)

dzvjs

In the above equality we have used Einstein's convention. In particular, an 1°-form CU has the local description w=

Ldi1...i d.CU11

/\

Ad;z:1*,

Wil...¢,.

Wlaxll

7

7

a

7

1$¢1 N be a smooth map. The pullback by f defines a morphism of associative algebras f* : Q' ( N ) -> Q' ( M ) .

Exercise 2.3.10. Prove the above proposition.

Natural Constructions on Manifolds

73

2.3.3 Fiber bundles We consider useful at this point to bring up the notion of fiber bundle. There are several reasons to do this. On one hand, they arise naturally in geometry, and they impose themselves as worth studying. On the other hand, they provide a very elegant and concise language to describe many phenomena in geometry. We have already met examples of fiber bundles when we discussed vector bundles. These were "smooth families of vector spaces". A fiber bundle wants to be a smooth family of copies of the same manifold. This is a very loose description, but it offers a first glimpse at the notion about to be discussed. The model situation is that of direct product X = F X B, where B and F are smooth manifolds. It is convenient to regard this as a family of manifolds (Fb)beB. The manifold B is called the base, F is called the standard (model) fiber, and X is called the total space. This is an example of trivial fiber bundle. In general, a fiber bundle is obtained by gluing a bunch of trivial ones according to a prescribed rule. The gluing may encode a symmetry of the fiber, and we would like to spend some time explaining what do we mean by symmetry. Definition 2.3.11. (a) Let M be a smooth manifold, and G a Lie group. We say the group G acts on M from the left (respectively right), if there exists a smooth map < I > : G x M - > M , (g,m)»->9m, such that 1 E JIm and ®g(®;,m) = gI,m (respectively g(i>I,m) = I19m) Vg, h 6 G, m 6 M. In particular, we deduce that Vg 6 G the map g is a diffeomorphism of M . For any m 6 M the set G-m={©gm; g a G } is called the orbit of the action through m. (b) Let G act on M. The action is called free if Vg 6 G \ {1}, and 'v'm E M Tgm 7; m. The action is called ejective if, Vg 6 G \ {1}, ®9 7; JIm.

It is useful to think of a Lie group action on a manifold as encoding a symmetry of that manifold. Example 2.3.12. Consider the unit 3-dimensional sphere

S2 = {(x,y,z) 6 R3; $1:2 +3/2 + 2 2 = l}. Then the counterclockwise rotations about the z-axis define a smooth left action of S1 on S2. More formally, if we use cylindrical coordinates (T, 9, 2) Hz: = TCOSH, y = r s i n 9 , z=0, then for every go 6 IR mod 27r 2 S1 we define RS0 : S2 -> S2 by RSOITQQQZ) = (r, (9 -|- (pl mod Qtr, z). The resulting map R : S1 X S2 -> S2, (go, p) +-> R(plp) defines a left action of S1 on S2 encoding the rotational symmetry of S2 about the z-axis.

74

Lectures on the Geometry of Manifolds

Example 2.3.13. Let G be a Lie group. 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. One says V is a G-module. For example, the tautological linear action of SO(n) on R" defines a linear representation of SO(n). Example 2.3.14. Let G be a Lie group. For any g 6 G denote by Lg (resp. Rg) the left (resp. right) translation by g. In this way we get the tautological left (resp. right) action of

G on itself. Definition 2.3.15. A smoothjber bundle is an object composed of the following: (a) (b) (c) (d) (e)

a smooth manifold E called the total space, a smooth manifold F called the standard fiber, a smooth manifold B called the base, a surjective submersion 'IT : E -> B called the natural projection, a collection of local trivializations, i.e., an open cover (I/1a) of the base B, and diffeomorphisms \1/0, : F X Ua -> 7r"1(Uo) such that

We v be : : b, */(f, be 6 F i.e., the diagram below is commutative. 7T o

F XU0

'PQ '7T 1

X

UQ,

U/0J

UQ We can form the transition (gluing) maps \I/0 . F x Uag - > F x Ua ,where l/'ag = UQ O UP, defined by Wag = upa 1 o W . According to (e), these maps can be written as 1 b a 0 ( f » be = (Toflilblf, be, where Tg- QUO TQQCGF. The group G is called the symmetry group of the bundle. (g) There exist smooth maps gag : Uog -> G satisfying the cocycle condition G

.gas

I

X

1 E G, .Eva

I

.975 • QBQ) V@,B,v,

and such that

TQBUPI =

T9o¢=(b)

We will denote a G-fiber bundle by ( E , or, F, B, G).

-

75

Natural Constructions on Manifolds

The choice of an open cover (Ua) in the above definition is a source of arbitrariness since there is no natural prescription on how to perform this choice. We need to describe when two such choices are equivalent. Two open covers (Url and (W), together with the collections of local trivializations

ea . F x u

-> for-HUa) and

w

:F x

v -> w-1(v;)

are said to be equivalent if, for all Q, 11, there exists a smooth map Tai

. UQ VW

*

G,

such that, for any m e UmM/Q, and any f Q F , we have 0}@i(f,x)

Z

(Ta¢(93lf»

A G-bundle structure is defined by an equivalence class of trivializing covers. As in the case of vector bundles, a collection of gluing data determines a G-fiber bundle. Indeed, if we are given a cover (U0,)0,€A of the base B, and a collection of transition maps QQB : Ua n UB -> G satisfying the cocycle condition, then we can get a bundle by gluing the trivial pieces Uo, X F along the overlaps. More precisely, if b 6 Uamltg, then the element (f, IQIQFXUa is identified with the element (g5O,(bl • f , be G F X UB7T

Definition 2.3.16. Let E -> B be a G-fiber bundle. A G-automorphism of this bundle is a diffeomorphism T : E -> E such that or o T = or, i.e., T maps fibers to fibers, and for any trivializing cover (I/'ol (as in Definition 2.3.15) there exists a smooth map go, : Uo, -> G such that Q

1Tq/aU,

be

(9o (6) f,b ), w>,f.

Definition 2.3.17. (a) A fiber bundle is an object defined by conditions (a)-(d) and (f) in the above definition. (One can think the structure group is the group of diffeomorphisms of the standard fiber). 7T (b) A section of a fiber bundle E -> B is a smooth map S : B -> E such that 7T O S ]DB» i.e., 5(5) G

-1(6>, v i G B.

Example 2.3.18. A rank T' vector bundle (over K = R, (I) is a GL(v", K) -fiber bundle with standard fiber KT, and where the group GL(r, K) acts on KT' in the natural way. Example 2.3.19. Let G be a Lie group. A principal G-bundle is a G-fiber bundle with fiber G, where G acts on itself by left translations. Equivalently, a principal G-bundle over a smooth manifold M can be described by an open cover 'Ll of M and a G-cocycle, i.e., a collection of smooth maps

_ Q u v : U M V - G U,Ve'LL, such that V x 6 UHVV1W(U,V,W€"L()

9uv(33l9vw($l9wu(22l = 1 6 G.

76

Lectures on the Geometry of Manifolds

Exercise 2.3.20 (Alternate definition of a principal bundle). Let P be a fiber bundle with fiber a Lie group G. Prove the following are equivalent. (a) P is a principal G-bundle. (b) There exists a free, right action of G on G, P

G - > P , (19,9)+->19~9,

X

such that its orbits coincide with the fibers of the bundle P, and there exists a trivializing

cover

{w@ : G x

Ua ->7r-1(Ua)

}

7

such that \I/

0(h9/'~I,)= *I'0(h>u)'9» V g , h € G ,

'u,6

U0.

Exercise 2.3.21 (The frame bundle of a manifold). Let M n be a smooth manifold. Denote by F(M) the set of frames on M, i.e., F ( M ) = {(m,X1,

,Xnl, m

6 M, Xi 6 TmM and span(Xl,

,X,,) = T m M } .

(a) Prove that F(M) can be naturally organized as a smooth manifold such that the natural projections : F ( M ) -> M, (m, Xi, ,X,,) »-> m is a submersion. (b) Show F ( M ) is a principal GL(n, R)-bundle. The bundle F ( M ) is called the frame bundle of the manifold M. Hint: A matrix T (To 6 GL(n, K) acts on the right on F ( M ) by ~17 ...,x,,)

»-> (my lT-1l1Xi9

(T'1)X¢).

Example 2.3.22 (Associated fiber bundles). Let 77 : P -> G be a principal G-bundle. Consider a trivializing cover lU0la€A9 and denote by gag : Up Fl UB -> G a collection of gluing maps determined by this cover. Assume G acts (on the left) on a smooth manifold F X

F - > F , (Q, f ) »-> Tlglf-

The collection Tag = 'T(90@) : Uog -> 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. This new bundle is independent of the various choices made (cover (UQ) and transition maps QQBI. (Prove this!) It is called the bundle associated to P via T and is denoted by P XT F.

Exercise 2.3.23. Prove that the tangent bundle of a manifold M n is associated to F(m> via the natural action of GL(n, R) on R" Exercise 2.3.24 (The Hopf bundle). If we identify the unit odd dimensional sphere S271-1 with the submanifold 217

7

Zn) E

(Zn

l20l2

-I-

+ IonI 2

w

Natural Constructions on Manifolds

77

then we detect an S1-action on S2n-1 given by

et

- (z1,

zn) = (eWzl,

eweTL).

The space of orbits of this action is naturally identified with the complex projective space CP"l1. (a) Prove that p : S272-1 -> (0, oo) defines a balanced open Nf := { ( t , m ) 6 R X M; Itl < f(m) }~ Definition 3.1.1. A loealflow is a smooth map Q : N -> M, (t, m) +-> t(m), where N is a balanced neighborhood of {0} X M in R X M, such that (a) q>0(m) = m, Vm 6 M. (b) q>t(s(ml)

When N

1

@+s(m) for all s , t 6 R m 6 M such that ( s , m l , (s -I- t , m ) , l¢,q>S(mll 6 N.

R XM ,

Q?

is called allow.

The conditions (a) and (b) above show that a flow is nothing but a left action of the additive (Lie) group (R, +) on M .

Example 3.1.2. Let A be a e x n real matrix. It generates a flow iA on R" by t i>Aa3

e t A as

to

(2 ) k=0

79

-Ak k!

ac.

80

Lectures on the Geometry of Manifolds

Definition 3.1.3. Let Q :N-> M be a local flow on M . The infinitesimal generator of Q is the vector field X on M defined by X(p)

d dt

= X(P)

t(p), V P G M ,

llllllllll

i.e., X (p) is the tangent vector to the smooth path t »-> t(p) at t = 0. This path is called the flow line through p. Exercise 3.1.4. Show that X is a smooth vector field. Example 3.1.5. Consider the flow €¢A on R" generated by a n is the vector field XA on R" defined by d dt

XAIU/I

e

tA

u

X

n matrix A. Its generator

Au.

t O

Proposition 3.1.6. Let M be a smooth n-dimensional manzfold. The map X : {Local flows on M} -> Vect (M),

i> »->

X,

is a surjection. Moreover; If @¢ : N -> M (i=],2) are two loealflows such that Xi1 X2, then 1 = CD2 on N1 D No. Proof. Surjeetivily. Let X be a vector field on M . An integral curve for X is a smooth curve by : (a, b) -> M such that

WI

Z

X(7(t))-

In local coordinates ( f ) over on open subset U C M this condition can be rewritten as

Wt) =x(:¢1(t),...,x"(z)), vi

1,

ANa

(3.1. 1)

where W) = (;r31(tl, ...,x"(t)), and X = Xfi 3:1% a The above equality is a system of ordinary differential equations. Classical existence results (see e.g. [12, 60]) show that, for any precompact open subset K C U, there exists € > 0 such that, for all at G K, there exists a unique integral curve for X , 'Ye : ( - 5 , 5) -> M satisfying 'yas(0l

= LU.

(3.l.2)

Moreover, as a consequence of the smooth dependence upon initial data we deduce that the map @K:3\fK=l-(5)

X

K ->M, (ac,il*->'yx(tl,

is smooth. Now we can cover M by open, precompact, local coordinate neighborhoods (KQIQQA and as above, we get smooth maps Qa : Na = (-eyea) X Ka -> M solving the initial value problem (3. 1.1-2). Moreover, by uniqueness, we deduce 7

QQ - QQ on No VWN5.

81

Calculus on Manifolds

Define

N:

U No

7

06.A

andseti1'> : N-> M,@= paa onU\fo. The uniqueness of solutions of initial value problems for ordinary differential equations implies that i1'> satisfies all the conditions in the definition of a local flow. Tautologically, X is the infinitesimal generator of i1'>. The second part of the proposition follows from the uniqueness in initial value problems. The family of local flows on M with the same infinitesimal generator X 6 Vect(M) is naturally ordered according to their domains,

lfbl I N1 -> M)

4

l@2:N2->Ml

if and only if N1 C N2. This family has a unique maximal element which is called the local flow generated by X, and it is denoted by i1'>x .

Exercise 3.1.7. Consider the unit sphere S2 = {(a2,y,z)

e R3;

£1:2

+y2 +z2 = 1

For every point p 6 S2 we denote by X (p) e TOR3, the orthogonal projection of the vector k = (0, 0, 1) onto TpS2. (a) Prove that p »-> X (p) is a smooth vector field on S2, and then describe it in cylindrical coordinates (z, 0), where as

rcos0, y

= T' sin 0 ,

T'

lx2_l_y2l1/2.

(b) Describe explicitly the flow generated by X .

3.1.2 The Lie derivative Let X be a vector field on the smooth n-dimensional manifold M and denote by = ®X the local flow it generates. For simplicity, we assume is actually a flow so its domain is R x M. The local flow situation is conceptually identical, but notationally more complicated. For each t 6 R, the map t is a diffeomorphism of M and so it induces a push-forward map on the space of tensor fields. If S is a tensor field on M we define its Lie derivative along the direction given by X as • l hm - ((@i>s)m-sm) V m 6 M. (3.l.3) LxS1n t->0 t Intuitively, LxS measures how fast is the flow ® changing! the tensor S. If the limit in (3.13) exists, then one sees that LxS is a tensor of the same type as S. To show that the limit exists, we will provide more explicit descriptions of this operation.

lArnold refers to the Lie derivative Lx as the 'fisherman's derivative". Here is the intuition behind this very suggestive terminology. We place an observer (fisherman) at a fixed point p E M, and we let him keep track of the the sizes of the tensor S carried by the flow at the point p. The Lie derivatives measures the rate of change in these sizes.

82

Lectures on the Geometry of Mamfolds

Lemma 3.1.8. For any X 6 Vect (M) and

..

Above, (

7

o f :z L x f

f

6 C°°(M) we have

(df,x) df(X)) denotes the natural duality between T * M and T M ,

..I ,

: C°°(T*M)

COO(T*M)

I

I

X

C°°(TM) -> C°°(M),

C°°(TM) 9 ( 0 , X ) »-> a ( X ) 6 C°°(M).

X

In particular Lx is a derivation of cOO ( M l . Proof. Let Qt = QQ be the local flow generated by X. Assume for simplicity that it is defined for all t. The map Qt acts on cOO (M) by the pullback of its inverse, i.e.,

* = (-t)*. Hence, for point p E M we have . 1 Lx f (p) hm - (f(p) - f(-tp)) t->0 t

d dt

f(ltp) t

Z

p.

0

Exercise 3.1.9. Prove that any derivation of the algebra COO(M) is of the form Lx for some X 6 Vect(M), i.e., Der (C'°°(M)) E Veczt (M).

Lemma 3.1.10. Let X, Y 6 Vect (M). Then the Lie derivative of Y along X i5 a new vectorjield LxY which, viewed as a derivation O f f ° ° ( M ) , coincides with the commutator of the two derivations of GOO ( M ) defined by X and Y , i.e.,

L x Y f = [x, Yu, o f 6 C>°°(m). The veetorjield [X, Y ] = LxY is called the Lie bracket o f X and Y . In particular the Lie bracket induces a Lie algebra structure on Vect (M). Proof. We will work in local coordinates ( w ) near a point m E M so that

xa,

X

and Y =

Wa .

We first describe the commutator [X, Y1. If f € C°°(M), then .

[X, Ylf

.of

(

( XZ3x 71 ) Y .7 33123. .

XzYJ

.

82f .

.

exz 3;I:]

)

( ywxj ) Xi -"'m = 7(-t) = 7(0) - 1(0)t + 0(9) =

(we) with

I

(x0 -

\i

+ o(t2))

t

xi_ Then 7

and .

YJ

Yi(-> Note that is* t

T(0)M

txfi

35/j 351

.-l-Ot2. ( )

(3.l.5)

-> Tv(-t)M is the linearization of the map - tx + O(t2))

(of) »-> (

7

so it has a matrix representation *t

t

]1

1

+ 0(t2).

. 331]

(3.l.6)

M

In particular, using the geometric series (11-A)

=]l-i-A-i-A2+

7

where A is a matrix of operator norm strictly less than 1, we deduce that the differential (Dt = (®*

t)-1

: Tv(_t)M ->

T"(0)M7

has the matrix form

11 -I-t

I 3:z:w j

s

+ 0(t2l.

(3.l.7)

o

Using (3.l.7) in (3.1.5) we deduce

Yk -

k

(@ty-m)

YJ axk

. ay'
tx is a flow. (Exercise) Set

Q

R In other

exp(tX) : = @ ( 1 ) .

We thus get a map exp : T1G 2 LG -> G, X »-> exp(Xl

called the exponential map of the group G.

-

Fact 3. il'>tx (g) = g exp (ISX), i.e., §
0

k!

Xi.

Exercise 3.1.20. Prove the statements left as exercises in the example above.

Exercise 3.1.21. Let G be a matrix Lie group, i.e., a Lie subgroup of some general linear group GL(N, K). This means the tangent space T1G can be identified with a linear space of matrices. Let X, Y E T1G, and denote by exp(tX ) and exp(tY) the 1-parameter groups with they generate, and set g(s, t) = exp(sX) exp(tY) eXP(-SX) exp(-tY). (a) Show that g.s,t

= 1 -1- [X,Y]algst -|- O((s2

-|-

t2)3/2) as s , t -> 0,

where the bracket [X, Y]al9 (temporarily) denotes the commutator of the two matrices X and Y.

88

Lectures on the Geometry of Manifolds

(b) Denote (temporarily) by [X, Ylgeovrz the Lie bracket of X and Y viewed as left invariant vector fields on G. Show that at 1 6 G

[X, Ylalg = [X, Ylgeom(c) Show that Q(n) C n, R) (defined in Section 1.2.2) is a Lie subalgebra with respect to the commutator [- , -]. Similarly, show that _u(n), M) C n, (C) are real Lie subalgebras of M, (Cl, while (C) is even a complex Lie subalgebra of M, (C). (d) Prove that we have the following isomorphisms of real Lie algebras,

we,

Lo(n)

2

Q(t1)»

L U ( n ) 2 Mtl),

LsU(n) 2

LW) and £sL(n,

2

QW, (C)-

Remark 3.1.22. In general, in a non-commutative matrix Lie group G, the traditional equality

exp(tX) exp(tY) = exp(t(X

-|-

Y))

no longer holds. Instead, one has the Campbell-Hausdorfformula exp(tXl -exp(tY)

= exp ( ¢d1(x, y)

+ t2d2(x, y) + t3d3(x, y) +

- -)

7

where do are homogeneous polynomials of degree k in X, and Y with respect to the multiplication between X and Y given by their bracket. The do's are usually known as Dynkin polynomials. For example,

d1 IX, y)

d3(X, Y)

l

-21X 7Y

X + Y , d2(X,Y)

L

_ l

-([X, [X, YH -1- [Y, [Y, XH) etc.

12 For more details we refer to [58, 113]. '

3.2 Derivations of Q' (M) 3.2.1

The exterior derivative

The super-algebra of exterior forms on a smooth manifold M has additional structure, and in particular, its space of derivations has special features. This section is devoted precisely to these new features. The Lie derivative along a vector field X defines an even derivation in Q°(M). The vector field X also defines, via the contraction map, an odd derivation iX , called the interior derivation along X , or the contraction by X ,

ix More precisely, ix

: =tr (X

(X)

cu), Vi 6

Q"(M).

is the (r - 1)-form determined by

lix(.4JllX1, . . . 7X,"_1)

I

Wlx,x1, . . . 7X7~-1)7 VXI,

7XT°-1 6 Vect (M).

89

Calculus on Manifolds

The fact that iX is an odd .9-derivation is equivalent to ix(w /\ 17) = (1lXw) /\ 17 -|- (-l)deg"Jw /\ ( i X n ) , V i , 17 6 Q * ( M ) .

Often the contraction by X is denoted by XJ .

Exercise 3.2.1. Prove that the interior derivation along a vector field is a .9-derivation.

Proposition 3.2.2. (61) Hx, iY]S = ixia -|- iyix = 0. (b) The super-commutator Of Lx and i n as .9-derivations of Q* ( M ) is given by [Lx,iy]5 = LXiY - iYLX = i[x,y]The proof uses the fact that the Lie derivative commutes with the contraction operator, and it is left to the reader as an exercise. The above s-derivations by no means exhaust the space of s-derivations of Q' (M)- In fact we have the following fundamental result. Proposition 3.2.3. There exists an odd 5-derivation d on the 3-algebra ofdy erentialforms gr( - I uniquely characterized by the following conditions. (a) For any smooth function f E QOIMI, df coincides with the deferential o f f . (b) d2 0. (c) d is natural, i.e., for any smooth function $5 : N -> M , and for any form w on M , we have Z

dg25*w = Cl5*dw('I

\.

w, dl =

0).

The derivation d is called the exterior derivative.

Proof. Uniqueness. Let U be a local coordinate chart on M n with local coordinates 1 ,as"'). Then, over U, any 1°-form w can be described as 7 (.Ui1...i_d.CU

CU

z'

1

/\

/\

doi r

1$11 -> Q1(El. is a linear connection. This concludes the proof of the proposition. The tensorial operations on vector bundles extend naturally to vector bundles with connections. The guiding principle behind this fact is the product formula.

. .

If E ii = 1, 2) are two bundles with connections V , then E1 (X) E2 has a naturally induced connection VE1®E2 uniquely determined by the product rule, VE1®E2 (U1 ®

+ U1

(X)

V21IQ.

A connection on a bundle E induced a connection V* on the dual bundle E* determined by the identity

X('U, u) where

(Vxv,u)

-|-

( v , V X u ) , Vu € C°°(E),

..,

(

.

url = (Vlul) ® U2

)=c*°°(E*)

X

U

€ C°°(E*), X € Vect (M),

C°°(E)->C°°(M)

is the pairing induced by the natural duality between the fibers of E* and E. Any connection VE on a vector bundle E induces a connection v E n d ( E ) on End (E) 2 E* (X) E defined by K" X UO( (respectively to : EB trivialization of a bundle E over an open set Ua (respectively UB ), then the transition map "from Q to B" over Uo n UB is 96a = pa o ta 1. The standard basis in K , denoted by (6,-), induces two local moving frames on E: Z

€ayi

I

to 1ldz) and

€[3,¢

= to lldzl.

On the overlap U o U 5 these two frames are related by the local gauge transformation 85,1 = go l e a , i

Z

gaBea,'1l-

This is precisely the opposite way the two trivializations are identified. The above arguments can be reversed producing the following global result. Proposition 3.3.5. Let E -> M be a rank r smooth vector bundle, and (Ua) a trivializing cover with transition maps gal? : Uo, VW UB -> G L ( r , K ) . Then any collection of matrix valued 1-forms Fo, E Q1(End

FB

(goédgoel

U ) satisfying

+ gogfogae =

-(dg

go 1 + QBQFQQQ 1 over Ua D UP,

(33.6)

uniquely defines a covariant derivative on E.

Exercise 3.3.6. Prove the above proposition. We can use the local description in Proposition 3.3.5 to define the notion of pullback of a connection. Suppose we are given the following data.

.. .

A smooth map f : N -> M. A rank T' K-vector bundle E -> M defined by the open cover (Ua), and transition maps Quo : Ua VW UP -> GL(]K'°). A connection V on E defined by the l-forms Fa 6 Q1 l E n d l U o , ) l satisfying the gluing conditions (3.3.6).

Then, these data define a connection f*v on f*E described by the open cover f-1(Ua), transition maps QBQ o f and l-forms f*II0. This connection is independent of the various choices and it is called the pullback of V by f

Example 3.3.7 (Complex line bundles). Let L -> M be a complex line bundle over the smooth manifold M . Let {Ua} be a trivializing cover with transition maps zag : Ua O U5 -> Q* = GL(1, (I). The bundle of endomorphisms of L, End (L) = L* (X) L is trivial since it can be defined by transition maps (zofil -1 ®ZoB = 1. Thus, the space of

~

100

Lectures on the Geometry

of Manzfolds

connections on L, .A (L) is an affine space modeled by the linear space of complex valued 1-forms. A connection on L is simply a collection of (C-valued 1-forms wa on UO( related on overlaps by

we =

3.3.2

d

ZAg

zag

+w0' = dlogzog +w0'

Parallel transport

As we have already pointed out, 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. We will see in this subsection that such a procedure is all we need to define covariant derivatives. More precisely, we will show that once a covariant derivative is chosen, it offers a simple way of identifying different fibers. Let E -> M be a rank r K-vector bundle and V a covariant derivative on E. For any smooth path by : [0, 1] -> M we will define a linear isomorphism T'Y : E'y(0) -> E'YU) called the parallel transport along by. More exactly, we will construct an entire family of linear isomorphisms To

: Ev(0)

->

Et(t)°

One should think of this To as identifying different fibers. In particular, if U 0 6 ECU) then the path t +-> ut - Ttu0 6 E7U) should be thought of as a "constant" path. The rigorous way of stating this "constancy" is via derivations: a quantity is "constant" if its derivatives are identically 0. Now, the only way we know how to derivate sections is via V, i.e., ut should satisfy V2ut dt

0,

d

where

it

The above equation suggests a way of defining To. For any 'U,0 e E7(0)° and any t e [0, 11, define Ttuo as the value at t of the solution of the initial value problem

V dtd u(tl »U,(0)

0

(3.3.7)

U0

The equation (3.3.7) is a system of linear ordinary differential equations in disguise. To see this, let us make the simplifying assumption that W) lies entirely in some coordinate neighborhood U with coordinates (LV1, ..., at" , such that E IU is trivial. This is always happening, at least on every small portion of 7. Denote by l€O()1§a§v° a local moving frame trivializing E In so that u = uaeo. The connection 1-form corresponding to this moving frame will be denoted by F E Q1(End ( I I . Equation (33.7) becomes (using Einstein's convention) -|-

Ft5ug

0

u'""(0)

up

(3.3.8) 7

101

Calculus on Manifolds

where

d F *y J F E QUO ( End(11 W) =

(x1(t)»---»22"(tl)

7

then Ft is the endomorphism given by Ft€,3

LEAF 'Bea-

The system (3.38) can be rewritten as -|-

0

pfgfgimg

7

(3.3.9)

US'

we)

This is obviously a system of linear ordinary differential equations whose solutions exist for any t. We deduce

TWo .

(3.3.l0)

This gives a geometric interpretation for the connection 1-form F: for any vector field X, the contraction

-ixr = -F(x) e c°°(End(E)) describes the infinitesimal parallel transport along the direction prescribed by the vector field X, in the non-canonical identification of nearby fibers via a local moving frame. In more intuitive terms, if fy(t) is an integral curve for X , and To denotes the parallel transport along 'Y from E»V(0) to E7(t)' then, given a local moving frame for E in a neighborhood of '1/(0), To is identified with a t-dependent matrix which has a Taylor expansion of the form

To = ]1 - F015 -|- O(t2),

t very small,

(3.3.11)

with F0 I (1§xF) Io )If the connection V is also compatible with a metric h, then (3.3.2) shows the parallel transport To : E t ) -> E a ) is an isometric. 3.3.3

The curvature of

eonneetion

Consider a rank k smooth K-vector bundle E -> M over the smooth manifold M, and let V : Q0IE) -> o1(El be a covariant derivative on E.

Proposition 3.3.8. The connection V has a natural extension to an operator

do : oriel -> QT'-I-IIEI uniquely defined by the requirements,

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Lectures on the Geometry

la)

V,

dV IQ0(E)

lb) Vi 6 Q'°(m), 77 6 QSIE)

dv(w /\ 77) = do /\ 77 + (-l)"°w /\ den.

Outline of the proof

Existence. F o r m 6 S2'°(M), u 6 Q0(E) set

dv(w

(X)

u) = do

(X)

'LL -|-

(-l)7°wVu.

(3.3.l2)

Using a partition of unity one shows that any 17 6 Q""(E) is a locally finite combination of monomials as above so the above definition induces an operator Q"°(E) -> QQ"-I-1 (E). We let the reader check that this extension satisfies conditions (a) and (b) above. Uniqueness. Any operator with the properties (a) and (b) acts on monomials as in (3.3.l2) so it has to coincide with the operator described above using a given pa ition of unity. Example 3.3.9. The trivial bundle M has a natural connection V0 - the trivial connection. This coincides with the usual differential d : Q0(M) (X) K -> Q1(M) (X) K. The extension do" is the usual exterior derivative. There is a major difference between the usual exterior derivative d, and an arbitrary do_ In the former case we have d2 = 0, which is a consequence of the commutativity [63 7 31135 ] = 0, where l ) are local coordinates on M. In the second case, the equality (do)2 = 0 does not hold in general. Still, something very interesting happens.

Lemma 3.3.10. For any smooth function f 6 C°°(M), and any w 6 Q"(E) we have V _ V (d )2(f(»J) - f { ( d

Hence (do)2 is a bundle morphism A'°T*M

(X)

)2(»J}-

E -> AT'+2T*m ® E.

Proof. We compute (dV)2(f(»J) = dvwf A w + f d v w ) I

-if

if

A dew

dew

f(d")2(»

I

f(d")2 Q2(El, the operator (do)2 can be identified with a section of

HomK (E, A2T*M ®R E ) 2 E*

(X)

A2T*M

~

®1R

E = A2T*M

®1R EndK

(E).

Thus, ldv)2 is an EI1dK (E)-valued 2-form.

Definition 3.3.11. For any connection V on a smooth vector bundle E -> M, the object (do)2 6 Q2 lEI1d]K (E)) is called the curvature of V, and it is usually denoted by F(vl.

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Calculus on Manifolds

Example 3.3.12. Consider the trivial bundle Km. The sections of this bundle are smooth K"-valued functions on M. The exterior derivative d defines the trivial connection on KM , and any other connection differs from d by a M(K)-valued 1-form on M. If A is such a form, then the curvature of the connection d -|- A is the 2-form F(A) defined by F(A)u = (d -|- A)2u = (DA -|- A /\ A)u,

Vu E 0°°(m, KU.

The A-operation above is defined for any vector bundle E as the bilinear map

Qflmd (E))

X

Qk(End (E)) -> Q9+k(End lm),

uniquely determined by ((»J'°

(X)

no (X) AB, A, B a c ° ° ( E n d ( E ) ) .

wi

A)/\(t1S (X) B)

We conclude this subsection with an alternate description of the curvature which hopefully will shed some light on its analytical significance. Let E -> M be a smooth vector bundle on M and V a connection on it. Denote its curvature by F = F(V) 6 §22(End (E)). For any X , Y 6 Vect (M) the quantity F(X, Yl is an endomorphism of E. In the remaining part of this section we will give a different description of this endomorphism. For any vector field Z, we denote by in AZ"(E) -> Q-1(18l the COO(M) - linear operator defined by

iz(w ® u) = (izzy) (X) u, Vi E Q"(M), u 6 Q0(E).

For any vector field Z, the covariant derivative VZ : cOO (E) -> cOO (E) extends naturally as a linear operator Q"l(E) -> AZ"(E), which we continue to denote by Vi, uniquely defined by the requirements Vzlw®ul

The operators do, in, Vz titles.

lLzwl®u+w®Vzu.

AZ°(E) -> i"

QUO

-|-

satisfy the usual super-commutation iden-

(Mz

ix?iy -I- 1yix

= Vi.

0.

VXiY - iYVX = i[x,y]-

(3.3.13) (3.3.14) (3.3.15)

For any 'LL 6 Q0(E) we compute using (33.13)-(3.3.15)

F ( x , Y)u = iy¢x(dv)2u = ¢Y(zx¢zv)vu = bY(VX - do¢x)vu = ( z y v x ) v u - liydvlvx'LL = ( V x i y - z'[Xy])Vu - V;/Vxu = (VxVy - V;/Vx - V[x,y]l"~'-

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Hence F(X,Y) If in the above formula we take X on M , and we set Vi := Va m i , Vi

FU

Fez'

I

[Vx,Vy] - VlX,Y--

a

(3.3.16)

and Y a where ( a ) are local coordinates then we deduce pa 9

7

Flux, gmj

[Vow VJ]-

(3.3.l7)

Thus, the endomorphism F,-i measures the extent to which the partial derivatives Vi, Vi fail to commute. 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. Maybe an intelligent choice will restore the classical commutativity of partial derivatives SO we should concentrate from the very beginning to covariant derivatives V such that F(V) = 0.

Definition 3.3.13. A connection V such that F(V)

= 0 is called flat.

Proposition 3.3.14. If V is a connection on a real vector bundle that is compatible with a metric h, then FM 6 Q2 ( End_ III) where End_ ( E ) denotes the bundle of endomorphisms of E that are skew-symmetric with respect to h. Proof. Let X, Y 6 Vect ( M ) and u, U 6 COO(M). Using (33.2) we deduce

XYh(u, u) = Xh(Vyu, u) + Xh(u, Viv) = hlVxV§/U, 'UI + h(Vy'LL, VxUI + h(Vx'lL, Vyvl + h(u, VxVyUI. Hence [X, Y]h(u, UI = XY/z(u, UI - Y X h ( u v )

= h([VX,

V5/]'LL,

UI + h(u, [Vx

(3.3.18) 7

V3/]Ul.

On the other hand,

[X, Y]h(u, 'UI = h(V[Xy]'Ll,, 'UI

-|-

h(u, V[x,y]'U)-

(3.3.19)

The desired conclusion is reached after subtracting (33.19) from (3.3.18).

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. In general, the answer is negative, and this has to do with the global structure of the bundle. In the second half of this book we will discuss in more detail this fact.

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Calculus on Manifolds

3.3.4 Holonomy

The reader may ask a very legitimate question: why have we chosen to name curvature, the deviation from commutativity of a given connection. In this subsection we describe the geometric meaning of curvature, and maybe this will explain this choice of terminology. Throughout this subsection we will use Einstein's convention. Let E -> M be a rank r smooth K-vector bundle, and V a connection on it. Consider local coordinates (ac1, ac") on an open subset U C M such that E In is trivial. Pick a moving frame (el, ..., of E over U. The connection 1-form associated to this moving frame is

F

7

It is defined by the equalities (Vi

I=

1


790 = II, Vs, so $8 = 0. The same goes for CQ. Thus we have dTf,¢5 dIs,t

S

F12

dareaPst ds Loosely speaking, the last equality states that the curvature is the "amount of holonomy per unit of area". The result in the above proposition is usually formulated in terms of holonomy.

Definition 3.3.17. Let E -> M be a vector bundle with a connection V. The holonomy of V along a closed path 'Y is the parallel transpo along by. We see that the curvature measures the holonomy along infinitesimal parallelograms. A connection can be viewed as an analytic way of trivializing a bundle. We can do so along paths starting at a fixed point, using the parallel transport, but using different paths ending at the same point we may wind up with trivializations which differ by a twist. The curvature provides an infinitesimal measure of that twist.

Exercise 3.3.18. Prove that any vector bundle E over the Euclidean space Rn is trivializable. Hint: Use the parallel transport defined by a connection on the vector bundle E to produce a bundle isomorphism E -> E0 X R", where E0 is the fiber of E over the origin.

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3.3.5 The Bianchi identities

Consider a smooth K-vector bundle E -> M equipped with a connection V = VE . We have seen that the associated exterior derivative do : Q/>(E) -> Qp+1 (E) does not satisfy the usual (do)2 = 0, and the curvature is to blame for this. The Bianchi identity describes one remarkable algebraic feature of the curvature. Recall that VE induces a connection in any tensor bundle constructed from E. In particular, it induces a connection in E* (X) E = End (E) which we denote by V E n d ( E l . This extends to an "exterior derivative" End(E) DE do : op(End (E)) -> Op+1(End (E)).

~

Proposition 3.3.19 (Bianchi identity). Let E -> M be a K-vector bundle on M and V E a connection on E. Then D E F W E ) = 0.

Roughly speaking, the Bianchi identity states that (dV)3 is 0.

Proof. We will use the identities (3.3.13)-(3.3.15). For any vector fields X , Y , Z we have ZxDE =

VExNd(E)

- DE¢x .

Hence,

(DEFIIX, y, z)

ZzZyZxDEF = ¢Z¢y(v§"'*lE) - DEix)F ¢Z(vEXnc*(E)iy

- i[x,y])F - in(v§nd(E) - D E / ) ¢ x F

land(E)iziy

-

-(vi"d(E)¢ZiX

i[x,z]iy

- ii[x,y]lF

- i[y,z]ix -

VEz"dlE)¢y@x)F

+ i[y,z]ix + i[2 X ] i ¥ ) F -(v§ GL(1,]R).

The fiber at p G M of this bundle consists of 1°-densities on Ep (see Subsection 2.2.4).

Definition 3.4.1. Let M be a smooth manifold and or > 0. The bundle of 1°-densities on M is

When

T'

l

IAIRI =IAI'°(TMl. we will use the notation IAIN = IAI}w. We call IAIN the density bundle o f

M. Denote by C°°(IAIM) the space of smooth sections of IAIN, and by C3°(IAIM) its subspace consisting of compactly supported densities. It helps to have local descriptions of densities. To this aim, pick an open cover of M consisting of coordinate neighborhoods (Url . Denote the local coordinates on Uo by (we). This choice of a cover produces a trivializing cover of TM with transition maps T

where n is the dimension of M. Set 600 of functions nO( 6 cOO (Ua) related by

330

( ) j 3565

7

1 R and all e 6 C°°(det T M ) . In particular, any smooth map (b M -> N between manifolds of the same dimension induces a pullback transformation

: 0°O>(IAln) -> c°°(IAnm),

(l)*

described by

(gb*,u)(6) = ,up(det (M . e) = ldetq15*| 1u(e), We 6 C°°(detTM). Example 3.4.2. (a) Consider the special case M = ]R'"'". Denote by e l , ..., en the canonical basis. This extends to a trivialization of TR" and, in particular, the bundle of densities comes with a natural trivialization. It has a nowhere vanishing section ldvv,l defined by ldv,,,l(€1 /\

AAn) = 1.

In this case, any smooth density on R" takes the form ,LL = f ld'un |, where f is some smooth function on ]R"'. The reader should think of ldv'nl as the standard Lebesgue measure on R" . If go : R" -> R" is a smooth map, viewed as a collection of n smooth functions 651

-

Q51(CC

1 7°°°7$N)7 • • •

,sin

1 -

7°°'7$N)7

then, (25*(ldvnll = det

lwas( e(a2) )|.

Observe that lwl = | - wl, so this map §2M(Ml -> @OO(IAlml is not linear. (c) Suppose g is a Riemann metric on the smooth manifold M. The volume density defined by g is the density denoted by ldvgl which associates to each e E C°°(det TM) the pointwise length ac »->

le(.)lg~

If (Um ( 3 ) ) is an atlas of M , then on each UO( we have top degree forms d¢¢¢=dw1A»--Adg"

7

113

Calculus on Manifolds

ldac°OI.

to which we associate the density described as

In the coordinates .

II m

..doZ ga,.7 TM

®

dgj0 j

(51:23

) the metric g can be

.

We denote by l9al the determinant of the symmetric matrix I.go restriction of ldvgl to UO( has the description

lga;ij)1§i,j R

M

uniquely defined by the following conditions. (a)

FM

is invariant under

do eomorphisms,

i.e., for any smooth manifolds M, N

same dimension n, any dQ§"eomorphism cl) : M -> N, and any ,LL 6

'n

05° (IAIM),

,a_

/M

(b) is a local operation, i.e., for any open set U C M, and any ,u 6 supp 1,L C U , we have

'u (c) For any p 6

of the

we nave

o3°(IAIM) with

,u.

08° (R'nl we have

/

Rn

aldun I

/

Rn

p(:13)d;u,

where in the right-hand-side stands the Lebesgue integral of the compactly supportedfunction p. is called the integral on M .

FM

Proof. To establish the existence of an integral we associate to each manifold M a collection of data as follows. (i) A smooth partition of unity A C 08° (Ml such that Va 6 A the support supp a lies entirely in some precompact coordinate neighborhood UQ, and such that the cover (UO, ) is locally finite. (ii) For each Uo we pick a collection of local coordinates ( x ) , and we denote by id;ual (n = dim M) the density on UO( defined by II

(

.-I

lda3a I

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of Manzfolds

Lectures on the Geometry

For any ,u E 0°O(|A|), the product as, is a density supported in UQ, and can be written as 04/1

=

;U#adaoc

7

where ua is some smooth function compactly supported on Uo, The local coordinates allow us to interpret uo as a function on R". Under this identification Idwf; I corresponds to the Lebesgue measure ldvnl on R", and uOt is a compactly supported, smooth function. We set IUa¢ld53ol.

f]n

VULG

Finally, define A M

/

,u

2/

def

,u

M

QUA

up/,. UQ

The above sum contains only finitely many nonzero terms since supp ,L/, is compact, and thus it intersects only finitely many of the U' S which form a locally finite cover. To prove property (a) we will first prove that the integral defined as above is independent of the various choices: the partition of unity A C 0go>(M), and the local coordinates

ll";laeA-

• Independence of coordinates. Fix the partition of unity A, and consider a new collection of local coordinates (ya) on each UQ. These determine two densities id:z;;l and respectively Idyll |. For each ,LL 6 05°(IAIM) we have

aden any ldYo¢lv

co = where

#3 #Z 6

05'°(UQ) are related by

The equality

R"

uiildwol

Rn

my Idyo I

is the classical change in variables formula for the Lebesgue integral.

Independence of the partition of unit y. LetA,8 C C§°(M) two partitions of unity on M. We will show that

¢

A

/ / M

8

M

Form the partition of unity

A*8 : Note that supp QB C Uog

{0¢B; ( a , B ) € .A

X

8}(IAI N) we have

ow-

(¢*@)¢*u (b'1(Ua)

UQ

= a s)

(

The collection (l forms a partition of unity on M. Property (a) now QEA follows by summing over a the above equality, and using the independence of the integral on partitions of unity. To prove property (b) on the local character of the integral, pick U C M, and then choose a partition of unity 8 C 0golu) subordinated to the open cover IVBIBQQ3- For any partition of unity A C C'§°(M) with associated cover IVaIQEA we can form a new partition of unity A * 8 of U with associated cover Vol = Vo, n VB- We use this partition of unity to compute integrals over U. For any density ,u on M supported on U we have

/ 2/ ,LL

M

a

Vo

co

22/

0GABGB

QB# :: V06

2/

0,B€.A*8

QB# VQB

/

,u.

U

Property (c) is clear since, for M = Rn, we can assume that all the local coordinates chosen are Cartesian. The uniqueness of the integral is immediate, and we leave the reader to fill in the details. 3.4.2

Orientabilily and integration of dzjferentialforms

Under some mild restrictions on the manifold, the calculus with densities can be replaced with the richer calculus with differential forms. The mild restrictions referred to above have a global nature. More precisely, we have to require that the background manifold is oriented. Roughly speaking, the oriented manifolds are the "2-sided manifolds", i.e., one can distinguish between an "inside face" and an "outside face" of the manifold. (Think of a 2-sphere in R3 (a soccer ball) which is naturally a "2-faced" surface.) The 2-sidedness feature is such a frequent occurrence in the real world that for many years it was taken for granted. This explains the "big surprise" produced by the famous counter-example due to MObius in the first half of the 19th century. He produced a 1-sided surface nowadays known as the MObius band using paper and glue. More precisely, he

116

Lectures on the Geometry

of Manzfolds Il

l

l

`

\

\

\ 1

I ,r / / / /

Fig. 3.2

The Mobius band.

glued the opposite sides of a paper rectangle attaching arrow to arrow as in Figure 3.2. The 2-sidedness can be formulated rigorously as follows.

Definition 3.4.4. A smooth manifold M is said to be orientable if the determinant line bundle det TM (or equivalently det T * M ) is trivializable. We see that det T*M is trivializable if and only if it admits a nowhere vanishing section. Such a section is called a volume form on M. We say that two volume forms W i and W2 are equivalent if there exists f 6 C°°(M) such that C02

étwl .

This is indeed an equivalence relation, and an equivalence class of volume forms will be called an orientation of the manifold. We denote by O ' r ( M ) the set of orientations on the smooth manifold M. A pair (orientable manifold, orientation) is called an oriented manifold. Let us observe that if M is orientable, and thus Or(Ml 73 (D, then for every point p 6 M we have a natural map

Or(M) -> O'r(TpM), O'r(M) 9 or »-> orp 6 O'r°(TpM), defined as follows. If the orientation or on M is defined by a volume form co, then wp 6 det Tp M is a nontrivial volume form on T p M , which canonically defines an orientation o r , on T p M . It is clear that if W i and W2 are equivalent volume form then 0'rw1,p = orw2 ,pThis map is clearly a surjection because or_,vp = - o p , for any volume form CU.

Proposition 3.4.5. If M is a connected, orientable smooth manifold M , then for every p 6 M the map O'r(M) 9 or »-> o f p E Or(TpM)

is a bisection.

117

Calculus on Manifolds

Proof. Suppose or and or' are two orientations on M such that O'T'p = o p . The function

M 9 q *-> 6(q) = o r / o r q 6 {d:1} is continuous, and thus constant. In particular, 6(q) = 6(p) = 1, 'v'q 6 M. If or is given by the volume form w and or' is given by the volume form w' , then there exists a nowhere vanishing smooth function p : M -> R such that w' = pa. We deduce

sign »0(q) = €(q), Vq 6 M . This shows that the two forms w I and w are equivalent and thus or = or'

.

The last proposition shows that on a connected, orientable manifold, a choice of an orientation of one of its tangent spaces uniquely determines an orientation of the manifold. A natural question arises.

How can one decide whether a given manyL'old is orientable or not. 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. The orientability question can be given a very satisfactory answer using topological techniques. However, it is often convenient to decide the orientability issue using ad-hoc arguments. In the remaining part of this section we will describe several simple ways to detect orientability.

Example 3.4.6. If the tangent bundle of a manifold M is trivial, then clearly TM is orientable. In particular, all Lie groups are orientable. Example 3.4.7. Suppose M is a manifold such that the Whitney sum Then M is orientable. Indeed, we have det(Bk ® TM)

det

Bk

(X)

4 ® TM

is trivial.

det TM.

Both det Bk and det(Rk ® TM) are trivial. We deduce det TM is trivial since det TM = det(Rk ® TM) (det Rel *

~

(X)

This trick works for example when M 2" S". Indeed, let 1/ denote the normal line bundle. The fiber of 1/ at a point p 6 S" is the 1-dimensional space spanned by the position vector of p as a point in R", (see Figure 3.3). This is clearly a trivial line bundle since it has a tautological nowhere vanishing section p »-> p 6 Vp- The line bundle 1/ has a remarkable feature: U

® TS" = IRT),-l-1

Hence all spheres are orientable. Important convention. The canonical orientation on R" is the orientation defined by the volume form day1 /\ /\ dat", where x1, ..., x" are the canonical Cartesian coordinates. The unit sphere S" C lR"+1 is orientable. In the sequel we will exclusively deal with its canonical orientation. To describe this orientation it suffices to describe a positively oriented basis of det TpM for some p 6 S". To this aim we will use the relation IRR-l-1 up ® TpS"

---

i v

118

of Manzfolds

Lectures on the Geometry

4

I I |

/

/

\

I

\

/

| |

\ \

/ /

\

|

\

/

- _ _ / _

X.-

I

\

< ,

*-

/

"\u.

°-~.

_ " L _ _ ' \

-

>

- - - _ ; . _'°°~.,

\

I

\

I

\

I I

L

`\ \

"

I I I I

.

.

l/

Fig. 3.3

The normal line bundle to the round sphere.

An element cu E det TpS" defines the canonical orientation if p-» A w 6 det Rn-i-1 defines the canonical orientation of R"+*. Above, by p-» we denoted the position vector of p as a point inside the Euclidean space lR"+1. We can think of 15' as the "outer" normal to the round sphere. We call this orientation outer normaljirst. When n = l it coincides with the counterclockwise orientation of the unit circle S1 . Lemma 3.4.8. A smooth manifold M is orientable zfand only

of there exists an open cover

lUa)0 E A ) and local coordinates (5131 , ..., kg) on Uo, such that /

det

l

a;

> 0 OHU0 VWUQ.

j

(3.4.3)

35135 I

\

Proof. 1. We assume that there exists an open cover with the properties in the lemma, and we will prove that det T*M is trivial by proving that there exists a volume form. Consider a partition of unity 8 C c3° (Ml subordinated to the cover (UoJQQA i.e., there exists a map SO : 8 -> A such that 9

suppl C U(/)(B)

VI

E 8.

II

w

a s

Define Bongo(,B))

B

---

where for all a Q A we define wa := do1 A A dog. The form w is nowhere vanishing since condition (3.43) implies that on an overlap U O re Uom the forms was, ..., com differ by a positive multiplicative factor. 2. Conversely, let CU be a volume form on M and consider an atlas (Uo (x13))~ Then w INa= ,L/'fadac1 /\

---

- - - /\ dog,

119

Calculus on Manifolds

where the smooth functions gluing condition

,Uo

are nowhere vanishing, and on the overlaps they satisfy the

A 015

det

I

,UB

I I I I I I I I I I

.

la

---

A permutation up of the variables 51:1 , ..., x will change day1 A A dog by a factor €((/2) so we can always arrange these variables in such an order so that /Ja > 0. This will insure the positivity condition

A as > 0. The lemma is proved.

We can rephrase the result in the above lemma in a more conceptual way using the notion of orientation bundle. Suppose E -> M is a real vector bundle of rank r on the smooth manifold M described by the open cover l U a l a € A and gluing cocycle 7

gg0 : U05 -> GL(r, R).

The orientation bundle associated to E is the real line bundle ® (E) -> M described by the open cover lUa)0EA9 and gluing cocycle

650 := sign det 950 : U0g -> R* = GL(l,]R). We define orientation bundle ( 9 M of a smooth manifold M as the orientation bundle associated to the tangent bundle of M, ® M @(TM). The statement in Lemma 3.4.8 can now be rephrased as follows. Corollary 3.4.9. A smooth manifold M is orientable If and only If the orientation bundle @M

is trivializable.

From Lemma 3.4.8 we deduce immediately the following consequence.

Proposition 3.4.10. The connected sum of Iwo orientable manifolds is an orientable manfold.

Exercise 3.4.11. Prove the above result. Using Lemma 3.4.8 and Proposition 2.2.75 we deduce the following result.

Proposition 3.4.12. Any complex manifold is orientable. In particular, the complex Grassmannians Grk ( m l are orientable. Exercise 3.4.13. Supply the details of the proof of Proposition 3.4. 12.

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Lectures on the Geometry

of Manzfolds

The reader can check immediately that the product of two orientable manifolds is again an orientable manifold. Using connected sums and products we can now produce many examples of manifolds. In particular, the connected sums of g tori is an orientable manifold. By now the reader may ask where does orientability interact with integration. The answer lies in Subsection 2.2.4 where we showed that an orientation or on a vector space V induces a canonical, linear isomorphism 2O1" : det V* -> IAIV, see (2.2.l3). Similarly, an orientation or on a smooth manifold M defines an isomorphism C°°(detT*m) -> C°°(IAlml.

ZOO'

For any compactly supported differential form

Ld

on M of maximal degree we define its

integral by z0,,.w.

'm

'm

We want to emphasize that this definition depends on the choice of orientation. We ought to pause and explain in more detail the isomorphism 2O7' : cOO (det T * M ) -> cOO (IAIN). Since M is oriented we can choose a coordinate atlas ( u , I ) such that I'

Q

ii

`l

dim M ,

> 0, n

det

(3.4.4)

1§,'i,j§M

and on each coordinate patch UQ the orientation is given by the top degree form doa dol A---Adccg.

A top degree differential form LU is described by a collection of forms We

padsa

Po E CYOO(U0),

and due to the condition (3.4.4) the collection of densities Ila

I

Padua' E C°°(U0,, I

III lm)

satisfy ,ua = ,up on the overlap Uog. Thus they glue together to a density on M, which is precisely ZOT'LU. Example 3.4.14. Consider the 2-form on R3, w = :cdy /\ do, and let S2 denote the unit sphere. We want to compute j S2 ( ; I $2 , where S2 has the canonical orientation. To compure this integral we will use spherical coordinates (T, go, H). These are defined by (see Figure 3.4.) ac

y Z

At the point p

(1,0,0) we have c'91,"=3,,=15',

SO

dr

r sin go COS H 7" sin up sin 0 TCOS up 3023y(p:-(9Z,

that the standard orientation on S2 is given by d o /\ do. On S2 we have T' E 0 so that Andy /\ do

152 =

sin go COS 9 (cos 9 sin cpd -|- sin 9 COS godc,o)) /\ (- sin (p)dgo

= sin3 go COS2 9d(p /\ d9.

1 and

121

Calculus on Manifolds

II

z

up

O

r

.< \

9

/

/

|

\

L

/

I I

\ \

P

I I I I I

/ /

\

I

\ \

/

|

\ \

/

|

\ \

v

I

/

/

/

X

Fig. 3.4

Spherical coordinates.

The standard orientation associates to this form de density sin3 (ac:os2 0ldgod0l, and we deduce 2tr

11'

lm] x [0,27/]

sin3 up cos2 0ld(pd0|

sing cpdcp 0

COS2

l

0d6

0

_

47r volume of the unit ball 83 C R3. 3 As we will see in the next subsection the above equality is no accident. '

Example 3.4.15 (Invariant integration on compact Lie groups). Let G be a compact, connected Lie group. Fix once and for all an orientation on the Lie algebra LG. Consider a positively oriented volume element w 6 det LG. We can extend w by left translations to a left-invariant volume form on G which we continue to denote by w. This defines an orientation, and in particular, by integration, we get a positive scalar II

U

CU.

G

Set dVG = in so that e

we

1.

(3.4.5)

G

The differential form dVG is the unique left-invariant n-form (n = dim G) on G satisfying (3.4.5) (assuming a fixed orientation on G). We claim dVG is also right invariant. To prove this, consider the modular function G 9 h »-> A(h) 6 R defined by

Rfldvg )

A(h)dVG.

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Lectures on the Geometry

of Manzfolds

The quantity A(h) is independent of h because RhdVG is a left invariant form, so it has to be a scalar multiple of dVG. Since (R;»,1/'L2)* = (R/12 Rhll* = Rh1Rh2 we deduce A(h1h2) = A(/21lA(h2)

Wu,

122

6 G.

Hence h +-> Ah is a smooth morphism G -> (M {0},~). Since G is connected A(G) C R+, and since G is compact, the set A(G) is bounded. If there exists QUO Q G such that A(x) go 1, then either A(x) > 1, or A(a:-1) > 1, and in particular, we would deduce the set lAl5'3nllne Z is unbounded. Thus A E 1 which establishes the right invariance of dVG. 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 so 6 G define To

((Lg)*T)mdVG(9)G

Then as »-> T, defines a smooth tensor field on G. We claim that T is left invariant. Indeed, for any h G G we have

(Lhl*T

/ /

G(Lh)*((L9)*TldVG(9)

u

hg

G((Lhg)*T)dVG(9l

-TZ

(Lu)*TLh_1dVG(fu,)

(Lh_1dVG

Z

G

If we average once more on the right we get a tensor GEQH->

( (Rg)i'

)¢zvG,

G

which is both left and right invariant.

Exercise 3.4.16. Let G be a Lie group. For any X G LG denote by ad(X) the linear map LG->»CG defined by

LG 9 Y»-> [x,y]

QLG.

(a) If w denotes a left invariant volume form prove that V/XGLG LxW = - t r ad(X)w.

(b) Prove that if G is a compact Lie group, then tr a d ( X ) = 0, for any X

Q

LG.

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Calculus on Manifolds

3.4.3

Stokes' formula

The Stokes' formula is the higher dimensional version of the fundamental theorem of calculus (Leibniz-Newton formula) b

if

f(&),

f(1»)

where f : [0, b] -> R is a smooth function and df = f ' (t)dt. In fact, the higher dimensional formula will follow from the simplest l-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 n

(;u1,...,a:") Q R" ;ac

1

_ 0}.
V2 is an isomorphism of G-modules. According to the previous discussion, any other nontrivial G-morphism L : VI -> V2 has to be an isomorphism. Consider the automorphism T = S-1L : $1 -> V1. Since $1 is a complex vector space T admits at least one (non-zero) eigenvalue A. The map A]1V1 - T is an endomorphism of G-modules, and her (Ally, - T) go 0. Invoking again the above discussion we deduce T E A1V1, i.e., L AS. This shows dirnHornG(V1, V2) = 1.

~

=

Schur's lemma is powerful enough to completely characterize S1 sentations of S1.

Mod, the repre-

Example 3.4.39 (The irreducible (complex) representations of S1). Let V be a complex irreducible S1-module S1

X

V9(eW,u)l-->T0v€V,

130

of Manzfolds

Lectures on the Geometry

-

where TO1 TO2 = T@1+92 mod 2'rr. In particular, this implies that each T9 is an S1automorphism since it obviously commutes with the action of this group. Hence T9 = A(0)]1V which shows that d i V = 1 since any 1-dimensional subspace of V is S1invariant. We have thus obtained a smooth map

A:S1->(C*

7

such that

Aw"

girl

= >(e")A(@@>,

Hence A : Sl -> KI* is a group morphism. As in the discussion of the modular function we deduce that l,\l E 1. Thus, A looks like an exponential, i.e., there exists a Q R such that (vergCy./)

A(@i*'> = exp('iaH),

we 6 R.

Moreover, exp(27r'ia) = 1, so that a Q Z. Conversely, for any integer n 6 Z we have a representation

Sl &l Aut (el

'0 (6'a

. 0

,zl +-> ezn

z.

The exponentials exp('£n0) are called the characters of the representations

,On-

Exercise 3.4.40. Describe the irreducible representations of T"-the n-dimensional torus.

Definition 3.4.41. (a) Let V be a complex G-module, g +-> T(g) 6 Aut ( V ) . The character of V is the smooth function

==

Xv : G -> C, xv(9) trT(g). (b) A elassfunction is a continuous function f : G -> (C such that f(h9h

_1

_

) - f(9) V 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 XU their characters. Then the following hold.

(0) XU1@U2 = XU1 -|- XU2' XU1®U2 = XU1 • XU1(19) XU,ll) = dim Ui(c) XU.* = Y M -the complex conjugate Of XU. (d) G

XU (g)dvG (go

:

dim UG

7

where UG denotes the space of G-invariant elements of Up, HG

{as E

u , Hz: = T¢l9)93 we E G}.

(6) G

XU1 (9) 'Yi/5 (9)dVG(9) = dim HOMG(U2, U1)-

131

Calculus on Manifolds

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 a unitary operator on U. The action of G on U* is given by T(9-1)*. Since T(g) is unitary, we have T(g-1)* = T(g). This proves (c). Proof of (d). Consider

P : U -> U,

Pa

T(g)u dvG(g). G

Note that PT(h) = T(h)P, WE E G, i.e., P E HOMG(U, U). We now compute T(h)Pu

G

T(h9)u dVG(9)

TWO Rh-1dVG('y),

G

T('y)'u, dvG('y) = Pa. G

Thus, each Pa is G-invariant. Conversely, if as 6 U is G-invariant, then

/

Pa:

G

T(9):vd9

/

:IJ

G

dVG (9)

Xv

i.e., UG = Range P. Note also that P is a projector, i.e., P2 = P. Indeed,

P2u

T G

l9)p

Gaga

P

G

u 'U»dvG(9 ) -- P

Hence P is a projection onto UG, and in particular d i l l ; UG = t r P

trT(g) dvG(g)

Xu(g) dvG(g).

G

G

Proof of (6). G

XU1 '36-U2dVGl9l

G

XU1

-xu;dvG(9l

XU1®U; dvG(9l G

ding (Hom (up, u1))G

XHom(U2,U1) G

dim() HOIHG(U2, U1),

since Home coincides with the space of G-invariant morphisms.

Corollary 3.4.43. Let U , V be irreducible G-modules. Then XU» XV)

G

~

l ,U = V 0 ,U %V

XU °Yvd9 = 61/v

Proof. Follows from Theorem 3.4.42 using Schur's lemma. Corollary 3.4.44. Let U , V be two G-modules. Then U 'E V if and only z f x u = XV.

Proof. Decompose U and V as direct sums of irreducible G-modules

V

e €91(Nj)-

Hence XU = Z:MZx\ and Xv = 2 nix . The equivalence "representation" ~: >» "characters" stated by this corollary now follows immediately from Schur's lemma and the previous corollary.

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Lectures on the Geometry

of Manzfolds

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 and denote by Xv its character. The complex exterior algebra ATV* is a complex G-module, as the space of complex multi-linear skew-symmetric

maps

V

X

V->(C.

X

Denote by bglvl the complex dimension of the space of G-invariant elements in One has the equality

be)

G

A/]R.

The vector space A,,1V* is a real G-module. We complexity it, so that A; V ®(C is the space of R-multi-linear, skew-symmetric maps

V x - ~ - x V ->(C, and as such, it is a complex G-module. The real dimension of the subspace gklvl of Ginvariant elements in Akv* will be denoted by bk(V), so that the Poincaré polynomial of

J;(v) = €BIlg3k is

Ikb7°klyl.

Pf1:(v) (t) k

On the other hand, bk (V) is equal to the complex dimension of Ak V* (X) C. Using the results of Subsection 2.2.5 we deduce

A;v

(X)

(C

r

a

Age* (Xm: A;V*

laB+y=kA;V*

(X)

I

AW*

•

(3.4.9)

k

Each of the above summands is a G-invariant subspace. Using (3.4.7) and (3.4.9) we deduce Pa;(v) (t)

07fz ( T ( g )

k

Gi+j

)(ti(T(9)

det( iv -|- tT(g)l dell iv

G

)t¢+j avG(gl

k

+ t T ( g ) l dvG(g)

(3.4.10)

'detl NV + tT (g ))1 2 dog).

We will have the chance to use this result in computing topological invariants of manifolds with a "high degree of symmetry" like, e.g., the complex Grassmannians.

3.4.5

Fibered ealeulus

In the previous section we have described the calculus associated to objects defined on a single manifold. The aim of this subsection is to discuss what happens when we deal with an entire family of objects parameterized by some smooth manifold. We will discuss only the fibered version of integration. 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 libration and so we will not deal with it. The interested reader can learn more about this operation from [55], Chapter 3, Sec. 5.

134

Lectures on the Geometry

of Manzfolds

Assume now that, instead of a single manifold F, we have an entire (smooth) family of them (Fe)be B . In more rigorous terms this means that we are given a smooth fiber bundle p : E -> B with standard fiber F. On the total space E we will always work with split coordinates ( a , by ), where w) are local coordinates on the standard fiber F, and (yj ) are local coordinates on the base B (the parameter space). The model situation is the bundle E=R'1Re.m

B, (x,y)

Z

I

p

>y.

We will first define a fibered version of integration. This requires a fibered version of orientability.

Definition 3.4.45. Let p : E -> B be a smooth bundle with standard fiber F. The bundle is said to be orientable if the following hold. (a) The manifold F is orientable, (b) There exists an open cover (Ua), and trivializations P'1(Ual the gluing maps

W 5 o \ P a 1 : F x Uag

F

X

U05

> F x UQ, such that

IU0B = UQ VW Url

are fiberwise orientation preserving, i.e., for each y G Ugh, the diffeomorphism

F9f

F

preserves any orientation on F . Exercise 3.4.46. If the base B of an orientable bundle p : E -> B is orientable, then so is the total space E (as an abstract smooth manifold). Important convention. Let p : E -> B be an orientable bundle with oriented basis B. The natural orientation of the total space E is defined as follows. If E F X B then the orientation of the tangent space T(f,b)E is given by QF X we, where we 6 det TfF (respectively we 6 det TbB) defines the orientation of TfF I

(respectively TbB). The general case reduces to this one since any bundle is locally a product, and the

gluing maps are fiberwise orientation preserving. This convention can be briefly described as orientation total space = orientation fiber /\ orientation base. The natural orientation can thus be called the fiber-first orientation. In the sequel all orientable bundles will be given the fiber-first orientation.

Let p : E -> B be an orientable fiber bundle with standard fiber F.

Proposition 3.4.47. There exists Cl linear operator p* E/B

:

Q;ptlE) -> Qgpt (B) ,

T'

= dim F,

135

Calculus on Manifolds

uniquely defined by its action on forms supported on domains D of split coordinates

~

D = RR x

Run

2> Run , (33; y) »-> y-

I f s = fda! /\ dy'], f 6 CCpt(]R"+M), then

0 7 III ¢ T' E/B (IR f d f I l dyJ 7 III = T' The operator I5/8 is called the integration-along-fibers operator

•

The proof goes exactly as in the non-parametric case, i.e., when B is a point. One shows using partitions of unity that these local definitions can be patched together to produce a well defined map

-> Q;ptlBl°

QZpt

E/B

The details are left to the reader.

Proposition 3.4.48. Let p : E -> B be an orientable bundle with an T-dimensional standardjiber F. Then for any LU 6 Qe*pt ( E ) and 17 6 welpt(B) such that deg w+deg n = dim E we have

dEuJ = l-1l7dB

w. E/B

If B is oriented and CU 17 at as above then

f

I(/ )

CU I\p*('7)

w

E

B

E/B

(Fubini)

Av-

The last equality implies immediately the projection formula p*(w /\ PW =

P )

(3.4.11)

/\ 77-

Proof. It suffices to consider only the model case p:E=Rx]RM->]RM=B,

and w

fdail

/ \ b Y ] . Then

a

E/B

(f

Adatl A d 3 / J + (-1>III

3:6z

ii

/

.

f ax'

dew

dEad

(atgy) 3, y,

j

@f

Rt

Hz

35

div /\ do]

-|-

(-1)|I|

@f dat] /\ dye /\ dye. byJ

(L

@f . d I é9y3 QUO

/\ dye /\ dyJ.

The above integrals are defined to be zero if the corresponding forms do not have degree Stokes' formula shows that the first integral is always zero. Hence

dEw E/B

I

(-1) I/I 3

0}.

Remark 3.4.S2. Let p : (E, BE) -> B be a B is orientable and the basis B is oriented as well, then on BE one can define two orientations. (i) The fiber-first orientation as the total space of an oriented bundle 3E->B. (ii) The induced orientation as the boundary of E.

These two orientations coincide.

Exercise 3.4.S3. Prove that the above orientations on BE coincide.

137

Calculus on Manifolds

Theorem 3.4.S4. Let p : ( E , B E ) -> B be an orientable 3-bundle with an 1°-dimensional interiorjiben Then for any w 6 Q;pt(E) vm have dEad - (-1)*d8 I3E/B

IE/8

w (Homotopy formula). E/B

The last equality can be formulated as

/

AE/B

/

dE - (-l)"d8

E/B

E/B

This is "the mother of all homotopy formulae". It will play a crucial part in Chapter 7 when we embark on the study of DeRham cosmology.

Exercise 3.4.55. Prove the above theorem.

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Chapter 4

Riemannian Geometry

Now we can finally put to work the abstract notions discussed in the previous chapters. Loosely speaking, the Riemannian geometry studies the properties of surfaces (manifolds) "made of canvas". These are manifolds with an extra structure arising naturally in many instances. On such manifolds one can speak of the length of a curve, and the angle between two smooth curves. In particular, we will study the problem formulated in Chapter 1: why a plane (flat) canvas disk cannot be wrapped in an one-to-one fashion around the unit sphere in R3. Answering this requires the notion of Riemann curvature which will be the central theme of this chapter.

4.1 Metric properties 4.1.1

Definitions and examples

To motivate our definition we will first try to formulate rigorously what do we mean by a "canvas surface". 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. Alternatively, this means that the only operations we can perform are those which do not change the lengths of curves on the surface. Thus, one can think of "canvas surfaces" as those surfaces on which any "reasonable" curve has a well defined length. Adapting a more constructive point of view, one can say that such surfaces are endowed with a clear procedure of measuring lengths of piecewise smooth curves. Classical vector analysis describes one method of measuring lengths of smooth paths in R3. If by [0, 11 -> R3 is such a paths, then its length is given by 1

length (w)

l(t)udt, 0

where l'Y(t)l is the Euclidean length of the tangent vector WI, We want to do the same thing on an abstract manifold, and we are clearly faced with one problem: how do we make sense of the length lf}/(t)l? Obviously, 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. The simplest way to do achieve this is to assume that each tangent 139

140

Lectures on the Geometry

of Manzfolds

space is endowed with an inner product (which can vary from point to point in a smooth way).

Definition 4.1.1. (a) A Riemann manifold is a pair (M, g) consisting of a smooth manifold M and a metric g on the tangent bundle, i.e., a smooth, symmetric positive definite (0, 2)tensor field on M. The tensor g is called a Riemann metric on M. (b) Two Riemann manifolds (Mi, oil (i = 1, 2) are said to be isometric if there exists a diffeomorphism QUO : M1 -> M2 such that (l5*g2 = 91If ( M , g ) is a Riemann manifold then, for any ;136 M , the restriction g,,:TxMxTxM->H

..

..

is an inner product on the tangent space T x M . We will frequently use the alternative notation ( I's = gx( , . The length of a tangent vector 'U 6 TxM is defined as usual, 7

gLC:l'U,'U)1/2

Ivlx

If by [a, b] -> M is a piecewise smooth path, then we define its length by 1('y)

b

l'ltllv(t)dt°

a

If we choose local coordinates

g

(al , . . . ,a:") on M, then we get a local description of g as l id azj (a rd:d x i gi

9

$1

78

I

Proposition 4.1.2. Let M be a smooth manifold, and denote by RM the set of Riemann metrics on M . Then RM is a non-empty convex cone in the linear space of symmetric (0, 2)-tensors. Proof. The only thing that is not obvious is that (Qm is non-empty. We will use again partitions of unity. Cover M by coordinate neighborhoods ( U 6 ) o € A . Let (kg) be a collection of local coordinates on UQ. Using these local coordinates we can construct by hand the metric ga on Ua by go

;

(d;u1)2-I-°~ + (dw:;)2.

Now, pick a partition of unity 8 C 05° (M) subordinated to the cover qUo)oeA, i.e., there exits a map go : 8 -> A such that VB 6 8 suppl C U¢(g)~ Then define g

I39w

.

.z + i

zZ

_

u-I-w

•

z

establishes an isometric (D , 8 ) % ( m , h ) .

Example 4.1.8 (Left invariant metrics on Lie groups). Consider a Lie group G, and denote by LG its Lie algebra. Then any inner product (w ) on LG induces a Riemann metric h = (w °>9 on G defined by

-

Ilglx»yl

I

(X»Y)9

T1G is the differential at g k G of the left translation map L9 One checks easily that the correspondence

1.

G99'-9 is a smooth tensor field, and it is left invariant, i.e.,

L g h h

'v'g€G.

If G is also compact, we can use the averaging technique of Subsection 3.4.2 to produce metrics which are both left and right invariant.

143

Riemannian Geometry

4.1.2 The Levi-Civita eonneetion To continue our study of Riemann manifolds we will try to follow a close parallel with classical Euclidean geometry. The first question one may ask is whether there is a notion of "straight line" on a Riemann manifold. In the Euclidean space R3 there are at least two ways to define a line segment. (i) A line segment is the shortest path connecting two given points. (ii) A line segment is a smooth path 'Y : [0, 1] -> R3 satisfying

w)

0.

(4.l.1)

Since we have not said anything about calculus of variations which deals precisely with problems of type (i), we will use the second interpretation as our starting point. We will soon see however that both points of view yield the same conclusion. Let us first reformulate (4.1.1). As we know, the tangent bundle of R3 is equipped with a natural trivialization, and as such, it has a natural trivial connection V0 defined by

v931 = 0

iv"£,j, where

a

a

7

Vi

Va

9

i.e., all the Christoffel symbols vanish. Moreover, if 90 denotes the Euclidean metric, then

V05

k

90(V06j, 60 - 90(6j, V06

0,

i.e., the connection is compatible with the metric. Condition (4.l.l) can be rephrased as V(¢)(tl

0,

(4.l.2)

so that the problem of defining "lines" in a Riemann manifold reduces to choosing a "natural" connection on the tangent bundle. Of course, we would like this connection to be compatible with the metric, but even so, there are infinitely many connections to choose from. The following fundamental result will solve this dilemma. Proposition 4.1.9. Consider a Riemann manifold ( M , g). Then there exists a unique symmetric connection V on TM compatible with the metric g, i.e.,

T=0, V g = 0 . The connection V is usually called the Levi-Civita connection associated to the metric g.

Proof. Uniqueness. We will achieve this by producing an explicit description of a connection with the above two properties. Let V be such a connection, i.e., V g = 0 and

V x Y - V ; / X : [X,Y], VX,Y 6 V e c t ( M ) .

For any X, Y, Z 6 Vect (M) we have

Z g ( X , Y) = 9(VzX, Y) -|- g(X, VzYI since Vg = 0. Using the symmetry of the connection we compute Zg(X,Y) - Y g ( Z , X ) +Xg(Y,Z) = 9(VzX»Y) - 9 ( V y Z , X l *l'glVxY,Zl

144

Lectures on the Geometry

of Manzfolds

+ g ( X , VzY ) - 9(Z, VyXI + 9(Y» VxZ) 9([Z» YL X) + 9([X» YL Z) + g([Z» XL Y)

-|-

2g(VxZ, Y)~

We conclude that

§{X9(Y» Zl - YQ (Z ,X )

QIVxZ, Y)

-|-

Zg(X,Y)

-9([x,y],z) + g([Y» ZLX) - 9([ z,x ],yl }.

(4.l.3)

The above equality establishes the uniqueness of V. Using local coordinates (x1 . . . ,ac") on M we deduce from (4.l.3), with X a , Y = 3 k = 3 , , k , Z = 3 =j 3x3 la that 1

Q(

glee

i3j,31 R, IQ(x,yl := -tr (ad(x) ad(y))

x , y 6 L.

The Lie algebra ,G is said to be semisimple if the Killing pairing is a duality. A Lie group G is called semisimple if its Lie algebra is semisimple.

Exercise 4.1.18. Prove that SO(n) and SU(n) and SL(n, RI are semisimple Lie groups, but U(n) is not.

147

Riemannian Geometry

Exercise 4.1.19. Let G be a compact Lie group. Prove that the Killing form is positive semi-definite1 and satisfies (4.1.7).

Hint:

Use Exercise 4.1.16.

Exercise 4.1.20. Show that the parallel transport of X along exp(tY) is (L@xp(%y)>* l R e x p ( § Y ) l * X °

Example 4.1.21 (Geodesics on flat tori, and on SU(2)). The n-dimensional torus T n S1 X X S1 is an Abelian, compact Lie group. If (91 7 • . • 7 @") are the natural angular r\/

coordinates on T", then the flat metric is defined by II m

(do1)2 +

+ (do")2_

The metric g on T" is bi-invariant, and obviously, its restriction to the origin satisfies the skew-symmetry condition (4.l.7) since the bracket is 0. The geodesics through 1 will be the exponentials

. . . a ( ¢ ) t+->(e°"1t 7 .. . e t ) 7

end.

If the numbers as are linearly dependent over Q, then obviously ' y 0 1 1 . . . O n (t) is a closed curve. On the contrary, when the a's are linearly independent over Q then a classical result of Kronecker (see e.g. [59]) states that the image of ' y o i l . . . o n is dense in Tn!!! (see also Section 7.4 to come). The special unitary group SU(2) can also be identified with the group of unit quaternions {a+bfi+cj+dk;

a2+b2+02+d2

1},

so that SU(2) is diffeomorphic with the unit sphere S3 C R4. The round metric on S3 is bi-invariant with respect to left and right (unit) quaternionic multiplication (verify this), and its restriction to (1 0 0, 0) satisfies (4.l.7). The geodesics of this metric are the 1parameter subgroups of S3, and we let the reader verify that these are in fact the great circles of S3, i.e., the circles of maximal diameter on S3. Thus, all the geodesics on S3 are closed. 7

4.1.3

7

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. In particular, 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. The global topology of the manifold is responsible for this "failure". lThe converse of the above exercise is also true, i.e., any semisimple Lie group with positive definite Killing form is compact. This is known as Weyl's theorem. Its proof, which will be given later in the book, requires substantially more work.

148

Lectures on the Geometry

of Manzfolds

Locally however, things are not "as bad". Local Riemannian geometry is similar in many respects with the Euclidean geometry. For example, locally, all of the classical incidence axioms hold. In this section we will define using the metric some special collections of local coordinates in which things are very close to being Euclidean. Let (M, g) be a Riemann manifold and U an open coordinate neighborhood with coordinates (ac1, ..., mTL). We will try to find a local change in coordinates (al +-> (ye) in which the expression of the metric is as close as possible to the Euclidean metric 90 = (S d g d y j . Let q 6 U be the point with coordinates (0,...,0) . Via a linear change in coordinates we may as well assume that 9¢i( 0 'ye 5x(tl = 'Yq,xl5tl-

Hence, there exists a small neighborhood V o f 0 6 Tqll/I such that, for any X 6 V, the geodesic 'Yq,X (to is defined for all ltl < 1. We define the exponential map at q by

expq : V C TqM -> M, X »-> 'Yq,X(1). The tangent space TqM is a Euclidean space, and we can define Dq(1°) C T q M , the open "disk" of radius T' centered at the origin. We have the following result.

Proposition 4.1.22. Let ( M ,

go and q 6

M as above. Then there exists T' > 0 such that the

exponential map

expq : D q(al -> M i5 a dyeomorphism onto. The supreme of all radii

T'

with this property i5 denoted by

DmM)-

Definition 4.1.23. The positive real number pm(q) is called the injectivily radius o f M at q. The infimum Pm is called the injectivily radius o f M .

i18f Pm (Q)

149

Riemannian Geometry

The proof of Proposition 4.1.22 relies on the following key fact.

Lemma 4.1.24. The Fréchet deferential at 0 6 TqM of the exponential map DO expo . TOM

Texpq(0)M

I

TqM

i5 the identity TqM -> TqM. Proof. Consider X 6 TqM. It defines a line t »-> TX in TqM which is mapped via the exponential map to the geodesic 'Yq,X (t). By definition (Do expq)X =

'

q,Xl0)

X.

Proposition 4.1.22 follows immediately from the above lemma using the inverse function theorem.

Now choose an orthonormal frame (el, ..., en) ofTqM, and denote by (as1, ..., asn) the resulting Cartesian coordinates in TqM. For 0 < 7° < pM(q), any poi f t p 6 expq(D q(7")) can be uniquely written as P

: :

expo(we),

SO that the collection (ml, ..., mol provides a coordinatization of the open set expq(D q(¢~)) C M. The coordinates thus obtained are called normal coordinates at q, the open set expq (D ql'r°)l is called a normal neighborhood, and will be denoted by B AQ) for reasons that will become apparent a little later.

Proposition 4.1.25. Let W ) be normal coordinates at q 6 M, and denote by g'in the expression of the metric tensor in these coordinates. Then we have g fzjlql

($1j

and

0 such that, Wm E B ,~(q), we have e < Pm ( m ) and B € ( m ) D B rlql. In particular; any two points of B ,.(q) can be joined by a unique geodesic of length < e. We must warn the reader that the above result does not guarantee that the postulated connecting geodesic lies entirely in B ,~(q). This is a different ball game. Proof. Using the smooth dependence upon initial data in ordinary differential equations we deduce that there exists an open neighborhood V of (q, Of 6 TM such that exp X is well defined for all (m, X ) 6 V. We get a smooth map

F:V->M

X

M

(m,X)»->(m,expmXl.

We compute the differential of F at (q, 0). First, using normal coordinates (wi) near q we get coordinates W; Xi ) near (q, 0) 6 TM. The partial derivatives of F at (q, Of are /

D(q,0)F

Q

1

\

3 Z

\

_|_

/

exp,,n(tX) is a geodesic of length < 5 joining m to expm(X). It is the unique geodesic with this property since F : V -> U is injective. We can now formulate the main result of this subsection. Theorem 4.1.28. Let q, T' and 5 as in the previous lemma, and consider the unique geodesic by : [0, 1] -> M of length < e joining two points of B ,~(q). I f s : [0, 1] -> M is piecewise smooth path with the same endpoints as by then

/ Wtlldt /'
0, we obtained a new, rescaled metric PA = I\2g. A simple computation shows that the Christoffel symbols and Riemann tensor of 9> are equal with the Christoffel symbols and the Riemann tensor of the metric g. In particular, this implies

R1cg = Rlcg However, the sectional curvatures are sensitive to metric rescaling. For example, if g is the canonical metric on the 2-sphere of radius l in R3, then QA is the induced metric on the 2-sphere of radius A in R3. Intuitively, the larger the constant A, the less curved is the corresponding sphere. In general, for any two linearly independent vectors X, Y Q Vect ( M ) we have

K9AlX7 y) = A-2Kg(x, y). In particular, the scalar curvature changes by the same factor upon rescaling the metric. If we thing of the metric as a quantity measured in meter2 , then the sectional curvatures are measured in m€[€t-2.

169

Riemannian Geometry

II

Exercise 4.2.9. Let x,y,z, W 6 TpM such that span(X,Y) dimensional subspace of TpM prove that Kp(X, Y ) K,,(z, W)-

span(Z,W) is a 2-

According to the above exercise the quantity K p ( X , Y) depends only upon the 2-plane in TpM generated by X and Y. Thus Kp is in fact a function on Gp2 (p) the Grassmannian of 2-dimensional subspaces of T p M .

Definition 4.2.10. The function Kp curvature o f M at p.

Gr 2(Pl -> R defined above is called the sectional

Exercise 4.2.11. Prove that Gr 2(Ml = disjoint union of

GF2 (p)

p6 M

can be organized as a smooth fiber bundle over M with standard fiber Gr 2lRnl» n dim M, such that, if M is a Riemann manifold, G r 2 ( M ) 9 (p; or) »-> Kp(7r' is a smooth map. 4.2.2 Examples

.. ..

Example 4.2.12. Consider again the situation discussed in Example 4.1.15. Thus, G is a Lie group, and ( 7 ) is a metric on the Lie algebra LG satisfying

(ad(X)Y, z)

= - 0.

170

Lectures on the Geometry

of Manzfolds

Denote the Killing form by I [[X»Z]»Y]) 1 4

12 l

12 ([X» E )

l 4

[Y,

(adlyl[x, E,~], (adlXlEz', adlYlEj)

_l Z

l tr 4 (ad(y)ad(x) )

l zI{lX7Yl_

In particular, on a compact semisimple Lie group the Ricci curvature is a symmetric positive definite (0, 2)-tensor, and more precisely, it is a scalar multiple of the Killing metric. We can now easily compute the scalar curvature. Using the same notations as above we get (E

S

ET

1 4

lislE»,;,

In particular, if G is a compact semisimple group and the metric is given by the Killing form then the scalar curvature is 1

s(/~z) = 4 dim G.

Remark 4.2.13. 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. Although the results are not as "crisp" as in the bi-invariant case many nice things do happen. For details we refer to [96].

Example 4.2.14. Let M be a 2-dimensional Riemann manifold (surface), and consider local coordinates on M, (ac1, 5132). Due to the symmetries of R, Rijkl

-Rijlk

Rklij

7

we deduce that the only nontrivial component of the Riemann tensor is R sectional curvature is simply a function on M

K

l -R1212

191

i

l 55(9), where

191

R1212-

The

detlgijl-

In this case, the scalar K is known as the total curvature or the Gauss curvature of the surface.

171

Riemannian Geometry

In particular, if M is oriented, and the form do1 A day2 defines the orientation, we can construct a 2-form l

5(9)

l

Kd'Ug

l g

27rV

191

R1212d5U1

/\

d;132.

The 2-form e(g) is called the Euler form associated to the metric g. We want to emphasize that this form is defined only when M is oriented. We can rewrite this using the pfaffian construction of Subsection 2.2.4. The curvature R is a 2-form with coefficients in the bundle of skew-symmetric endomorphisms of TM SO we can write

R A

1

dog, A

®

vlgl

[

0

£1212

0

R2112

Assume for simplicity that (xl , x2) are normal coordinates at a point q E M. Thus at q, lgl = l since 31, M. There is an alternative way of looking at N. Choose

U , V € V e c t ( S ) , N € COO(Nsl.

If we write g(¢

7

¢

)=(

.. )» 7

(~(u,v>,n)

Z

then

( (v%v)" , n)=(v£¥'vcn) -(v,vgyfn)=-(v,lvgyIn)).

=U-(V,N) We have thus established

(v, (vgyIn)T) = (n(U,v),n) = (n(v,U),n) = -(U, (wvv)-

(42.12)

The second fundamental form can be used to determine a relationship between the curvature of M and that of S. More precisely we have the following celebrated result.

Theorem Egregium (Gauss). Let RM (resp. RS) denote the Riemann curvature of (M, g) (resp. (S, g Is)» Then for any X, Y, Z, T E Vect (S) we have

Ix

(RM

aT)

'Y)z

= II>x2) ( '

Proof. Note that

T N (x l ' (Ya e N ( )

(4.2.13)

7

YT

D

VXMY

= viiI + n(x, y).

RM(X, Y)X

= WxM, vym]z - V

We have

= VxMW?Z + MY, 2 ) ) -

y Z

W" ( viz + MX, z) ) - Vi

12 - NIX, ii, z).

Take the inner product with T of both sides above. Since N(¢, .) is N5-valued, we deduce using (4.2.10)-(4.2.l2)

(RM(x, y)z, T I

ii, VW, T

(VXMV§,Z,T)

-(v¥v§K'z T ) - (VYMMX, z>,T) - (Vi),n(y, T) ) - ( V < , y ] Z 7 T ) ~

This is precisely the equality (4.2.13). 2 The

+ (VXMN(Y, z ) , T )

first fundamental form is the induced metric.

178

Lectures on the Geometry

of Manzfolds

The above result is especially interesting when S is a transversally oriented hypersurface, i.e., S is a a codimension 1 submanifold such that the normal bundle Ns is trivial.3 Pick an orthonormal frame n of Ns, i.e., a length 1 section of Ns, and choose an orthonormal moving frame (Xl, ..., XT7,_1) of TS. Then (Xl, ..., X m _ 1 , n) is an orthonormal moving frame of ( T M ) 15, and the second fundamental form is completely described by II

..

Nn(X,Y)

(N(X,Y),n).

N," is a bona-fide symmetric bilinear form, and moreover, according to (4.2. 12) we have

n,,(x,y)=-(vXMn,y)

(4.2.14)

(VyMN,X),

In this case, Gauss formula becomes

(RS(x, y>z, T )

(RM(x, y)z, T )

Let us further specialize, and assume M

Run,

n,,,z, T )

N,,,(X, Z)

n,,(x, z) Nn(Y, 2)

(4.2.l5)

In particular, the sectional curvature along the plane spanned by X, Y is

(R$(X,Y)Y,X) = N,-,(X,X) - No(Y,Y) - INn(X7Y)l 2 Remark 4.2.20. (a) The equality (4.2.l5) is a truly remarkable result. On the right-handside we have an extrinsic term (it depends on the "space surrounding S"), while in the left-hand-side we have a purely intrinsic term (which is defined entirely in terms of the internal geometry of S). Historically, the extrinsic term was discovered first (by Gauss), and very much to Gauss surprise ('?!'?) one does not need to look outside S to compute it. This marked the beginning of a new era in geometry. It changed dramatically the way people looked at manifolds and thus it fully deserves the name of The Golden (egregium) Theorem of Geometry. We can now explain rigorously why we cannot wrap a plane canvas around the sphere. Notice that, when we deform a plane canvas, the only thing that changes is the extrinsic geometry, while the intrinsic geometry is not changed since the lengths of the"fibers" stays the same. Thus, any intrinsic quantity is invariant under "bending". In particular, 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".

Example 4.2.21. Suppose that S is a co-oriented hypersurface of Run. The choice of COorienting unit normal vector field n can be viewed a smooth map n : S -> STT1-1 where S771-1 denotes the unit sphere in Run. Observe that, for any p E S, we have an equality of subspaces in RM

TpS 3 Locally,

I

Trap)

Sm

1

all hypersurfaces are transversally oriented since Ns is locally trivial by definition.

179

Riemannian Geometry

The second fundamental form N of S is symmetric bilinear form on TpS. Using the metric on TpS we can identify it with a symmetric operator

N : T,,s -> Tps -_ To) S771

1 7

(nx,Y )

N(X, Y).

Using the equality (4.2.14) we deduce {NX,Y)

I

No(X,Y) =

-(VIM No X ) = - ( V

Na

Y

This shows that - N is the differential of the map § : S - > S M - 1 7 S3»p»->n(p)-E

Sm

1

The map 9 is called the Gauss map of the hypersurface S C lR71. The differential of the Gauss map is known as the shape operator of the hypersurface. We will have more to say about the Gauss map in Section 9.2.1. Suppose now that S is the sphere of radius 7° in RM centered at the origin, and n is the our unit normal vector field along S. In this case the Gauss map is l

S9;z:»->-:z3€SM 1 T'

Thus l

Do 9

-].TxS. T'

1 We deduce from (4.2.15) that the sectional curvature of S along any 2-plane is F'

Example 4.2.22. (Quadrics in R3). Let A : R3 -> R3 be a self-adjoint, invertible linear operator with at least one positive eigenvalue. This implies the quadric Q A = { u € R 3 ; (Au,u)=1}, is nonempty and smooth (use the implicit function theorem to check this). Let 'LLO Then

TuoQA={$€R3; ( A )

0}

6

QA.

I./4'LL()l-L»

QA is a transversally oriented hypersurface in R3 since the map QA 9 u »-> Au defines a 1 nowhere vanishing section of the normal bundle. Set n = IAuI Au. Consider an orthonormal frame (so, e l , 62 of R3 such that 60 = n(u0). Denote the Cartesian coordinates in R3 with respect to this frame by (x0 ,ac1,a:2), and set S2={u€]R3;

l'u,l=1},

2933+->n(;ulES2

7

depends on how Z) is embedded in R3 so it is an extrinsic object. Denote by Nn the second fundamental form of 2 --> R3, and let (x1, as2 l be normal coordinates at q 6 Z] such that ' orientation TqX] - 31 /\ 32.

Consider the Euler form so on Z] with the metric induced by the Euclidean metric in R3 Then, taking into account our orientation conventions, we have 27T€2(31,

32)

=

R1212 =

det

[N,,(31, 31) Nnlala 32) :Nnla27 31) :Nnla27 32)

(4.2.17)

Now notice that

37, nn

Z

-N,,( Elf is another isomorphism determined by the frame _f of Ea' related to Q via a transition map g : 6 -> GLm(R),

g.

Then

Ti(2% y)

=

9-1(;3)T§(22»

y)9(y)-

We denote by do the differential with respect to the x variable. We deduce

o (fL y

-doT( f ) x , y l x = y

-(

dx9l1(;vl )x=y • T§lYv y) ~9(y)

9-1 (33) ( dmT§(a3, y)) Ix=yg(:u)

9'1(y)d9(y)9'1(y) -9(y) + 9`1(y)F(§)y9(yl 9(y)'1d9(yl + 9'1(y)F(§)y9(y)Thus

F(§ 9) = stldg + 9-1F(§)9-

186

Lectures on the Geometry

of Manzfolds

Proposition 3.3.5 shows that the collection of locally defined 1-forms Hg), Q local frame of E, defines a connection VC on E. This connection is compatible with the metric 7 -)c because the local connections VI are such when Q is an orthonormal frame. Proposition 4.2.29. Any correlate K on M induces a canonical metric ( - , - )c on T M . The associated Levi-Civita connection coincides with the connection V C described in a local frame Q by

vC where the m

y

d+

7

m-matrix F Q ) is given by (4.2.2]).

Proof. It suffices to prove that the connection VC is symmetric, i.e., its torsion is 0. Once we choose local coordinates ( x ) on 6 we have a natural frame of El @ t o ;

With these choices

Cx'y

'a

(mot

is represented by the matrix S§(a:, zu) with entries 8a@(x, y) = c92'*,:u5K(a2, y).

x) the inverse of (305(3?= wll- Then F1(X) is an m x m matrix

We denote by

F

UI

File,£ul =

a Fmm) )1$0,BS"1° -c9:/52

-3:IJ;

l5e(x,x)-15£(,;,y)ll S2(a:, $2)-1 Se(;z3, 1/3)

S§(;z3, 5C>-16x1S£(£U7 311

S§(:1;°,;r3l'1(9Q3 S£(;r3,a;°l - S£(;u,;ul'13x. Sgl5U)Yly=x

ex S£(a2, yl -|-

SglXa

1

SglXv a;')'1

Sglx, :z;'l-1 R

X

is the correlate determined by an embedding M RN . Denote by e1, . . . , eN the canonical basis of the dual Hom(RN , R). Denote by 'Pa the restriction to M of the linear function ea : RN -> R. Now choose N independent random variables Ai, . . . , An with mean 0 and variance 14 and form the random linear combination N

ZAQ/Q.

\11

a

1

The random object \I/ is known as a random function or random field on M . The value of \I/ at a fixed point p is the random variable

WP)

2 Aowo(p

I

a

Its mean or expectation is

Z \P1/:(P)E[A0 1

R, K ( p, q ) = IE[*P(pl\P(q)]. In other words, K ( p , q ) computes the covariance of the random variables \If(pl and \I/(q) X

Note that

K(p,q

IE ( 2 A ~ v ( p ) ) a

ICA

B Q/ B( q I

B

h i m pwe( p) M A Y B 1 04,5 04,5

This implies that ]E[AiAj] variable X .

=

Sag

>}

§:\1'o(q) Ql l M is a smooth path such that the following hold. (i) W ) = Pi, 'i ( i i ) SLY)

= 0, 1.

< SL (if), for any smooth path if : [0, 1] -> rejoining Po to P1-

Then the path by satisfies the Euler-Lagrange equations

d dt

a Lu, , 'YI 37

a

_L

('y(t), I(t)) 6 TM. Note that (SOn = co, and at endpoints (So = 0.

Exercise 5.1.6. Prove that if t »-> X (t) 6 T ' y ( t ) M is a smooth vector field along by, such that X (15) = 0 for t 0, l then there exists at least one variation rel endpoints a such that (So = X. Hint: Use the exponential map of some Riemann metric on M .

204

Lectures on the Geometry

of Manzfolds

Y /

pa

Fig. 5.1

Deforming a path rel endpoints.

We compute (at s = 0) 1 -1 d d aL l aL 0 = -SL((I8) = • .da'Ldt + ..(sadt. L(t, 0457 045) = ds ds 0 3q'l 0 3119 Integrating by parts in the second term in the right-hand-side we deduce u

1

/{

aL. - -d

(5.l.2)

aL

( l}

(SQidt_ (5. 1.3) dt off 0 fig'l The last equality holds for any variation a. From Exercise 5.1.6 we deduce that it holds for any vector field 6afidi along *y. At this point we use the following classical result of analysis.

If f (to is a continuous

function on [ 0 , ] ] such that 1

f (t)glt)dt

= 0 Vg 6 C0 (0,1),

0

then f is identically zero. Using this result in (5.1.3) we deduce the desired conclusion. Remark 5.1.7. (a) In the proof of the least action principle we used a simplifying assumption, namely that the image of by lies in a coordinate neighborhood. This is true locally, and for the above arguments to work it suffices to choose only a special type of variations, localized on small intervals of [0, In terms of infinitesimal variations this means we need to look only at vector fields along by supported in local coordinate neighborhoods. We leave the reader fill in the details. (b) The Euler-Lagrange equations were described using holonomic local coordinates. The minimizers of the action, if any, are objects independent of any choice of local coordinates, so that the Euler-Lagrange equations have to be independent of such choices. We check this directly.

11

205

Elements of the Calculus of Variations

If (of) is another collection of local coordinates on M and ( i j induced on TM, then we have the transition rules azi SO

Ii)

are the coordinates

31-k

n

,q"),

=u'(q1 7

7

gawk q

7

that

a

3337

9

3q

go'Z

3367

a 3qj

a q(t). Thus, when we express the Euler-Lagrange equations for a minimizer 70 of this action, we may as well assume it is parameterized by arclength, i.e., to I = 1. The Euler-Lagrange equations for L1 are 39i' - '

d

g k j q'j

gfzj q 'in

Along the extremal we have 9ziqq.7 equations can be rewritten as

d

-.

3113 q'q.7

2V5lij q

iv

l (arclength parametrization) so that the previous

lg1 R associates to each state several meaningful quantities.

.. .

aL The generalized momenta: Pi or The energy: H = pin - L. aL The generalizedforee: F = Gq'L

l l l l l l l l l - - _

l l l l l l l l l - - _

This terminology can be justified by looking at the lagrangian of a classical particle in a potential force field, F = -VU, 1 L = -m 2

Ill 2

U(q).

'

The momenta associated to this lagrangian are the usual kinetic momenta of the Newtonian mechanics PI

my

7

while H is simply the total energy 1

H

•

hmM+UM%

It will be convenient to think of an extremal for an arbitrary lagrangian L(t, ii, q) as describing the motion of a particle under the influence of the generalized force. Proposition 5.1.10 (Conservation of energy). Let 'y(tl be an extremal of dent lagrangian L = L(q, ql. Then the energy is conserved along 7, i.e., II

WI

o'

d

-HW, dt

time indepen-

Proof. By direct computation we get d dt

d dt l p f i £ d dt

of;

avi (1 of;

{ ( go ) llllllllll

AL 3% q

}

X, (t, s) »-> I`%(8), such that

Fi l 8 ) = ' Y f z ( 3 l ' é = 0 , l , and (5 »-> Ft(.s)) 6 Q(X,£60) Vt 6 [0, 11.

We write this as '70

r\./$0

'Yl-

Note that a loop is more than a closed curve, it is a closed curve + a description of a motion of a point around the closed curve. Example 6.1.4. The two loops m m though they have the same image.

I ->@,v/(=(t)

exp(2k1rt), k

l , 2 are different

Definition 6.1.5. (a) Let '711 'Y2 be two loops based at 130 E X . The product o f 'Y1 and 'Y2 is the loop 'Y1(28l»

'Y1 * 'Y2(3)

The inverse of a based loop

by

')/2(2S

-

II,

0 S sl/2 l

/2 < 3 < l

is the based loop 'Y defined by 'Y

(so = v(1

(c) The identity loop is the constant loop

eH30

(5)

S •

$0.

229

The Fundamental Group and Covering Spaces

Intuitively, the product of two loops 'Yi and 'Y2 is the loop obtained by first following 'Yi (twice faster), and then 'Y2 (twice faster) . The following result is left to the reader as an exercise.

Lemma 6.1.6. Let

and

050 "mo 011, 50-x051

70-x071

be three pairs of homotopic based

loops. Then

* @0"'m0011 * 51. * 00 (c) to * (d) (010 * 50) * 'Y02x00£0 * (Bo * 'Yol-

(0) (b)

O10

Ag

-IB()ex.

e000-12000.

Hence, the product operation descends to an operation "~" on Q(X, x 0 ) / - a c 0 , the set of homotopy classes of based loops. Moreover the induced operation is associative, it has a unit and each element has an inverse. Hence (Q(X, 5 1 3 0 ) / - x 0 , -) is a group.

Definition 6.1.7. The group (Q(X,;u0)/~_,,30, .l is called the fundamental group (or the Poincaré group) of the topological space X, and it is denoted by W1 (X, 5150). The image of a based loop by in tf1 (X, xo) is denoted by iv] The elements of al (X, xo) are the "unshrinkable loops" discussed at the beginning of this chapter. The fundamental group w1(X,x0) "sees" only the connected component of X which contains xo. To get more information about X one should study all the groups {f/T1 (X, 3?)} a:€X -

Proposition 6.1.8. Let X and Y be two topological spaces. Fix two points, £130 6 X and Yo 6 Y . Then any continuous map f : X -> Y such that f (1130) = Yo induces o morphism of groups

f* : tfllX,€col

-> W1(Y»3/ol,

satisfying the following functorialily properties. (al (lx)* (b) If

]]'1r1lX,a:0)~

(X,

5130)

L (Y, 3/ol i> (Z,

are continuous maps, such that f (;130l = 3/0 and

(C) Let f0,f1 : (X, 320) -> (Y,

f0 is homotopic to al rel Fe(ac) such that, Fig(£cl

E

y0) be two

ZUNI

g(y0) =

Z0,

then (9 o f)*

I

Q* o

f*~

base-point-preserving continuous maps. Assume

X -> Y , (t,x) +-> Yo. Then (f0)* = (f1l*-

5130, i.e., there exists a continuous map F : I X

f l y ) for t = 0, l and Fe(to)

Proof. Let 'Y E QlX,$0l_ Then f('yl E

E

so, y0)» and one can check immediately that

v;ftov' => f('y)

I\.!

y0

f('y"l.

Hence the correspondence QIXMUo) 9 'Y

*

f('y) 6 Q(Y»Y0)

230

Lectures on the Geometry

of Manzfolds

descends to a map f : tf1 lx, 5130) -> 7I'1 LY, y0l. This is clearly a group morphism. The statements (a) and (b) are now obvious. We prove (c). Let to, fl : (X, k g ) -> (Y, y0) be two continuous maps, and Ft a homotopy rel 5130 connecting them. For any by 6 Q(X, xo) we have

fo('Y)

The above homotopy is realized by 8t

N-:yo

50

f1('Y)

51.

Ftl'}').

I

A priori, 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. However, if X is connected, and thus path connected since it is locally so, all the fundamental groups '/T1 (X, ac), cz: 6 X are isomorphic. X

0


Fig. 6.2

>

X

1

>

Connecting base points.

Proposition 6.1.9. Let X be a connected topological space. Any continuous path a [0, l ] -> X joining :UO to £111 induces on isomorphism go*

defined by, CY* (M ) :

a

:7r1(X,:z:0l ->

* by * a],° see

771lX,£U1l,

figure 6.2.

Exercise 6.1.10. Prove Proposition 6.1.9. Thus, the fundamental group of a connected space X is independent of the base point modulo some isomorphism. We will write Try (X, pt) to underscore this weak dependence on the base point.

Corollary 6.1.11. Two homotopically equivalent connected spaces have isomorphic fundamental groups.

Example 6.1.12. (a) 7l'1lRTL,ptl (b) 'rr1(annulus)

Tf1(pt,pt) = { l } -

7T1lS1l.

Definition 6.1.13. A connected space X such that '/T1 (X, pt) connected.

{l} is said to be simply

The Fundamental Group and Covering Spaces

Exercise 6.1.14. Prove that the spheres of dimension

231

_ 2 are simply connected.

Exercise 6.1.15. Let G be a connected Lie group. Define a new operation "*" on Q(G, l) by (Q * BMS) 0(s) . ms), where denotes the group multiplication. (a) Prove that a * B ""1 a * 5 (b) Prove that '/T1 (G, l ) is Abelian.

-

Exercise 6.1.16. Let E -> X be a rank T' complex vector bundle over the smooth manifold X, and let V be a flat connection on E, i.e., F(V) = 0. Pick ac() 6 X, and identify the fiber Exp with CC". For any continuous, piecewise smooth by 6 Q(X, xo) denote by T'Y = T l v l the parallel transport along by, so that T'Y 6 GL (U). (a) Prove that 0 - 3 3 o B => Ta = TB.

= To o Ta. Thus, any flat connection induces a group morphism

T,B*a

(b)

T : 7r1(X,:z:0) -> GL((C")

*y

»->T'7- 1 .

This morphism (representation) is called the monodromy of the connection. Example 6.1.17. We want to compute the fundamental group of the complex projective space CIP". More precisely, we want to show it is simply connected. We will establish this by induction. For n = l, (CIP1 = S2, and by Exercise 6.1.14, the sphere S2 is simply connected. We next assume (c1p'" is simply connected for k < n and prove the same is true for n. Notice first that the natural embedding (jk-l-1 C"-*1 induces an embedding (c1p'" CIP". More precisely, in terms of homogeneous coordinates this embedding is given by

~

[ z 0 , . . . : Zn]»-> [z0,...,z;,,0,...,0] € UP". Choose as base point pt = [1,0, . . . ,0] 6 CIP", and let by 6 Q((ClP",pt). We may assume by avoids the point P = [0, . . . , 0, 1] since we can homotop it out of any neighborhood of P. We now use a classical construction of projective geometry. We project by from P to the hyperplane ii" = Gp"-1 C--> CIP". More precisely, if C = [z0, . . . , zn] 6 UP", we denote by ¢r(§) the intersection of the line PQ with the hyperplane J . In homogeneous coordinates 7 2 N - 1 7 0] when Zn ¢ 1). : : iz0I1 "ala • • • Z n - 1 ( 1 - ZnlvOl (= 2107 Clearly or is continuous. For t 6 [0, l] define [z0(l - HzTL), . . . ,zn_1(l - toTL), (1 - t)zn]. 7Tt lcl Geometrically, '/To flows the point Q along the line PC until it reaches the hyperplane 32 Note that 'ii,¢(§) = C, Vt, and VC 6 %. Clearly, '/To is a homotopy rel pt connecting *y = two (by) to a loop 'Y1 in Qi" 2 (CIPn-1 based at pt. Our induction hypothesis shows that 'Y1 can be shrunk to pt inside Qi" . his proves that CIP" is simply connected. 7

232

6.1.2

Lectures on the Geometry

of Manzfolds

Of categories and functors

The considerations in the previous subsection can be very elegantly presented using the language of categories and functors. This brief subsection is a minimal introduction to this language. The interested reader can learn more about it from the monograph [74, 89]. A category is a triplet € = ( Ob (€), H o m ( € ) , o) satisfying the following conditions. (i) Ob (6) is a set whose elements are called the objects of the category. (ii) Hom((3) is a family of sets Hom (X, Y), one for each pair of objects X and Y. The elements of Hom (X, Y) are called the morphisms (or arrows) from X to Y. (iii) o is a collection of maps O

Hom(X, Yl

X

Hom(Y, Z) -> Hom(X, Z), ( f , g) +-> g O f,

which satisfies the following conditions.

(Cl) For any object X, there exists a unique element Hx wHom(X,X) such that, fo]1x (C2)

'v'f

=f

go]lx = g

Vf

6 Hom(X,Y),

Vg 6 Hom(Z,X).

G Hom(X,Y), g E Hom(Y, Z), h 6 Hom(Z, W) h(9of)=(ho9)of-

Example 6.1.18. • Top is the category of topological spaces. The objects are topological spaces and the morphisms are the continuous maps. Here we have to be careful about one foundational issue. Namely, the collection of all topological spaces is not a set. To avoid this problem we need to restrict to topological spaces whose subjacent sets belong to a certain Universe. For more about this foundational issue we refer to [74]. (Top, *) is the category of marked topological spaces. The objects are pairs (X, *), where X is a topological space, and * is a distinguished point of X . The morphisms

.

(X, *I L (Y, o) are the continuous maps f : X -> Y such that f(*) = . ]pVect is the category of vector spaces over the field IF. The morphisms are the IF-linear maps. Gr is the category of groups, while Ab denotes the category of Abelian groups. The morphisms are the obvious ones. RMOd denotes the category of left R-modules, where R is some ring.

. . .

Definition 6.1.19. Let 81 and (32 be two categories. A covariant (respectively contravariant) functor is a map St: Ob (81)

X

Ho1n(€1) -> Ob 1621

X

Ho1n(€2),

( X , f ) »-> (5'(X)»5'(f))» such that, if X

i> Y, then 5"(X) 5"(fQ 5"(Y) (respectively 5'(X)

(i) 5"(]1x) = ]1;~" (V - > U ) .

®V

R Veet

*V*-)

7

The fundamental group construction of the previous is a covariant functor '/T1

(Top,

*I -> Gr.

In Chapter 7 we will introduce other functors very important in geometry. For more information about categories and functors we refer to [74, 89].

6.2 Covering Spaces 6.2.1

Definitions and examples

As in the previous section we will assume that all topological spaces are locally path connected.

Definition 6.2.1. (a) A continuous map '7l' : X -> Y is said to be a covering map if, for any y 6 Y, there exists an open neighborhood U of y in Y, such that 7r-1 (U) is a disjoint union of open sets $2 C X each of which is mapped homeomorphically onto U by 7r. Such a neighborhood U is said to be an evenly covered neighborhood. The sets ld are called the sheets of 'fr over U. (b) Let Y be a topological space. A covering space of Y is a topological space X, together with a covering map '7T : X -> Y. (c) If 7T : X -> Y is a covering map, then for any y 6 Y the set 7r"1 (kg) is called the fiber over y. Example 6.2.2. Let D be a discrete set. Then, for any topological space X, the product X X D is a covering of X with covering projection or(;u, d) = at. This type of covering space is said to be trivial. Exercise 6.2.3. Show that a libration with standard fiber a discrete space is a covering. Example 6.2.4. The exponential map exp : R -> S1, t »-> exp(2trit) is a covering map. However, its restriction to (0, OO) is no longer a a covering map. (Prove this!)

Exercise 6.2.5. Let (M, g) and (M, go be two Riemann manifolds of the same dimension such that (M, go is complete. Let Q25 : M -> M be a surjective local isometric, i.e., is smooth and

l'UI9 Prove that ¢ is a covering map.

I

ID(25('v)l§

Vu E TM.

234

Lectures on the Geometry

of Manzfolds

The above exercise has a particularly nice consequence. Theorem 6.2.6 (Cartan-Hadamard). Let (M, g) be a complete Riemann manifold with non-positive sectional curvature. Then for every point q E M , the exponential map

expq . TqM -> M is a covering map.

Proof. The pull-back h = €Xpq (9) is a symmetric non-negative definite (0, 2)-tensor field on TqM. It is in fact positive definite since the map expq has no critical points due to the non-positivity of the sectional curvature. The lines t »-> to through the origin of TqM are geodesics of h and they are defined for all t 6 R. By Hopf-Rinow theorem we deduce that (TqM, h) is complete. The theorem now follows from Exercise 6.2.5. Exercise 6.2.7. Let c? and G be two Lie groups of the same dimension and Q25 : 68 -> G a smooth, surjective group morphism. Prove that go) is a covering map. In particular, this explains why exp : R -> S1 is a covering map. Exercise 6.2.8. Identify S3 C R4 with the group of unit quaternions

S 3 = { q 6 H ; lql=1}~ The linear space R3 can be identified with the space of purely imaginary quaternions

R3 = Im]H[ = {a:'£ + y j + z k } . (a) Prove that q:z:q-1 6 lm H, */q 6 S3. (b) Prove that for any q 6 S3 the linear map Tq . lm H -> Imll-Il az »-> q;uql1 is an isometric so that Tq 6 SO(3). Moreover, the map S3 9 q»->Tq 6 SO(3)

is a group morphism. (c) Prove the above group morphism is a covering map. Example 6.2.9. Let M be a smooth manifold. A Riemann metric on M induces a metric on the determinant line bundle det TM. The sphere bundle of det TM (with respect to this metric) is a 2 : l covering space of M called the orientation cover o f M . Definition 6.2.10. Let X1 3 Y and X2 3 Y be two covering spaces of Y . A morphism of covering spaces is a continuous map F : XI -> X2 such that 7r2 o F = '/VI, i.e., the diagram below is commutative.

x1

F

)

XI

X YA If F is also a homeomorphism we say F is an isomorphism of covering spaces.

The Fundamental Group and Covering Spaces

235

7r

Finally, if X -> Y is a covering space then its automorphisms are called deck transformations. The deck transformations form a group denoted by Deck (X, w).

~=z.

Exercise 6.2.11. Show that Deck l]Rg_'§l'S1l

Exercise 6.2.12. (a) Prove that the natural projection S" -> RIP" is a covering map. (b) Denote by the tautological (real) line bundle over RP". Using a metric on this line bundle form the associated sphere bundle S Q L ) -> RIP". Prove that this fiber bundle defines a covering space isomorphic with the one described in part (a).

u

6.2.2 Unique lzfting properly 'lr

Definition 6.2.13. Let X -> Y be a covering space and F : Z - > Y a continuous map. A ly? of f is a continuous map F : Z -> X such that 7T o F =f,i.e., the diagram below is commutative.

X /VF

F'//

.

'ii

/

/

Z

f

>Y

Proposition 6.2.14 (Unique Path Lifting). Let X 1> Y be a covering map, by : [0, l] -> Y a path in Y and :UQ a point in thejiber over Yo = 7/(0), £130 G tr"1 (yo). Then there exists at most one lift offs, F : [0, 1] -> Y such that r(0) = 500. Proof. We argue by contradiction. Assume there exist two such lifts, Fl, F2 : [0, 1] -> X . Set

II

S

t € [0, 1]; 1"1(t)

F2(t) }.

The set S is nonempty since 0 6 S. Obviously S is closed so it suffices to prove that it is also open. We will prove that for every S C S, there exists 5 > 0 such that 18 - 6, S -|- 51 [0, 11 C S. For simplicity, we consider the case S = 0. The general situation is entirely similar. We will prove that there exists To > 0 such that [0, kg] C S. Pick a small open neighborhood U of kg such that 7T restricts to a homeomorphism onto rr(U). There exists To > 0 such that vi( [0, r01 C U,1l = l , 2. Since 'rroII1 = VTOF2, we deduce F1 1107T'01- F2 l[0:T01° The proposition is proved.

l

Theorem 6.2.15. Let X l> Y be a covering space, and f : Z -> Y be a continuous map, where Z is a connected space. Fix Z0 6 Z , and 5130 6 7r"1(y0), where Yo = f(z0). Then there exists at most one lzft F : Z -> X of f such that F (20) = kg.

236

Lectures on the Geometry of Manu'olds

Proof. For each Z 6 Z let aZ be a continuous path connecting Zo to Z. If Fl, F2 are two lifts of f such that F1 (to) = F2(z0) = xo then, for any Z 6 Z, the paths F1 = F(aZ), and F2 = F2 (az) are two lifts of by = f (aZ) starting at the same point. From Proposition 6.2.14 we deduce that F1 E FQ, i.e., F1 (z) = F2(z), for any z 6 Z.

6.2.3 Homotopy lu'ting properly Theorem 6.2.16 ( Homotop y lifting propert y). Let X l> Y be a covering space, f : Z -> Y be a continuous map, and F : Z -> X be a ly? o f f . If

h : [0,l] is a homotopy of f ( ho (z)

X

Z -> Y

(t,z) »-> ht(z)

f (z) ), then there exists a unique [Ut of h

H : [0, l] x Z -> X

(t , z) *-> He(zl»

such that H0(z) Proof. For each Z 6 Z we can find an open neighborhood UZ of Z 6 Z, and a partition 0 = to < to < . . . < tn = l, depending on z, such that h a p s [ti_1, to] XUZ into an evenly covered neighborhood of h i i _ 1 (z). Following this partition, we can now successively lift h IIXUz to a continuous map H = Hz : I X Uz -> X such that H0(§) = F(§), VC 6 Uz. By unique lifting property, the liftings on I X Uz, and I X Uz, agree on I X (UZ, n UZ2), for any 217 212 6 Z, and hence we can glue all these local lifts together to obtain the desired lift H on I X Z . Corollary 6.2.17 (Path lifting property). Suppose that X l> Y is a covering map, y0 6 Y , and '7 : [0, l] -> Y is a continuous path starting at to. Then, for every £130 G 7r"1(y0l there exists a unique lzft F : [0, l] -> X offs starting at 1/30. Proof. Use the previous theorem with f : {pt} -> Y , f (pi = 1/(Ol, and htlPtl

'Yltl~

Corollary 6.2.18. Let X l> Y be a covering space, and Yo E Y. Iffy0, 71 € Q(Y, 3/ol are homotopic rel 3/0, then any lifts Fo, F1 which start at the same point also end at the same point, i.e., poll) = I`1(1l. Proof. Lift the homotopy connecting 'Yo to 'Yi to a homotopy in X . By unique lifting property this lift connects F0 to Fl. We thus get a continuous path p¢(1) inside the fiber ,/T-1 (yo) which connects I`0(1) to F1(1l~ Since the fibers are discrete, this path must be constant. 'IT

*y

Let X -> Y be a covering space, and Yo 6 Y. Then, for every ac 6 QW, yo), denote by Fx the unique lift of by starting at x. Set ac-*y

:=

Fas

(1).

By Corollary 6.2.18, if U.) 6 Q(Y, yo) is homotopic to by rel Yo, then ac-fy

ac-oJ.

6 7T-1(?/0)»

and any

237

The Fundamental Group and Covering Spaces

Hence, zz:-7 depends only upon the equivalence class [by] E at

. (['y] . M)

Try (Y,

y0l. Clearly

(co . ['y]l . [~J],

i

and £8

•

bY()

My

so that the correspondence 7r

-1

(Yu)

X 7T1(Y»3/ol 9

( J o ) +-> 2 3 ° ' y

Q 3 3 - 7 Q 7r

-1

(to)

defines a right action of 7r1(Y, y0) on the fiber 7r"1(y0). This action is called the monodromy of the covering. The map as »-> Hz: by is called the monodromy along by. Note that when Y is simply connected, the monodromy is trivial. The map 'IT induces a group morphism 7r*

W1(X7$0)

-)tFlly,y0l

5130 € t l ' - 1 l Y 0 l .

Proposition 6.2.19. or* is injective. Proof. Indeed, let by 6 Q(X, 1110) such that WW) is trivial in 7r1(Y, to). The homotopy connecting 7r('y) to ey0 lifts to a homotopy connecting by to the unique lift of 2Y0 at $150, which is efro . 6.2.4

On the existence

of l#ts

Theorem 6.2.20. LetX l> Y be a covering space, :cg E X, Yo = tr(a:0l 6 Y , f : Z -> Y a continuous map and 2 0 6 Z such that f (z0) = Yo~ Assume the spaces Y and Z are connected (and thus path connected). f admits a lzft F : Z -> X such that F(z0) = £130 If and only If

f* (7t1(27 Zgll C

7T* (7T1

lx,cc0ll

(6.2. 1)

Proof. Necessity. If F is such a lift then, using the functoriality of the fundamental group construction, we deduce f* = Tr* o F*. This implies the inclusion (6.2.l). Sufficiency. For any Z 6 Z, choose a path 'Ye from Zo to Z. Then oz f(vzl is a path from Yo to y = f(zl. Denote by A z the unique lift of aZ starting at 1130, and set F(z) = AZ(1). We claim that F is a well defined map. Indeed, let cuZ be another path in Z connecting Z0 to Z. Set AZ := f(wz), and denote by AZ its unique lift in X starting at 330. We have to show that Az(1) = AZ (1). Construct the loop based at Z0

Be

We

* 'Ye

Then f(5z) is a loop in Y based at Yo. From (6.2. 1) we deduce that the lift B Z of f(5z) at E X is a closed path, i.e., the monodromy along f(@z) is trivial. We now have

330

AZ(1)

I

BZ(1/2)

I

AGO)

I

A2(1).

238

Lectures on the Geometry

of Manzfolds

This proves that F is a well defined map. We still have to show that this map is also continuous. Pick Z 6 Z. Since f is continuous, for every arbitrarily small, evenly covered neighborhood U of f (z) 6 Y there exists a path connected neighborhood V of Z E Z such that f ( V ) C U. For any C E V pick a path 0 = UQ in V connecting z to Q. Let w denote the path w = 'Ye * UQ (go from Z0 to Z along Vi, and then from Z to C along up). Then F(§) = Q(1), where Q is the unique lift of f(w) starting at 5130. Since (f(C) E U, we deduce that 9(1) belongs to the local sheet E, containing F(z), which homeomorphically covers U. We have thus proved Z E V C F-1(E). The proof is complete since the local sheets E form a basis of neighborhoods of F(z). 'ii

Definition 6.2.21. Let Y be a connected space. A covering space X -> Y is said to be universal if X is simply connected. Y (i=0,]) be two covering spaces of Y . Fix Xi E Xi such Corollary 6.2.22. Let XI that p0(:v0) = p(:u1) = Yo E Y . If X0 is universal, then there exists a unique covering morphism F : X0 -> XI such that F(:z:0) = al. Proof. A bundle morphism F : X0 -> XI can be viewed as a lift of the map Po : X0 Y to the total space of the covering space defined by lU1» The corollary follows immediately from Theorem 6.2.20 and the unique lifting property.

Corollary 6.2.23. Every space admits at

most one universal covering space (up to isomor-

phis).

Theorem 6.2.24. Let Y be a connected, locally path connected space such that each of its points admits a simply connected neighborhood. Then Y admits an (essentially unique) universal covering space. Sketch of proof. Assume for simplicity that Y is a metric space. Fix Yo 6 Y. Let 90 denote the collection of continuous paths in Y starting at 3/0» It can be topologized using the metric of uniform convergence in Y. Two paths in TYoare said to be homotopic rel endpoints if they we can deform one to the other while keeping the endpoints fixed. This defines an equivalence relation on Ty0. We denote the space of equivalence classes by Y, and we endow it with the quotient topology. Define p : Y -> Y by

p(['y]) = 7(1)

W

Q

Ty/0°

Then l},pl is a universal covering space of Y.

Exercise 6.2.25. Finish the proof of the above theorem. Example 6.2.26. The map 0 2 3 5 1 is the universal cover of S1. More generally, exp : R"->T"

(to, . . . Jn) +-> (exp(27rit11, . . . ,exp(2¢ritn1)), ii

\

-1,

The Fundamental Group and Covering Spaces

is the universal cover of T". The natural projection p of RIP".

STL

239

-> RIP" is the universal cover

Example 6.2.27. Let (M, g) be a complete Riemann manifold with non-positive sectional curvature. By Cartan-Hadamard theorem, the exponential map expq : TqM -> M is a covering map. Thus, the universal cover of such a manifold is a linear space of the same dimension. In particular, the universal covering space is contractible! ! ! We now have another explanation why exp : R" -> T" is a universal covering space of the torus: the sectional curvature of the (flat) torus is zero. Exercise 6.2.28. Let ( M , g l be a complete Riemann manifold and p : M -> M its universal covering space. (a) Prove that M has a natural structure of smooth manifold such that p is a local diffeomorphism. (b) Prove that the pullback p*g defines a complete Riemann metric on M locally isometric with g.

Example 6.2.29. Let ( M g ) be a complete Riemann manifold such that Ric: ( X , X ) > const.IXI2go

(6.2.2)

where const denotes a strictly positive constant. By Myers theorem M is compact. Using the previous exercise we deduce that the universal cover M is a complete Riemann manifold locally isometric with (M, g). Hence the inequality (6.2.2) continues to hold on the covering M. Myers theorem implies again that the universal cover M is compact!! In particular, the universal cover of a semisimple, compact je group is compact!!! 6.2.5

The universal cover and the fundamental group

Theorem 6.2.30. Let

X' 3> X

be the universal cover of 7TH

Proof. Fix £0 6

lX, oil

2

space X . Then

Deck (X -> x).

X and set £130 = Plfol- There exists a bijection Ev :Deck (X) -> p - o > ,

given by the evaluation Ev (Fl : F(€0). For any E Tr' 1 (xo), let 'kg be a path connecting go to g. Any two such paths are homotopic rel endpoints since X is simply connected (check this). Their projections on the base X determine identical elements in 711 ( X , 330). We thus have a natural map

-> tf1lX»1°ol F *->p(vF(€0))The map \11 is clearly a group morphism. (Think monodromy!) The injectivity and the surjectivity of \If are consequences of the lifting prope ies of the universal cover. \I/

Deck(X)

Corollary 6.2.31. If the space X has a compact universal cover, then w1 ( X , pt) isjinite.

240

Lectures on the Geometry

of Manzfolds

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 is a bijection onto Deck (X). Corollary 6.2.32 (H. Weyl). The fundamental group finite.

of a

compact semisimple group is

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 tel IS1)

Exercise 6.2.34. (a) Prove that 711 (]R]P'" ,pt) (b) Prove that W1 (Tn)

r\/

~=

Z2,'v'n

Zn

r\/

(Z,

>-

> 2.

Exercise 6.2.35. Show that the natural inclusion U(n - 1) U(n) induces an isomorphis between the fundamental groups. Conclude that the map det : U(n) -> S1 induces an isomorphism

~ 7T1(S1, 1) ~= Z.

7r1(U(n), 1) =

Chapter 7

Cohomology

7.1 DeRham cosmology 7.1.1

Speculations around the Poincaré 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

Q

311, llllllllllll-

3:6

p,

t(x3 E C. We begin with an a priori study of (7.1.4). Let u satisfy du a. Using the homotopy formula, d'£,=' + i»,='d,

L; we get

d(L,=anu - i,=*&) = 0.

dL;/u = d(di;~' -|- i,=°dlu = di;-'a

This suggests looking for solutions up of the equation

Lw

,

=

(p E

sz'°-1(c).

(7.1.5)

If go is a solution of this equation, then L,=*dgo = dL,=*§O = Di,=*&

I

L,=*(I - iR*d&

I

L,=*&.

Hence the form SO also satisfies L,,='(dgo - Q) = 0. Set w := dgo -01 : E l l d m l

Using the computations in Subsection 3.1.3 we deduce

L i m = do SO

7

that

lL,=WI)d$l3I = 0.

L I

243

Cohomology

We deduce that L,=-cuI = 0, and consequently, the coefficients W I are constants along the flow lines of ®t which all converge at 0. Thus, Ld]

const.

C]

Each monomial cfdacI is exact, i.e., there exist T1I E QL6-1(Cl such that dn; example, whenI = 1 < 2 < < Is day1 Add2 /\

Thus, the equality

LfrW

--

/\ dank = d(xldx2 /\

- Adel. •

= 0 implies w is exact. Hence there exists 17 E

d((0 - H)

efdaxl. For

Qk-1lcl such that

&7

i.e., the differential form 'LL := SO 17 solves (7.1.4). Conclusion: any solution of (7.l.5) produces a solution of (7.1.4). We now proceed to solve (7.1.5), and to this aim, we use the flow ®%. Define 0

l@t)* l'iv=*Oz)dt.

u

(7.l.6)

Here the convexity assumption on C' enters essentially since it implies that

who,

t*(C)cC

so that if the above integral is convergent, then u is a form on C. If we write lt)* l'iv='(}£l

WW)dwI 7

i

III=k-1

then 0

u(a:)

ECU I

u1(w)dt) do t

(7.1.7)

-OO

We have to check two things. A.

The integral in (7.1.7) is well defined. To see this, we first write ajdai J 7

a= III=IO

(Qt * ii,=°Ozldt

_OO

/

S

0

(')*(=~a)dt

(®0)*(if0)

Z

'i,=*&.

The Poincaré 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 do on R2 \ {0} where or, 9) denote the polar coordinates in the punctured plane. To write it in Cartesian coordinates (so, y) we use the equality tan 9

y

cz:

so that (1 + tank

me =

y -do: + $2

d y and :u

lllllllllll

l

+

dO

-ydfzz + ardy 1:2

7

i.e.,

do

-ydx i

£u2

+ :Ody

+ y2

a.

Obviously, do = d29 = 0 on R2 \ {0} so that a is closed on the punctured plane. Can we find a smooth function u on R2 \ {0} such that du = a? We know that we can always do this locally. However, we cannot achieve this globally. Indeed, if this was possible, then du

do = 2tr.

a

Sl

Sl

Sl

On the other hand, using polar coordinates u = u(7°' H) we get

au

du S1

$1

-do c" 1

(C'°»d)

"">c7n-I-1

->

7

such that range ldn_1l C her ldnl, i.e., dndn-1 = 0.

Definition 7.1.8. Let ->Cyn

1

dn-

HJ"

dn

>Cv'rL-I-1->-~-

be a cochain complex of R-modules. Set

Z"(C') := her dn B"(C) :: range (d,,_1). The elements of Z" (C) are called cocycles, and the elements of B" (C) are called coboundaries. Two cocycles c,c' E Zn*(C') are said to be eohomologous if C - c' E Bn*(C). The quotient module

H"(Cl

Z"(C)/B"(Cl

is called the n-th cosmology 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 i f H"(C') = 0 for all n > 0.

For a cochain complex (C', d) one usually writes

H'((1) = H . ( a d ) =

@ H"((>). n>0

Example 7.1.9 (The DeRham complex). Let M be an m-dimensional smooth manifold. Then the sequence

>i>Q1

-> i>szmlm) -> 0 0 ->&0(M (where d is the exterior derivative) is a cochain complex of real vector spaces called the DeRham complex. Its cosmology groups are the DeRham cosmology groups of the manifold. d

Example 7.1.10. Let (g, [., -]) be a real Lie algebra. Define d : mg* -> Ak-I-19*, by (d(»Jl(X07 X I ,

7

Xi)

¢+9(»v({X@» xi], XG)

,X

7

in

7

7

Xi),

0 Bn-l-1

aw

(b) Two cochain morphisms : A' -> B' are said to be cochin homotopic, and we write this Bn-1} such that

(Ma)

MG)

i

in

o

x(al

:|:

X

O

d(al.

250

Lectures on the Geometry

of Manzfolds

(c) Two cochain complexes (A', d), and (B°, 6) are said to be homotopic, if there exist cochain morphism QUO

such that

w

O

B and Tb: B ->A,

lA, and (A°,dA>

> (82,B) w > (CMC') -> 0

be a short exact sequence of cochin complexes of R-modules. This means that we have a commutative diagram

251

Cohomology

DA

0

>

An+1

¢n+ 1 ) Bn-i-1

DA

0

¢n+

1)

¢n

ATL

Be

Cm

Ion

71bn-1)

DA

Bn-l

0

0

(7.l.l0)

do

dB

> An-1

>

C171-l-1

do

dB

DA

0

do

dB

Ion

1 )

Cn

1

>

0

do

dB

in which the rows are exact. Then there exists a long exact sequence

- - - - > H"-1 ->

(7.l.11)

We will not include a proof of this proposition. We believe this is one proof in homological algebra that the reader should try to produce on his/her own. We will just indicate the construction of the connecting maps 3%. This construction, and in fact the entire proof, relies on a simple technique called diagram chasing. Start with at 6 H"(C). The cosmology class at can be represented by some cocycle c 6 Z"(C'). Since 1% is surjective there exists b E B" such that C b). From the commutativity of the diagram (7.l.l0) we deduce 0 = dC¢,,(b) w-1-1d8 (be, i.e., dB (be E her W,,+1 = range Cl5,;+1. In other words, there exists CL E An+1 such that Hn+1(A) and moreover, the sequence (7.1.11) is exact. Exercise 7.1.19.1 Suppose R is a commutative ring with l. For any cochain complex (K', do) of R-modules, and any integer n we denote by K[n]' the complex defined by K[n]"" = K"+M, dK[n] = (-1)"dK. We associate to any cochain map f : K' -> L. two new cochain complexes: (a) The cone ( c ( f ) ° , d c ( f ) ) where ki-i-1

c(fl'

Z

K[l]° ® L.

7

-do 0 f dL

]

Zi

dc(f)

H 1 Ki-I-l i

(b) The cylinder ( c l y u f ) , d c y u f )

Ki Ko

iv

ki-I-1

1

do 0 0

-:l]_K»i+1

Ki

H1

0 0 dL

19+1

-do go f L. -> C y l ( f ) , : K' -> C'yl(f)', a canonical projecWe have canonical inclusions sons B : C'yl(f)° -> L', 6 = (5(f):0(f)' -> K[l]°,and7r Cyi(f) -> 0W(f). C'yl(f)°

® 0(f)° 7

dc'yl(f)

i

t

(i) Prove that a,

Bf-, 6(f) are cochain maps, B o a = 11 L and a o B is cochain homotopic to

11cyI(f)-

(ii) Show that we have the following commutative diagram of cochain complexes, where the rows are exact. 7T 5(f) > > L. > C(f)' > 0 0 K[1]' a

>

0

K.

f

>

Cal(fl

'ii

C(f)

>

0

B

11 K

K.

]lC(f)

f

L.

(iii) Show that the connecting morphism in the long exact sequence corresponding to the short exact sequence

0-> KO

f

> C a l ( f l ">c*(f)->0

coincides with the morphism induced in cosmology by 6(f) : C ( f ) -> K111.. (iv) Prove that acyclic.

f

induces an isomorphism in cosmology if and only if the cone of

f

is

lThis exercise describes additional features of the long exact sequence in cohomology. They are particularly useful in the study of derived categories.

253

Cohomology

Exercise 7.1.20 (Abstract Morse inequalities). Let C' = QQH20 C" be a cochain complex of vector spaces over the field IF. Assume each of the vector spaces C" is finite dimensional. Form the Poincaré series 7

and TN dimly Hnlgl.

PH.(c) (t) n>0

Prove that there exists a formal series R(t) E ZW] with non-negative coefficients such that

pc- (t) = PH-(c)l'5) + (1 + t)R(t). In particular, whenever it makes sense, the graded spaces C* and H* have identical Euler characteristics

x(C")

Pa(-ll =

PH- (ml - l ) = x ( H ' ( 0 l ) -

Exercise 7.1.21 (Additivity of Euler characteristic). Let 0->A'->B'->C'->0 be a short exact sequence of cochain complexes of vector spaces over the field IF. Prove that if at least two of the cosmology modules H'(A), H°(B) and H'(C') have finite dimension over IF, then the same is true about the third one, and moreover

x(H°(B)) = x( H' (All

x(H°((1)).

Exercise 7.1.22 (Finite dimensional Hodge theory). Let /

iv:

69 v", d..

\

II

..

, / be a cochin complex of real vector spaces such that dim V" < oo, for all n. Assume that each V" is equipped with a Euclidean metric, and denote by dTL : Vn-I-1 -> V" the adjoint of dn. We can now form the Laplacians

d)

7120

An . V" -> V", An (a) Prove that &B,,,20An = (d -|- d*)2. (b) Prove that A'no = 0 if and only if dna = 0 and

0. In particular, her A'n C

Z"(V'l~ (c) Let that

C

6 Zn*(V). Prove that there exists a unique E 6 Z"(V) cohomologous to

C

such

III = min{ la/l; C - c' 6 Bn (C)}, where | | denotes the Euclidean norm in V" . (d) Prove that E determined in part (c) satisfies An? = 0. Deduce from all the above that the natural map her ATL -> H"lv°l is a linear isomorphism.

-

254

Lectures on the Geometry

of Manzfolds

Exercise 7.1.23. Let V be a finite dimensional real vector space, and UoGV. Define » do, = d g ) : Arv -> Ak:+1V o r > u 0 A w . (a) Prove that

do Aka do Ak-I-1V

is a cochain complex. (It is known as the Koszul complex.) (b) Use the Cartan identity in Exercise 2.2.55 and the finite dimensional Hodge theory described in previous exercise to prove that the Koszul complex is acyclic if U0 go 0, i.e.,

H'"(A'v, d(00l) = 0, We 7.1.4

Funetorialproperties

of

_ 0.

the DeRham eohomology

Let M and N be two smooth manifolds. For any smooth map QS : M -> N the pullback (l>* Q' Q'

(N) ->

(M)

is a cochain morphism, i.e., d)*dn = drab*, where do and respectively dn denote the exterior derivative on M, and respectively N . Thus, (l)* induces a morphism in cohomology which we continue to denote by d)*

H.

(N) -> H'(M). In fact, we have a more precise statement.

co*

Proposition 7.1.24. The DeRhom cohomology construction i5 a controvoriontfunctorfrom the category

of smooth

momfolds and smooth maps to the category

of Z-graded vector

spaces with degree zero morphisms.

Note that the pullback is an algebra morphism QS* Q. ( N ) -> Q' I M ) , ¢*(@A@)

(¢*00 A (¢*e), 'v'a,B 6 Q'

Z

IN,

and the exterior differentiation is a quasi-derivation, so that the map it induces in cohomology will also be a ring morphism.

Definition 7.1.25. (a) Two smooth maps Q50, Q51 : M -> N are said to be (smoothly) homotopic, and we write this Q50 8h Q51, if there exists a smooth map N

@:I X

M -> N

(t,m) »-> ®t(ml,

such that is = QUO-, form = 0, 1. (b) A smooth map (15 : M -> N is said to be a (smooth) homotopy equivalence if there exists a smooth map uP : N -> M such that N . FiSh

w

FiSh

Proposition 7.1.26. Let H' (Ml-

255

Cohomology

Proof. According to the general results in homological algebra, it suffices to show that the pullbacks . At this point, our discussion on the fibered calculus of Subsection 3.4.5 will pay off. The projection : I X M -> M defines an oriented 8-bundle with standard fiber I . For any w 6 Q ' ( N ) we have the equality I

Cbllw) - (l50(Wl = 'I>*((»Jl llxM -'1>*((»J) l0xM=

'1>*((»Jl-

(8IxM)/M

We now use the homotopy formula in Theorem 3.4.54 of Subsection 3.4.5, and we deduce @*

/ /

lull

(alxM)/M

dIxm@*lWl -

aM

(IXM)/M

@*oWl

(IXM)/M @*ldNL4J)

- dM

(IXM)/M

@*oWl.

(IXM)/M

Thus

x(wl

(IXM)/M

*(,,)

is the sought for cochain homotopy.

Corollary 7.1.27. Two homotopy equivalent spaces have isomorphic cosmology rings. Consider a smooth manifold M, and U, V two open subsets of M such that M = UUV. Denote by nU (respectively W) the inclusions U M (resp. V M). These induce the restriction maps nU

. Q .(M )_>

QUO

(U)

'

•

w»_> w

IU'

and

Q'(M) ->

QUO

iv), w e - > w l v

We get a cochin morphism QUO

(M) -> Q'lU) ®

QUO

(V), w +-> (zuw,zvw).

There exists another cochin morphism

6 : we) ® m y ) -> Q' (Um Vl» (we) +-> -¢*JIUnv

+'7IUrwv

Lemma 7.1.28. The short Mayer-Vietoris sequence

0 -> Q°(m) '">Q'(U) ® Q'(v> is exact.

6 >Q*(UVWV)->

0

256

Lectures on the Geometry

of Manzfolds

Proof. Obviously 7° is injective. The proof of the equality Range 7° = her 6 can be safely left to the reader. The surjectivity of 6 requires a little more effort. The collection {U, V} is an open cover of M, so we can find a partition of unity {(,oU, (,CV} C COO(M) subordinated to this cover, i.e., supp(/)u C U, supp(/)v C ii 0 S Note that for any w 6 Q*(U n V) we have

( p u , (PV

S l,

(PU

+l0v

= 1.

supp (Pvw C supp (PV C v, and thus, upon extending (PVLU by 0 outside V, we can view it as a form on U. Similarly, (PULU 6 Q*(V). Note that

6(-(0vw, (www) = ( i v + (,f)u)°J This establishes the surjectivity of 6.

w.

Using the abstract results in homological algebra we deduce from the above lemma the following fundamental result.

Theorem 7.1.29 (Mayer-Vietoris). Let M = U U V be an open cover of the smooth manifold M. Then there exists a long exact sequence

--- -> H' H"'(U) ® H W ) -> Hk=(U n vI -> Hk+1(m) -> 6

*r'

3

,

called the long Mayer-Vietoris sequence. The connecting morphisms 3 can be explicitly described using the prescriptions following Proposition 7.1.18 in the previous subsection. Let us recall that construction. Start with CU 6 Q"(U D V) such that do = 0. Writing as before cu = (Pvw + (PUT,

we deduce

d(govw) = d(-(pgw) on U D V. Thus, we can find 17 6 Qk+1(M) such that

tllu= dWvw)

'7lv= d(-¢UWl-

Then k w : T). The reader can prove directly that the above definition is independent of the various choices. The Mayer-Vietoris sequence has the following factorial property.

Proposition 7.1.30. Let QUO : M -> N be a smooth map and {U, V } an open cover of N . Then U/ = Q5-1(U)7 V' = cl5-1(V) form an open cover o f M and moreover the diagram below is commutative. 6 6' He+llnl > Hk(U) ® H'°(v) Hk(nl >HUNV) .°

d>*

,my

H'*

>

HW) ® HW)

6/

>

H'"

Hk+1lml

257

Cohomology

Exercise 7.1.31. Prove the above proposition. 7.1.5

Some simple examples

The Mayer-Vietoris theorem established in the previous subsection is a very powerful tool for computing the cohomology of manifolds. In principle, it allows one to recover the cohomology of a manifold decomposed into simpler parts, knowing the cohomologies of its constituents. In this subsection we will illustrate this principle on some simple examples.

Example 7.1.32 (The cosmology of spheres). The cosmology of S1 can be easily computed using the definition of DeRham cosmology. We have H0(S1) = R since S1 is connected. A 1-form 17 6 Q1 (51) is automatically closed, and it is exact if and only if T)

S1

Indeed, if 17

0.

dF, where F : R -> R is a smooth 21r-periodic function, then S1

77=F(27r)-F(0)=0.

f (0)d0, where f

Conversely if 17

: R -> R is a smooth 27r-periodic function and 27r

f (0)d0

0,

0

then the function

F(t)

f (s)ds 0

is smooth, 21r-periodic, and dF Thus, the map S1

17.

:Q1->R, 17+->

17, $1

induces an isomorphism H1 (51) ->R, and we deduce pS1(t)= 1 +

To compute the cosmology of higher dimensional spheres we use the Mayer-Vietoris theorem. The (n -|- 1)-dimensional sphere Sn-l-1 can be covered by two open sets USOuth

= Sn-I-1 \ {North

pole} and

North

= Sn-I-1 \ {South

pole}.

Each is diffeomorphic to lRTL+1. Note that the overlap N o r t h n USOuth is homotopically equivalent with the equator S". The Poincaré lemma implies that Hk+1lUnorthl

®

f\J

Hk+1(Usouthl

0

for k > 0. The Mayer-Vietoris sequence gives HklUnorthl

®

Hklljsouth)

Hklljno'rth

VW

Usouthl

Hk+1lsn+1l

0.

258

Lectures on the Geometry

of Manzfolds

For k > 0 the group on the left is also trivial, SO that we have the isomorphisms

H'"(s") 2 H'"(UnO,t,, re USOuthl 2 H' B be a smooth vector bundle. If the base B is ofjnite type, then so is the total space E. In the proof of this proposition we will use the following fundamental result.

Lemma 7.1.37. Let p : E -> B be a smooth vector bundle such that B is dyeomorphic to Rn. Then p : E -> B is o trivializable bundle.

Proof of Proposition 7.1.36. Denote by F the standard fiber of E. The fiber F is a vector space. Let (Uz~)19$1/ be a good cover of B. For each ordered multi-index I := {it < < ik} denote by UI the multiple overlap U m (W Uik. Using the previous lemma we deduce that each E, E IU1 is a product F X Up, and thus it is diffeomorphic with some vector space. Hence (ED is a good cover.

-- -

---

Exercise 7.1.38. Prove Lemma 7.1.37. Hint: Assume that E is a vector bundle over the unit open ball B C R". Fix a connection V on E, and then use the V-parallel transport along the half-lines L , [0,oo) 9 t +-> to E R", ac 6 R" \ {0}. @»We denote by M n the category _finite type smooth manifolds of dimension n. The morphisms of this category are the smooth embeddings, i.e., the one-to-one immersions M1 MQ, Mi E Mn.

Definition 7.1.39. Let R be a commutative ring with l. A contravariant Mayer-Wetoris functor (or M V-functor for brevity) is a contravariant functor from the category Mn, to

260

Lectures on the Geometry

of Manzfolds

the category of Z-graded R-modules 3'

®n€Z3

*

N

GradRl\/Iod

9'"(M),

M»->

7

n

with the following property. If {U, V } is a MV-cover of M 6 Mn, i.e., U, V, U re V 6 Mn, then there exist morphisms of R-modules

an : 9*'"(U re v) ->

3rrz-I-1(m),

such that the sequence below is exact r*

- > 3 r l M ) -> 9""(U) ® m y ) i> 5t""(Urwv)

where

,r,*

a

s 3rH+1lml ->

7

is defined by T*

5"('¢u) ® 5"('¢vl»

and 6 is defined by 5(a2 ® y) = 5"(2U0vl(3/I - 5"('¢Ur1vl(tl(The maps z, denote natural embeddings.) Moreover, if N 6 Mn is an open submanifold

of N , and {U, V } is an M V - c o v e r o f M such that {Uri N , V O N } is an MV-cover of N, then the diagram below is commutative. an gw+1lml

3'"lumvl

)

3'"(umvmnl

an

>

9I'17~+1

TNT

The vertical arrows are the morphisms 5"(z,) induced by inclusions. The covariant MV-functors are defined in the dual way, by reversing the orientation of all the arrows in the above definition.

Definition 7.1.40. Let S", 9 be two contravariant MV-functors, f } ' , § : M n - > Grad _R1\/Iod.

A Col'l'€spol'ld€l'lc€2 between these functors is a collection of R-module morphisms (25m

® (We =®

>

9"(m),

TLGZ

one morphism for each M below is commutative

Q

Mn, such that, for any embedding M1 9k(R") i5 an isomorphism for any k 6 Z, then cl) i5 a natural equivalence.

Proof. The family of finite type manifolds Mn has a natural filtration

Ml CM2 c - - - c M Q c - - - , where M; is the collection of all smooth manifolds which admit a good cover consisting of at most 7" open sets. We will prove the theorem using an induction over r. The theorem is clearly true for r = l by hypothesis. Assume ¢'; is an isomorphism for all M 6 m;-1. Let M 6 M; and consider a good cover {U1, . . . , U,.} of M. Then

{ U = U1 u---LJU,-_1,U,,-} is an MV-cover of M. We thus get a commutative diagram 5"'(u) ® 2""(u~)

>

£¥n*(Urw U,.)

\

9"(U n U,.)

9"(U) ® 9"(uI)

8

> 9:'l1-I-lljwl

N-I-IIUI @ 912-I-I(U)

>

a 97.-I-1(jwl

5""+(u) ®

\

9%-I-IIUI

The vertical arrows are defined by the correspondence go. Note the inductive assumption implies that in the above infinite sequence only the morphisms (ISm may not be isomorphisms. At this point we invoke the following technical result.

Lemma 7.1.42 (The five lemma). Consider the following commutative diagram of Rmodules.

f

AS

>

1

f

2

B_2

If f

AS 1

1

> B_ 1

is an isomorphismfor any i ¢ 0, then

>

AS to

>

A1

1

fl

B0 SO

is

B1

f0.

A2 f2

B2

262

Lectures on the Geometry

of Manzfolds

Exercise 7.1.43. Prove the five lemma. The five lemma applied to our situation shows that the morphisms ( z ) m must be isomorphisms. Remark 7.1.44. (a) The Mayer-Vietoris principle is true for covariant MV-functors as well. The proof is obtained by reversing the orientation of the horizontal arrows in the above proof. (b) The Mayer-Vietoris principle can be refined a little bit. Assume that 5* and 9 are functors from Mn to the category of Z-graded R-algebras, and go : 3' -> 9 is a correspondence compatible with the multiplicative structures, i.e., each of the R-module morphisms m are in fact morphisms of R-algebras. Then, if (be are isomorphisms of Z-graded R-algebras, then so are the QSm 's, for any M 6 Mn . (c) Assume R is a field. The proof of the Mayer-Vietoris principle shows that if 5* is a MV-functor and diIIlR 5** (Rn) < oo then dim 5**(M) < oo for all M 6 Mn. (d) The Mayer-Vietoris principle is a baby case of the very general technique in algebraic topology called the acyclic models principle.

Corollary 7.1.45. Any finite type manifold has #nite etti numbers. 7.1.7 The Kenneth formula We learned in principle how to compute the cosmology of a "union of manifolds". We will now use the Mayer-Vietoris principle to compute the cosmology of products of manifolds. Theorem 7.1.46 (Kenneth formula). Let M 6 M m and N E Mn. Then there exists a natural isomorphism of graded R-algebras

H.

(M

X

N)

H.

f

6969

( M ) ® H* ( n )

H'"(Ml

(X)

HQIN)

p-I-q=n

7120

In particular, we deduce ProN(¢) =

PMT

. PW).

Proof. We construct two functors

529 Mm -> Grad R All, 5':M+->

3""'l(Ml 'r">0

6969

'r>0

H'"(M) (X) H"(N)

p-l-q=r

and

H" ( M x N ) ,

9"(M)

§:M+-> 'r'>0

'r'>0

7

263

Cohomology

where

f* IHp(M2)

5I'(f)

®]1H0

We let the reader check the following elementary fact.

Exercise 7.1.47. 5* and 9 are contravariant MV-functors. For M 6 Mm, define ¢M : st(m) -> 9(n) by ( b e ( w ® t I ) = ( » J X 17

WmWAWV

(USG

H'(m),

where arm (respectively W N ) are the canonical projections M M X N -> N). The operation X

: H°(M)

(X)

H ' ( N ) -> H' (M

X

N)

T) X

6

H.

(MU,

N -> M (respectively

((»J®t7)*->¢~vx17,

is called the cross product. The Kenneth formula is a consequence of the following lemma.

Lemma 7.1.48. (a)

go is a correspondence of MV-functors.

(19) gz5Rm is an isomorphism.

Proof. The only nontrivial thing to prove is that for any MV-cover {U, V} of M the diagram below is commutative. 3

®p+q=THPlU n V ) ® H

®p+q=rHP+1lMl

Hq(N)

do

¢umv

HT' Q;ptlN)One can verify easily that ¢* is a cochain map so that it induces a morphism QS* : H;ptlm) -> Hc.ptlNl° In terms of our category Mn we see that Hgpt is a covariantfunetor from the category Mn, to the category of graded real vector spaces. As we will see, it is a rather nice functor.

Theorem 7.2.1.

.

Hcpt

is a covariant MV-functon and moreover Hkpt (Rn

)

07 k E,

or : (t;v0,v1) »-> (t;u0 -I-t('u1 - v0)).

Note that

38=({0}x(E®E))

LI ({l}

(ElBE)).

X

Define 'IrTt E ® E -> E a s the composition E®E%{t}XE®E I

X

(E®E)l>E.

Observe that 3% = tflaa= (-w0) U Tr1.

For (Q, B) 6

Qgpt

X

6 Q;pt(E®E) by

Q;ptlEl define a ® B QQB

(#")*@ A (We.

Q;

(Verify that the support of a GM? is indeed compact.) For a G we have the equalities p*p*(1 /\

B=

7r1(a G) B) G

Q;;v(E),

QptlEl' and /3 6

a I\p*p*!3 = tr0(a G) B) 6

QJptlEl

Q;;v(E).

Hence D(o¢,B)=1>*p*o¢/\!5'-

a Ap*p*@=w1( E ® E. The lemma holds with

X

'J~*(0¢®5).

in low, B)

5/E

Proof of the Poincaré lemma for compact supports. 0 H';t(U

nv)

-> H ' ; ( U ) ® H'¢,(v> -> H k p t l m l

i> H'§;;1 ->

The connecting homomorphism can be explicitly described as follows. If w E a closed form then dlS0 U(»J )

= ii(-(pvw) on U

Qkpt(M)

is

re VL

We set do := d((pUw). The reader can check immediately that the cosmology class of do is independent of all the choices made.

268

of Manzfolds

Lectures on the Geometry

If QUO : N M is a morphism of Mn then for any MV-cover of {U, V} of M {¢-1(U),¢-1(V)} is an MV-cover of N. Moreover, we almost tautologically get a commutative diagram

Hkptlnl

5 >

l ,l eptlm

H'3;1((l>(umv))

¢*

d>*

6

HC

l re

H';;;1(U

v)

This proves Hgpt is a covariant Mayer-Vietoris sequence.

Remark 7.2.4. To be perfectly honest (from a categorical point of view) we should have considered the chain complex

@Q

eptk

7

go

k Hept k(Uov) denotes the connecting morphism in the DeRham cohomology with compact supports, and 6+ denotes its transpose. The Poincaré lemma for compact supports can be rephrased

Pfklnaqnl

R, 0,

k 0 k>0 I

'

270

Lectures on the Geometry

of Manzfolds

The Kronecker pairing induces linear maps

DM : H'"(m) -> Hk(ml. Lemma 7.2.8.

€9k Dk i5 a correspondence of MV

functors.

Proof. We have to check two facts. Fact A. Let

mi

N be a morphism in M+. Then the diagram below is commutative. ¢>*

HWy)

H/"(M)

>

I

I

Dn

g,*

PIN)

PPM)

Fact B. If {U, V } is an MV-cover of M

6

M+, then the diagram bellow is commutative

Hkwrw) ®Urwv

l

a

He+llml Pm

HUOV)

>

l

Hk+1lml

(-1)'°6*

..

Proof of Fact A. Let w 6 H"'(N). Denoting by ( » ) the natural duality between a vector space and its dual we deduce that for any u € HZL/°(M) we have (Hk M is a smooth map. We denote by € k ( M ) the set of k-dimensional cycles in M.

Definition 7.3.2. (a) Two cycles (SQ, (150), (So, $51) E € k ( M ) are said to be concordant, and we write this (SO, QSo) -C (So, $51), if there exists a compact, oriented manifold with boundary 2, and a smooth map Q : Z] -> M such that the following hold. (al) 32 = (-5ol l_l $1 where -So denotes the oriented manifold SO equipped with the opposite orientation, and "LI" denotes the disjoint union. (a2) l Si (15i, i = 0, 1.

273

Cohomology

(b) A cycle (S, go) Q (`3k(Ml is called trivial if there exists a (k -|- 1)-dimensional, oriented manifold Z] with (oriented) boundary S, and a smooth map : Z] -> M such that @law = QS. We denote by 'I/c(M) the set of trivial cycles. (c) A cycle (S, go) Q (`3k(Ml is said to be degenerate if it is concordant to a constant cycle, i.e., a cycle (S', q25/) such that QS' is map constant on the components of S'. We denote by Do (M) the set of degenerate cycles.

Exercise 7.3.3. Let ls0a¢0l Ne (S17¢1l° Prove that (-50 LIS1» (250 UQ51) is a trivial cycle.

S

1

Fig. 7.1

A cobordism in R3

.

[So Zn ( M ) i5 well defined.

oWe urge the reader to supply a proof of this fact.

274

Lectures on the Geometry

of Manzfolds

(b) The binary operation -|- induces a structure ofAbelian group on Zn ( M ) . The trivial element i5 represented by the trivial cycles. Moreover;

-m

= [-5,¢] 6 mm).

Exercise 7.3.5. Prove the above proposition. We denote by ?€k(M) the quotient of Z k ( 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. Suppose M Q M+. Any k-cycle (S, (15) defines a linear map H k ( M ) -> R given by

H'" 9 w +->

q25*w. .J

?

Stokes formula shows that this map is well defined, i.e., it is independent of the closed form representing a cosmology class. Indeed, if LU is exact, i.e., w = du/, then

¢*dw/

dl)*wI

0.

w

w

>

>

In other words, each cycle defines an element in ( H k ( M ) )*. Via the Poincaré duality we identify the vector space (H1

The dual o f a point is Dirac's distribution.

275

Cohomology

0-cycle. Its Poincaré dual is a compactly supported n-form w such that for any constant A (i.e., closed 0-form) A

A,

i.e., 1.

CU

Rn

Thus opt 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, HM). Then 6[M] = l 6 H 0 ( M ) . @)For any

do erentialform w, we set (for

typographical reasons),

lwl

:= deg w.

Example 7.3.8. Consider the manifolds M 6 M+ and N 6 M+_ (The manifolds M and N need not be compact.) To any pair of cycles (S, co) 6 @p(M), and (T, QD) G €q(N) we can associate the cycle (S X T , go X up) 6 @p+q(M X N). We denote by 7rm (respectively 7rN) the natural projection M X N -> M (respectively M X N -> N). We want to prove the equality (SSxT

l-1l(M -P)q6 S X

I

IT 7

(7.3.1)

where cu X 17

Pick (w, 17) E Q' (Ml using Exercise 7. 1.50

MXN

(w

X

1rmw /\ 1rN17, 'v'(w,17) Q Q'

( N ) such that, deg

t/)A(6s

X

-|-

Q'(M) degn

X

QUO

(N)-

= (m+n)

X 5Tl=l-1l(m-p)lt1l

MXN

(WAS)

-

(p+ql~ Then,

(nAsT).

X

The above integral should be understood in the generalized sense of Kronecker pairing. The only time when this pairing does not vanish is when lwl = p and 1171 = q. In this case the last term equals (_1lq(tn-p)

(f

S

CU

A 65

)(/

T

T)

A IT

)

I

l_1)q("®-p)

= (-1)q(tf.-p)

This establishes the equality (7.3. 1).

(f

/

q25*w S

SXT

)(/ ) zp*T)

T

()

X

w* (cu A H)-

276

Lectures on the Geometry

of Manzfolds

Example 7.3.9. Consider a compact manifold M 6 M+. Fix a basis (Wil of H' (M) such that each We is homogeneous of degree = do. and denote by wi the basis of H ' ( M ) dual to (We) with respect to the Kronecker pairing, i.e., ( w ' i w j ) , . = I-1)l°JI°lw3l(w3

In M

X

55.

ii)

M there exists a remarkable cycle, the diagonal A = A m : M - > M X M ,

33+->(ac,ac).

We claim that the Poincaré dual of this cycle is ii i ( 1 ) I IU.)

(SA = $M

(7.3.2)

XC4J»,;.

Indeed, for any homogeneous forms a,B 6 Q' (M) such that 101 +I/3l MXM

(Q

X

Z(

BIT $m

1) Iwil

(Q

B) /\ (w

X

MXM

(-1)l-1(S)

t°(T¢» (x)M)

-OTlT¢(xlM)

278

Lectures on the Geometry

of Manzfolds

1

1/ Fig. 7.3

The intersection number o f the two cycles on T 2 is 1.

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

E 3vI+. Consider a compact, oriented, k-dimensional sub-

manifold S M, and (T, go) E €,,_k (]l/I) a ( n - it)-dimensional cycle intersecting S transversally, i.e., S Rh QS. Then

S-T

(7.3.3)

65 /\ (5:r,

where 6. denotes the Poincaré dual of o. The proof of the proposition relies on a couple of technical lemmata of independent interest. Lemma 7.3.13 (Localization lemma). Let M 6 M+ and (S, (be 6 € k ( M ) . Then, for any open neighborhood N of Q5(S ) in M there exists 65n 6 Q;*'"(m) such that (a) 65n represents the Poincaré dual 65 Q HCnpt '"(M),(b) supp $5n C N.

Proof. Fix a Riemann metric on M. Each point p 6 Cl5(S) has a geodesically convex open neighborhood entirely contained in N. Cover g25(S) by finitely many such neighborhoods, and denote by N their union. Then N Q M+, and (S, (15) E ( N ) . Denote by (SsN the Poincaré dual of S in N. It can be represented by a closed form in Qgptk (N) which we continue to denote by (SsN . If we pick a closed form w E Qk ( M ) , then w | N is also closed, and

WA5sN r

W A 5sN r

Cl5*w. T

>

279

Cohomology

Hence, (SsN represents the Poincaré dual of S in Hype '"(m), and moreover, supp (SsN C N.

Definition 7.3.14. Let M E M+, and S --> M a compact, k-dimensional, oriented submanifold of M. A local transversal at p E S is an embedding Q) : B C R"-'" -> m, B = open ball centered at 0 e t

such that S Hn Q) and (l5-1($)

k7

{0}~

I

Lemma 7.3.15. Let M Q M"L, S M a compact, k-dimensional, oriented submode zfold of M and ( B , (15) a local transversal at p 6 S. Then for any sujiciently "thin " closed neighborhood N of S C M we have s~(8,(l))

B¢*6SN

Proof. Using the transversality S Rh co), the implicit function theorem, and eventually restricting go to a smaller ball, we deduce that, for some sufficiently "thin" neighborhood N of S, there exist local coordinates (5151, . . . ,x") defined on some neighborhood U of p 6 M diffeomorphic with the cube

{laval < 1, of;} such that the following hold. (i)SVlU={:z:k:+1

=--~=

n

= 0 } , p = ( 0 , . . . 7 0).

(ii) The orientation of S n U is defined by dzc1 A - - - /\ dark. 'n-k ->M is expressed in these coordinates as (iii) The map QUO : B C R(y1»---,y""°)

cc

1

= 0, ..., co:k

=0,$ck+1

= y1

in 7

in

k

7

( i v ) N O U = {l:uil < 1 / 2 ; j = 1 , . . . , n } . Let e :l:1 such that edit1 /\ . . . /\ d s " defines the orientation of TM. In other words, e

For each f

o l

7

s~ (B, ¢).

. , xi) 6 S al U denote by Ps the (n - k)-"plane"

7 iN +1 l:1:il k } . {(§§£z:k . . . /\ (n 1 lWe orient each Pe using the A do", and set k)°foI'm d$k

Pe

7

55N .

..II

U(6)

7

Pe

Equivalently,

u(gl

B

(z5§5sN 7

where (25€ : B -> M is defined by 1

1

My , - - - , y " " " ) = (€,y ,---,y""")-

280

of Manzfolds

Lectures on the Geometry

To any function up

COO(S re U) such that

go(§l Q

suppgoc

no _< 1 / 2 , ¢_ M such that K O N = 0). Then, QS* (SsN is compactly supported in the union of the BE's and

68N /\ (ST

2/

¢* (SSN

M

T

(15I6sNBi

From the local transversal lemma we get 8

¢*6§ = S ( l m )

21%

(sJ)-

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 to 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 ( s ) 0, 1. 1f

6 ( M M ) and ((T¢,¢¢)

Q

(2,,-k(m) where M G m+,

(a) SO N o SO) To N o T19 (b) the cycles S are embedded, and (c) Si Hn To, then S 0 ° T 0 = $ 1 -$1.

Definition 7.3.17. The homological intersection pairing is the Z-bilinear map j = j m : j l f k l M ) X j{n_kl]ll)->]R, (M E JV[+) defined by

is

3(S, T)

(iT.

M

We have proved that in some special instances 3(S, T ) when M is compact, this is always the case.

Q

Z. We want to prove that,

Theorem 7.3.18. Let M 6 M+ be compact manifold. Then, for any (S, To 6 1TCk(Ml 17'(n_k III, the intersection number U(S,

To is an

X

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 7.8'.]8. Then

:](S,T)

I

(-1)"-k

5SXT /\

(YA,

MXM

where A is the diagonal cycle A : M -> M X M, as +-> (co, co). (It is here where the compactness of M i5 essential, since otherwise A would not be a cycle. )

282

of Manzfolds

Lectures on the Geometry

Proof. We will use the equality (7.3.l) (SSxT

I

(-1)"- '"6s

6:1

X

Then

( 1)"

k

/ /

M

ICes

(SSxTA(IA

MXM

X

IT) /\ PA

MXM

A * ((is

X

IT)

55

(iT.

M

Lemma 7.3.20 (Moving lemma). Let S E @1K7(M)) and T 6 € n _ k ( M ) be two cycles in M 6 M-I-. If S i5 embedded, then T is concordant to a cycle T such that S Hn T. The proof of this result relies on Sard's theorem. For details we refer to [62], Chapter 3. Proof of Theorem 7.3.18. Let l5,Tl E €klml X (211-k(M)~ Then,

:1 = (-1>"-'ms x T, Al. Since A is embedded we may assume by the moving lemma that (S U(S X T , A) Q Z.

X

T) Hn A, so that

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 XN FF : M - > M x N :Ur->(x,F(a:)). FF is called the graph off. Any point y E N defines an n-dimensional cycle M X {y}. Since N is connected all these cycles are concordant so that the integer FF (M X {y}l is independent of y.

Definition 7.3.21. The topological degree of the map F is defined by

degF := (M x {y}) - FF. Note that the intersections of FF with M X {y} correspond to the solutions of the equation Thus the topological degree counts these solutions (with sign).

Proposition 7.3.22. Let F : M -> N as above. Then for any n-form w E AZ"(N) n

F*w = deaF r

Jn

w.

Remark 7.3.23. The map F induces a morphism R

r\/

H**(n) 1 H " ( M )

f"*J

R

which can be identified with a real number. The above proposition guarantees that this number is an integer.

283

Cohomology

Proof of the proposition.

Note that if w E Q"(N) is exact, then P

P

'm

F*w = 0.

Thus, to prove the proposition it suffices to check it for any particular form which generates Hn(N). Our candidate will be the Poincaré dual dy of a point y 6 N. We have N

1,

'So

while equality (73.1) gives 5MX{9} =

(Ym

X

(So

-

1

X 53/~

We can then compute the degree of F using Theorem 7.3.18 deaF = deaF

/

(Sy

N

/

(1

X

(sys

$pF

MXN

FF(l

X

(sys

F*6y.

M

Corollary 7.3.24 (Gauss-Bonnet). Consider a connected sum ded in R3 and let 92 : E -> S2 be its Gauss map. Then

M

of g-tori

E

2g embed-

deg 92 = X(2) = 2 - 2g.

This corollary follows immediately from the considerations at the end of Subsection 4.2.4. Exercise 7.3.25. Consider the compact, connected manifolds MI,M1,N 6 M+ and the smooth maps F : MZ- -> N, i = 0, 1. Show that if F0 is concordant to F1 then deg F0 = deg Fl. In particular, homotopic maps have the same degree.

Exercise 7.3.26. Let A be a nonsingular n

n real matrix. It defines a smooth map Ax

X

FA : SN-1 -> SN-1

7

at »->

III

Prove that deg FA = sign det A. 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, RI to a diagonal matrix. F

Exercise 7.3.27. Let M -> N be a smooth map (M, N are smooth, compact oriented of dimension n). Assume y E N is a regular value of F, i.e., for all :u 6 F-1(y) the derivative D F : T,,m -> TyN is invertible. For QUO

Q

F-1(y) define

deg(F, so)

1, 1,

DasF preserves orientations otherwise

Prove that deg F =

deg(F, x). F(fv)=y

284

Lectures on the Geometry

of Manzfolds

Exercise 7.3.28. Let M denote a compact oriented manifold, and consider a smooth map F : M -> M. Regard H' (M) as a superspace with the obvious Z2 -grading ) H. AS' H even IMl €l9 H 0 d d lm

(

)

I

and define the Lefschetz number A(F) of F as the supertrace of the pull back

F* H' (M)-> H' (M). Prove that A(F) = A FF, and deduce from this the Lefschetz fixed point theorem: A(F) ¢ 0 => F has a fixed point.

Remark 7.3.29. For an elementary approach to degree theory, based only on Sard's theorem, we refer to the beautiful book of Milnor [97]. 7.3.4

The Thom isomorphism theorem

Let p : E -> B be an orientable fiber bundle (in the sense of Definition 3.4.45) with standard fiber F and compact, oriented basis B. Let dim B = m and dim F = r. The total space E is a compact orientable manifold which we equip with the fiber-first orientation. In this subsection we will extensively use the techniques of fibered calculus described in Subsection 3.4.5. The integration along fibers E/B

=

: Q;p(E) -> 9'-~'°(8)

p*

satisfies P* dE

(-1)dBP* 7

so that it induces a map in cosmology P* : H;,,(E) -> H'-w8).

This induced map in cohomology is sometimes called the Gysin map. Remark 7.3.30. If the standard fiber F is compact, then the total space E is also compact. Using Proposition 3.4.48 we deduce that for any w 6 Qs (El, and any 17 6 Q' (B) such that deg

-|-

degn = dim E,

we have

E»

so that, in this case, the Gysin map coincides with the pushforward (Gysin) map defined in Remark 7.2.9.

285

Cohomology

Exercise 7.3.31. Consider a smooth map f : M -> N between compact, oriented manifolds M, N of dimensions m and respectively n. Denote by if the embedding of M in M XN as the graph of f

M3a:»->(a;,f(a:)) € M

N.

X

The natural projection M X N -> N allows us to regard M X N as a trivial fiber bundle over N. Show that the push-forward map f* : H ' ( M ) -> H'+"""(N) defined in Remark 7.2.9 can be equivalently defined by

f* =7r* where 7r* : H ' ( M

X

o

7

N)-> H"""(N) denotes the integration along fibers while H ' ( m ) -> H.+"(m

if)*

X

al,

is the pushforward morphism defined by the embedding i f . Let us return to the fiber bundle p : E -> B. Any smooth section o : B -> E defines an embedded cycle in E of dimension m = dim B. Denote by 5O' its Poincaré dual in Hgpt

Using the properties of the integration along fibers we deduce that, for any w E Qm(B), we have 5O' /\ p*w E

I(/ B

)

5O' w. E/B

On the other hand, by Poincaré duality we get 5O' Ap*w = (-1y"fn', E

/ /

p*w /\ 6O'

E

(-am

O,*p*w=(-ll1°m

B

w.

(po)*~»:(-1)*m B

B

Hence

6O' = (-l)""' E Q0(B).

p*50 E/B

Proposition 7.3.32. Let p : E -> B a bundle as above. If it admits at least one section, then the Gysin map P* i5 surjective.

Hgpt(El -> H'-*(B)

286

Lectures on the Geometry

of Manzfolds

Proof. Denote by To the map : H' (B) ->

To

'+'°(E) Hept

w +-> ( - 1 ) m 5 , /\p*w = P*w /\ 50.

Then To is a right inverse for p*. Indeed w

(-1)'°""p*6o

w

(-l)"Mp* (60

p*w) =

p*

(fO'wl.

The map p* is not injective in general. For example, if (S, go) is a k-cycle in F, then it defines a cycle in any fiber 'ii-1 (6), and consequently in E. Denote by 68 its Poincaré dual in Hip-klgl. Then for any w E §2M"*(B) we have

B

lp*5sl /\ cu

d:

is /\ P*w E

/

D

(1)*1>*(»J

/

(p o ¢)*(»J = 0,

S

since p o (b is constant. Hence p* 55 0, and we conclude that that if F carries nontrivial cycles her p* may not be trivial. The simplest example of standard fiber with only trivial cycles is a vector space. Definition 7.3.33. Let p : E -> B be an orientable vector bundle over the compact oriented manifold B (dirndl = m, rank (E) = T). The Thom class of E, denoted by TE is the Poincaré dual of the cycle defined by the zero section go : B -> E, b +-> 0 6 Et. Note that TE

6

HeptlEl°

Theorem 7.3.34 (Thom isomorphism). Let p : E -> B as in the above definition. Then the map 7'

H°(B) ->

°+T(El Hcpt

w »-> TE Ap*w

is an isomorphism called the Thom isomorphism. Its inverse is the Gysin map

(-1)"'"p* : H;,..(E) -> H'-t(1-31. Proof. We have already established that in = (-1)'~"», To prove the reverse equality, To* = (-1)~"., we will use Lemma 7.2.2 of Subsection 7.2.1. For B 6 Q;pt(E) we have (P*P*7'El /\

where in ( E , IN) G (-l)T7716

=

5

TE /\ (p*p*B)

Q;ptlEl° Since P*P*TE

(TE

As*(p*@)

-|-

;

Z

(-ll'°d(\"(TE» in),

(-1)rm, we deduce

exact form => (-1)"r°mB =

Exercise 7.3.35. Show that TE : E.

»

TE

H ' ( M ) ->

OMB)

l

HZpt(E)~

.+dimMlEt Hcpt

is the push-

Remark 7.3.36. Suppose M 6 M+F, and S --> M is a compact, oriented, k-dimensional manifold. Fix a Riemann metric g on M, and denote by Ns the normal bundle of the embedding S --> M, i.e., the orthogonal complement of TN in (TM)IS. The exponential map defined by the metric g defines a smooth map exp:Ng->M,

287

Cohomology

which induces a diffeomorphism from an open neighborhood 6 of S in Ns, to an open (tubular) neighborhood N of S in M. Fix a closed form 65n E on-1 M denotes the canonical inclusion. Then exp* is the Poincaré dual of the cycle (S, Q) in Ns, where Q : S -> Ns denotes the zero section. This shows that exp* is a compactly supported form representing the Thom class of the normal bundle Ns . Using the identification between O and N we obtain a natural submersive projection 'or : N -> S corresponding to the natural projection Ns -> S. In more intuitive terms, 71' associates to each ac E N the unique point in S which closest to as. One can prove that the Gysin map

58N

so

H.

i*

(S) -> H ; , ; ; ( " ( m ) ,

is given by,

H. (S) 9 cu

|.

> exp* (SsN /\ tr*w 6 H l " " " ) ( M ) .

(7.3.5)

Exercise 7.3.37. Prove the equality (7.3.5). 7.3.5

Gauss-Bonnet revisited

We now examine a very special type of vector bundle: the tangent bundle of a compact, oriented, smooth manifold M. Note first the following fact. Exercise 7.3.38. Prove that M is orientable if and only if TM is orientable as a bundle. Definition 7.3.39. Let E -> M be a real orientable vector bundle over the compact, oriented, n-dimensional, smooth manifold M. Denote by TE 6 H"cpt the Thom class of E. The Euler class of E is defined by

e(El

CoTE

E H"(M),

where QUO : M -> E denotes the zero section. e ( T M ) is called the Euler class of M, and it is denoted by e ( M ) . Note that the sections of TM are precisely the vector fields on M. Moreover, any such section (7 : M -> TM tautologically defines an n-dimensional cycle in TM, and in fact, any two such cycles are homotopic: try a homotopy, affine along the fibers of TM. Any two sections 0 0 , 071 : M -> TM determine cycles of complementary dimension, and thus the intersection number Oo • 0 1 is a well defined integer, independent of the two sections. It is a number reflecting the topological structure of the manifold.

Proposition 7.3.40. Let 0 0 , 0 1 : M -> T M be two sections ofTer. Then e(M) M

Oo

• 01.

288

Lectures on the Geometry

of Manzfolds

In particular If dim M i5 odd then

e(M)

0.

M

Proof. The section 00, 01 are concordant, and their Poincaré dual in Hcnpt (TM) is the Thom class TM. Hence

(See /\ (So

0'0'0'1

.ITM

TM

/\

TM

JTM TM /\ (SCO

TM

/

/

CoTM

M

M

If dim M is odd then

e(M)

e(M).

O'0°(71=-0'1°O'0

M

M

Theorem 7.3.41. Let M be a compact oriented n-dimensional manifold, and denote by e ( M ) E H",(M) its Euler class. Then the integral of e ( M ) over M i5 equal to the Euler enaracteristie of M, n

e(M) M

= x(M)

(--)'"bk(m). k=0

In the proof we will use an equivalent description of X(M). Lemma 7.3.42. Denote by A the diagonal cycle A QQ-['nljlj

X

M).Thenx(M)=A~A.

Proof. Denote by AM E H;'pt(M X M) the Poincaré dual of A. Consider a basis (we) of H. (M) consisting of homogeneous elements. We denote by (w"L) the dual basis, i.e.,

M

i5 an embedding.

X

M

290

Lectures on the Geometry

of Manzfolds

exp("L(). Let U be a neighborhood of A C NA as in the above lemma, and set N Denote by the Poincare dual of A in "Ll, 6 He*pt(U), and by the Poincare dual of A in N , (SAN E HcnptlNl° Then

so

so

u

62/mX

so

u Hence

AM

6AN*A6AN N

The cohomology class (7.3.6) means that =

52

/\

am

x(M)-

MXM

so is the Thom class of the bundle NA -> A, which in view of TNA

6XA6X

= TM. We get

/ (so / CSTNA / A

A

e(M) M

M

CSTm

/

M

e(M).

xIMI-

If M is a connected sum of g tori then we can rephrase the Gauss-Bonnet theorem as follows.

Corollary 7.3.45. For any Riemann metric h

-e(h) =

1 shd'l)h

OH Cl

C0HH6C1601 Sum of g-tori Fig WE have

elxg) in H*(29)~

The remarkable feature of the Gauss-Bonnet theorem is that, once we choose a metric, we can explicitly describe a representative of the Euler class in terms of the Riemann curvature. The same is true for any compact, oriented, even-dimensional Riemann manifold. In this generality, the result is known as Gauss-Bonnet-Chern, and we will have more to say about it in the next two chapters. We now have a new interpretation of the Euler characteristic of a compact oriented manifold M. Given a smooth vector field X on M, its "graph" in T M ,

FX = {(a2,X(x)) E TxM , :UE M}, is an n-dimensional submanifold of TM. The Euler characteristic is then the intersection number

x(]M) F x • M , where we regard M as a submanifold in TM via the embedding given by the zero section. In other words, the Euler characteristic counts (with sign) the zeroes of the vector fields on M. For example if X(Ml 7é 0 this means that any vector field on M must have a zero! We have thus proved the following result. Corollary 7.3.46. I f x ( M ) 7é 0 then the tangent bundle T M is nontrivial.

The equality X(S2"l = 2 is particularly relevant in the vector field problem discussed in Subsection 2.1.4. Using the notations of that subsection we can write v(S2") 0. We have thus solved "half" the vector field problem.

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Cohomology

Exercise 7.3.47. Let X be a vector field over the compact oriented manifold M. A point as E M is said to be a non-degenerate zero of X if X (0) = 0 and det

for some local coordinates w ) near kg such that the orientation of TU0M is given by day1 A A do". Prove that the local intersection number of FX with M at xo is given by

---

ix()

(Fx» Ml

sign det

(

3xi 3337

M

ac

390

This is sometimes called the local index of X at to, and it is denoted by Il;(X7;L0l. From the above exercise we deduce the following celebrated result.

Corollary 7.3.48 (Poincaré-Hopf). If X i5 a vectorjield along a compact, oriented man#old M , with only non-degenerated zeros 5131, . . . ,ask, then

'i(X7£Cj).

x(M) j

Exercise 7.3.49. Let X be a vector field on Rn and having a non-degenerate zero at the origin. (a) Prove that for all r > 0 sufficiently small X has no zeros on S = {l:z:l = r}. (b) Consider F : S -> S n - 1 defined by l

F'"">

1

lx(w)l

X(a2).

Prove that i(X, 0) = deg F for all T > 0 sufficiently small. Hint: Deform X to a linear vector field. 7.4 Symmetry and topology The symmetry properties of a manifold have a great impact on its global (topological) structure. We devote this section to a more in depth investigation of the correlation symmetrytopology.

7.4.1 Symmetric spaces

Definition 7.4.1. A homogeneous space is a smooth manifold M acted transitively by a Lie group G called the symmetry group.

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Lectures on the Geometry

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Recall that a smooth left action of a Lie group G, on a smooth manifold M G X M - > M (g,m)*->9~m, is called transitive if, for any m 6 M , the map \Ifm:G9g+->g-mEM is surjective. For any point ac of a homogeneous space M we define the isotropy group at QUO by

U = {g Q G 9

11}-

Lemma 7.4.2. Let M be a homogeneous space with symmetry group G and ac, y 6 M. Then

(a) Ux i5 a closed subgroup (b) Ux 2 39.

off,°

-

Proof. (a) is immediate. To prove (b), choose g E G such that y = g at. Then note that Uy = g:]xg-l.

Remark 7.4.3. 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. For proofs we refer to [61, 128]. Fact 1. Any closed subgroup of a Lie group is also a Lie group (see Remark 1.2.31). In particular, the isotropy groups U33 of a homogeneous space are all Lie groups. They are smooth submanifolds of the symmetry group. Fact 2. Let G be a Lie group, and H a closed subgroup. Then the space of left cosets,

G / H : = { 9 ° H ; gaG}, can be given a smooth structure such that the map

G X (G/H) G/H lg1ȣ12Hl ** (9192) ' H is smooth. The manifold G/H becomes a homogeneous space with symmetry group G. All the isotropy groups subgroups of G conjugate to H. Fact 3. If M is a homogeneous space with symmetry group G, and at 6 M, then M is equivariantly diffeomorphic to G/jx, i.e., there exists a diffeomorphism (l5:M->G/jx,

such that My y) = 9 - (My)

We will be mainly interested in a very special class of homogeneous spaces. Definition 7.4.4. A symmetric space is a collection of data (M, h, G, (7, i) satisfying the following conditions. (a) (M, h) is a Riemann manifold. (b) G is a connected Lie group acting isometrically and transitively on M GxM9(g,m)»->g~m€M. (c) 0 : M X M -> M is a smooth map (m1,m2) +-> 0m1 (mg) such that the following hold.

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Cohomology

(cl) We 6 M of : M -> M is an isometric, and 0m(m) (CZ) (to 0 0 m = He.

m.

(63) D0mlTmM= -llTmm(04) ugh = gvmg-1.

(d)i:M

X

G->G,(m,g) +-> mg is a smooth map such that the following hold.

(dl) 'v'm E M, it : G -> G is a homomorphism of G. (do) iTTI o iTTI = 1G~ (d3) igm = gi771g-1, 9) E M x G.

am,

(e)

0771g0'TTI1(x)

=

0mg0771(x)

= im(g) ~a3, V m , x 6 M , g 6 G.

Remark 7.4.5. This may not be the most elegant definition of a symmetric space, and certainly it is not the minimal one. As a matter of fact, a Riemann manifold (M, g) is a symmetric space if and only if there exists a smooth map O' : M X M -> M satisfying the conditions (cl), (c2) and (c3). We refer to [61,70] for an extensive presentation of this subject, including a proof of the equivalence of the two descriptions. Our definition has one academic advantage: it lists all the properties we need to establish the topological results of this section. The next exercise offers the reader a feeling of what symmetric spaces are all about. In particular, it describes the geometric significance of the family of involutions of .

Exercise 7.4.6. Let ( M , hl be a symmetric space. Denote by V the Levi-Civita connecson, and by R the Riemann curvature tensor. (a) Prove that VR = 0. (b) Fix m E M , and let 'y(t) be a geodesic of M such that 7(0)

(tmv(t)

i

m. Show that

by(-tl.

Example 7.4.7. Perhaps the most popular example of symmetric space is the round sphere S" C lR7*+1. The symmetry group is SO(n -|- l), the group of orientation preserving "rotations" of lR3."'+1. For each m 6 Sn-1-1 we denote by of the orthogonal reflection through the 1-dimensional space determined by the radius Om. We then set i71,.(T) = G 0gh=gh-1g-1

and i :G

X

(G

X

G)->G

X

G ig(g1,g2)=(9919

,9g2g-1)-

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Lectures on the Geometry

of Manzfolds

We leave the reader to check that these data do indeed define a symmetric space structure on (G, in). The symmetry group is G X G.

Example 7.4.9. Consider the complex Grassmannian M = Gr kl@"l . Recall that in Example 1.2.22 we described Gr I;T*A

TAT* •

Note that U(n) * M M , and 90 is U(n)-invariant. Thus U(n) acts transitively, and isometrically on M. For each subspace S 6 M define RS

Z=

P5-PS

=2P$-1.

The operator Rs is the orthogonal reflection through S-L_ Note that Rs E U(n), and = 2. The map A »-> Rs *A is an involution of End+ ((Cn*). It descends to an involution of M. We thus get an entire family of involutions

Re

O'

:M

X

M -> M,

lP51,Ps2l *-> R$1 *PA2

Define i :M

X

U(n) -> U(nl, isT = RsTRs.

We leave the reader to check that the above collection of data defines a symmetric space structure on Gr ICl so* by •

To Ld

O

G

as(g)€gw dvGl9l-

Note that

T joe) and

dTjLd

I

Tjdw.

j

Each To is thus a cochain morphism. It induces a morphism in cosmology which we continue to denote by Tj- The proof of the lemma will be completed in several steps.

Step 1.

]lonH*(M), for all g E G . Let X f:I

X

M -> M

QLG

such that g =expX. Define

ft(m) := exp(tX) m = exp(tX)m.

The map f is a homotopy connecting Hm with

This concludes Step 1.

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Lectures on the Geometry

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Step 2.

To For t E [0, 1], consider Q5j,t : Be -> G defined as the composition KJ

By

exp-1

> B,

> Do"

-

t -)

exp) Dtr

Be,

G.

Q*(M)->§2*(M) by

Define T

TjtLd

G

(g)€¢ ,¢(t)w dVG (g) = T]q25.7 tw.

Old

(7.4.1)

We claim that T80 is cochin homotopic to T . To verify this claim sett := 6S, -oo < S < 0 and gS

= exp(eS exp-1(g))

Vg 6 B .

Then w

a(g]g)€gs€gw dvG(g).

Br

Br

For each g E B the map (5, m) »-> \P.S(m) by Xg its infinitesimal generator. Then d ds (Usual

/ /

011 (9jg)Lxg

is

ml defines a local flow on M.

We denote

KgS £9 w dVG (9)

Br

as(gjg)(dmixg + ixgdm)(€9s€9 w) dvG(g).

B,

Consequently,

I

II

/ (f -oo

as(g]gl(dixg + ixgd)(€gs£g w) dvG(g) ds.

B,

(An argument entirely similar to the one we used in the proof of the Poincaré lemma shows that the above improper integral is pointwise convergent.) From the above formula we immediately read a cochain homotopy X : §Z'(M) -> §2"1(Ml connecting U-oo to Uo. More precisely 0

{XlWl}la¢6MI=

/ (f -oo

B

@j(9j 9l

}L dvG(9l

1§xg€9s*9,°/

ds.

Now notice that

To10(.4J

(/

Oi (QICWG (9)

.

*

afgw,

81

while To,16,0 completed.

Tow. Taldng into account Step 1, we deduce T

as

Step 2 is

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Cohomology

The lemma and hence the proposition follow from the equality G

To

CLj]1H*(m)

average.

j

j

The proposition we have just proved has a greater impact when M is a symmetric space.

Proposition 7.4.14. Let ( M , h) be an, oriented symmetric space with symmetry group G. Then the following are true. (a) Every invariant form on M is closed. (b) If moreover M is compact, then the only invariant form cohomologous to zero is the trivial one.

Proof. (a) Consider an invariant ac-form w. Fix

Mg Q

M and set (D

um0w.

We claim (D

is invariant. Indeed, Vg e G Zgrl:

:

:

Z9afNow = (0m0g)*w = (gig0mog)*w = "t/°(9)'*'

*

um w

A

w.

Since Dam) ITeM= -]ITeM, we deduce that, at mo E M, we have (IJ = (-1)k:w. Both w and (If are G-invariant, and we deduce that the above equality holds at any point m = 9 mo. Invoking the transitivity of the G-action we conclude that

-

al:

(-1)'"w on M .

In paiticular, do; = (-l)k*dw on M. The (k + l)-forms do; = omodw above, we deduce

do

85, and do

CZ

are both invariant and, arguing as

(-llk+1dw'

The last two inequalities imply do = 0. (b) Let w be an invariant form cohomologous to zero, i.e., cu = do. Denote by * the Hodge *-operator corresponding to the invariant metric h. Since G acts by isometrics, the form 17 = *w is also invariant, so that d17 = 0. We can now integrate (M is compact), and use Stokes theorem to get wAs M

This forces w

E

daA=d: M

aAd17=0. M

0.

From Proposition 7.4.12 and the above theorem we deduce the following celebrated result of Elie Caftan ([25]) Corollary 7.4.15 (E. Cartan). Let ( M , h ) be a compact, oriented symmetric space with compact, connected symmetry group G. Then the chorology algebra H * ( M ) of M is isomorphic with the graded algebra Qinv ( M ) of invariant forms on M. 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.

298

Lectures on the Geometry of Manzfolds

7.4.3 The eohomology of eompaet Lie groups Consider a compact, connected Lie group G, and denote by LG its Lie algebra. According to Proposition 7.4. 12, in computing its cosmology, it suffices to restrict our considerations to the subcomplex consisting of left invariant forms. This can be identified with the exterior algebra A°LG. We deduce the following result. Corollary 7.4.16. H°(G) 5 H' (LG) -2 Air?/CG) where AULZG denotes the algebra of bi-invariantforms on G, while H' (LG) denotes the Lie algebra conomology introduced in Example 7.1 .I0. Using the Exercise 7.1.11 we deduce the following consequence. II

6

Corollary 7.4.17. If G is a compact semisimple Lie group then H l (G)

Proposition 7.4.18. Let G be a compact semisimple Lie group. Then H2(G) = 0. Proof. A closed bi-invariant 2-form w on G is uniquely defined by its restriction to LG, and satisfies the following conditions.

do = 0

' \I

I\ i

W([X0» X1]»X2) - W([X0»X2]» Xl) + W([X1» X2]» Xo) = 0,

and (right-invariance)

(Lx0W)(X1»X2) = 0 VX0 G LG ~:

°J([X0»X1]»X2) - W([X0»X2]»X1)

\

= 0.

Thus 0 VX0,X1,X2 E LG.

W([X0»X1]»X2)

On the other hand, since H1 (LG) = 0 we deduce (see Exercise 7.l.11) LG so that the last equality can be rephrased as w(X,Y)

[£G,£G],

0 VX,Y E i5G~

Definition 7.4.19. A Lie algebra is called simple if it has no nontrivial ideals. A Lie group is called simple if its Lie algebra is simple. Exercise 7.4.20. Prove that SU(n) and SO(m) are simple.

Proposition 7.4.21. Let G be a compact, simple Lie group. Then H3(G) H3 (G) is generated by the Caftan form a(X,Y,Z)

1

r\/

R. Moreover,

I=z([X,Y],Z),

where ii denotes the Killing pairing.

The proof of the proposition is contained in the following sequence of exercises.

Exercise 7.4.22. Prove that a simple

je

algebra is necessarily semi-simple.

299

Cohomology

Exercise 7.4.23. Let w be a closed, bi-invariant 3-form on a Lie group G. Then w(X,Y, [z,T]) = w([X,Y], z,T) VX,Y, z,T 6 LG.

Exercise 7.4.24. Let U.) be a closed, bi-invariant 3-form on a compact, semisimple Lie group. (a) Prove that for any X E LG there exists a unique left-invariant form 17X E Q1 (G) such that lixw)lY»Zl = *7xl[Y»Z])Moreover, the correspondence X »-> 17X is linear. Hint: Use Hl(Gl = H2(G) = 0. (b) Denote by A the linear operator LG -> LG defined by

f SO(3) described in the Subsection 6.2.1. Pay very much attention to the various constants. 7.4.4

Invariant forms on Grassmannians and Weyl's integral formula

We will use the results of Subsection 7.4.2 to compute the Poincaré polynomial of the complex Grassmannian GrkllCn). Set H = n - k. As we have seen in the previous subsection, the Grassmannian Grk(C"') is a symmetric space with symmetry group U(n). It is a complex manifold so that it is orientable (cf. Exercise 3.4.13). Alternatively, the orientability of GrkllCn) is a consequence of the following fact.

Exercise 7.4.26. If M is a homogeneous space with connected isotropy groups, then M is orientable. We have to describe the U(n)-invariant forms on GrkllCnl. These forms are completely determined by their values at a particular point in the Grassmannian. We choose this point to correspond to the subspace S0 determined by the canonical inclusion (Ck --> Q" . The isotropy of SO is the group H = U(k) X U(€). The group H acts linearly on the tangent space Vo = TSO Grk lm). If w is an U (n)-invariant form, then its restriction to V0 is an H-invariant skew-symmetric, multilinear map Vo

X

X

V0->R.

300

Lectures on the Geometry of Manzfolds

Conversely, any H-invariant element of A' Vo* extends via the transitive action of U(n) to an invariant form on Grk((C"). Denote by Aivw the space of H-invariant elements of A' V0*. We have thus established the following result.

Proposition 7.4.27. There exists an isomer rphism of graded R-algebras: H. We want to determine the Poincaré polynomial of the complexified Z-graded space, A'nv

(X)

(I

to dimqj Ajnv (X) (C

Pkg(t)

PGrk l(C'n,) (t)

.

j

Denote the action of H on V0 by

H 9 h »-> Th 6 Aut (Vo). Using the equality (3.4.10) of Subsection 3.4.4 we deduce

pklfl

| detmvo + tT,,)l2dvH(h),

(7.4.2)

H

where dVH denotes the normalized bi-invariant volume form on H. At this point, the above formula may look hopelessly complicated. Fortunately, it can be dramatically simplified using a truly remarkable idea of H. Weyl. Note first that the function

H 9 h »-> (/)(h) = ldetl0v0 + tT,,)l2 is a class function, i.e., up(ghg-1) = (,o(h), Vg, h E H. Inside H sits the maximal torus 'IT = Tk X Te, where

To

{ drag (61917

•

. . ,eW'°) E U ( k ) } andbT

= {dig

l€"1¢1

, . . . , e 1305e ) G

vi.

Each h E U ( k ) X U(€) is conjugate to diagonal unitary matrix, i.e., there exists g E H such that ghg'l 6 'Iii

We can rephrase this fact in terms of the conjugation action of H on itself

C :H

H -> H (9,h)*>C9(I1) gag The class functions are constant along the orbits of this conjugation action, and each such orbit intersects the maximal torus T. In other words, a class function is completely determined by its restriction to the maximal torus. Hence, it is reasonable to expect that we ought to be able to describe the integral in (7.4.2) as an integral over 'll`. This is achieved in a very explicit manner by the next result. Define As, : Tk -> (C by (610'L _ e ) , i As (91, . X

,@'°)

H

l LT GO 5G/'lr -2

£G»

can be rewritten as DH20q(x ® Y) = Ad90(Adt0 1 - ]1)Y + Ad90x, or, in block form, Dmoq

Ad

0

0

0

go

Adt-1 o

l£c/wr

The linear operator Adg is an m-orthogonal endomorphism of LG, so that det Adg - i t . On the other hand, since G = U(k) is connected, det Adg = det Ad1 = 1. Hence,

G).

det D20q = detlAdt-1

= LT

Now observe that LE/T

is equal to

{ X € u ( / ¢ l ; Xii = 0 v j = 1 , . . . , k } C _ H l k ) = £ G Given t = d i g (exp(f£911, . . . ,exp(i9/")) G To C U ( k ) , we can explicitly compute the eigenvalues of Adt-1 acting on LE/T. More precisely, they are {€XP('i(9z-9j)) ; 1
l C n

1

n+ 1

Ln( f)

Fofo),

L(f)

Tfdt.

We have to prove that

L(f) f i X .

lim Ln( f)

n->oo

(7.4.4)

It suffices to establish (7.4.4) for any f Q S, where S is a subset of X spanning a dense subspace. We let S be the subset consisting of the trigonometric monomials

@(@1, . . . , et) = exp(i6,7~

Geometrically, the partitions in T o are precisely those partitions whose Young diagrams fit inside a E X k rectangle. If A is such a partition then the Young diagram of the complementary of X is (up to a 180° rotation) the complementary of the diagram of A in the E X k rectangle (see Figure 7.5). For a proof of Lemma 7.4.35 we refer to [88], Section 1.4, Example 5. The true essence of the Schur polynomials is however representation theoretic, and a reader with a little more representation theoretic background may want to consult the classical reference [87], Chapter VI, Section 6.4, Theorem V, for a very exciting presentation of the Schur polynomials, and the various identities they satisfy, including the one in Lemma 7.4.35. Using (7.4.5), Lemma 7.4.35, and the definition of the Schur polynomials, we can describe the Poincaré polynomial of Grk (@") as 2

p t )

l I I

. l T k x T / 3 lApP/

tIAla$+(5k(§laX+5£(§)

CUT

/\

do.

The integrand in (7.4.6) is a linear combination of trigonometric monomials a11 e11 e , where the T'-S and S-S are nonnegative integers.

---

(7.4.6)

e

. 5`k

310

Lectures on the Geometry of Manzfolds

Note that if A, ,LL Q 9"k,e are distinct paltitions, then the terms "$\+al§) and have no monomials in common. Hence To

61/1+6(§)

0§+6(§)0N+6(§) df = 0.

Similarly, TO

&x+5(Q) Ab. In other words, S associates to each open set an Abelian group S(U), and to each inclusion U -> V, a group morphism ToU : sly) -> S(U) such that, if U *-> V -> W, then = TvU o , If s 6 S(V) then, for any U --> V, we set

g

S

lu= t~v"(s) 6 so).

I f f 6 S(U) then we define domf := U. The presheaves of rings, modules, vector spaces are defined in an obvious fashion. Example 7.5.1. Let X be a topological space. For each open set U C X we denote by C'(U) the space of continuous functions U -> R The assignment U »-> C'(U) defines a presheaf of R-algebras on X. The maps ToU are determined by the restrictions IN: C'(V) ->

co). If X is a smooth manifold we get another presheaf U +-> COO(U). More generally, the differential forms of degree k can be organized in a presheaf Qk (s). If E is a smooth vector bundle, then the E-valued differential forms of degree k can be organized as a presheaf of vector spaces

U »-> Q':=E(U) = Q'°(EIU). If G is an Abelian group equipped with the discrete topology, then the G-valued continuous functions C'(U, G) determine a presheaf called the constant G-presheaf which is denoted by Qx

-

Definition 7.5.2. A presheaf S on a topological space X is said to be a sheaf if the following hold. (a) If (UQ) is an open cover of the open set U, and f , 9 € 5 ( U ) satisfy

fIUO,=9IU¢,»VCY,

then f = 9If (b) (UQ) is an open cover of the open set U , and to 6 5(Ua) satisfy

flue=fnu

$U5

VUo, (W UB ,Q (D

then there exists f 6 S(U) such that, f Ina= fo,'v'a. Example 7.5.3. All the presheaves discussed in Example 7.5.1 are sheaves.

Example 7.5.4. Consider the presheaf S over R defined by

S(U) := continuous, bounded functions f : U -> R. We let the reader verify this is not a sheaf since the condition (b) is violated. The reason behind this violation is that in the definition of this presheaf we included a global condition namely the boundedness assumption.

314

Lectures on the Geometry

of Manzfolds

Definition 7.5.5. Let X be a topological space, and R a commutative ring with 1. We equip R with the discrete topology (a) A space of germs over X ( " s p a c e étalé " in the French literature) is a topological space 8, together with a continuous map 'IT : 8 -> X satisfying the following conditions. (al) The map

'IT is a local homeomoiphism that is, each point e 6 8 has a neighborhood U such that 'IT In is a homeomorphism onto the open subset 7r(U) C X. For every ac 6 X, the set fix := 7r"1(a3), is called the stalk at ac. (a2) There exist continuous maps

R x 8->€, and +:{('u,,v)€€

X

8; tr(u)=7r(u)

->€,

such that V T G R , 'v/:I:6X, V'u,,v€€,,, we have 1°-u6€a13, u + ' v € € : 1 3 ,

and with respect to the operations +, ~, the stalk

is an R-module, Van

ex

(b) A section of a space of germs 7r : 8 -> X over a subset Y C X is a continuous function S : Y -> 8 such that s(y) 6 So, /y 6 Y. The spaces of sections defined over Y will be denoted by 8(Y).

Example 7.5.6 (The space of germs associated to a presheaf). Let S be a presheaf of Abelian groups over a topological space X. For each :u 6 X define an equivalence relotion Nm on

U so),

U runs through the open neighborhoods of X ,

U 81:

by

f

Nx

g ~:

The equivalence class of f 6 at as. Set So

'»

EI open U 9 :u such that

Una: S(U)

flu= g l u .

is denoted by [f]x, and it is called the germ of

3 :=

{ [ f ] x ; d 0 > f 9:6 }» and

f

U So.

xiX

There exists a natural projection 7T : 3 -> X which maps [fix to ac. The "fibers" of this map are 1r"1(a:) = Sac - t h e germs a t e 6 X. Any f 6 S(U) defines a subset

co.

Up={[f]u;u€U We can define a topology in 3 by indicating a basis of neighborhoods. A basis of open neighborhoods of [f]as 6 S is given by the collection

{UQ ;

U 9

My

g

E

S(U) [9]w

i

lm }~

315

Cohomology

We let the reader check that this collection of sets satisfies the axioms of a basis of neighborhoods as discussed e.g. in [77]. With this topology, each f 6 S(U) defines a continuous section of 'IT over U

[ f ] : U9'u,»->[f]»,,, €Su. Note that each fiber S ac has a well defined structure of Abelian group

[fly

-|-

[g],, = [(f + g ) lu]x U 9 an is open and U C dom O doing.

(Check that this addition is independent of the various choices.) Since 'IT : f ( U ) -> U is a homeomorphism, it follows that 'IT : S -> X is a space of germs. It is called the space of germs associated to the presheaf S.

If the space of germs associated to a sheaf S is a covering space, we say that S is a sheaf of locally constant functions (valued in some discrete Abelian group). When the covering is trivial, i.e., it is isomorphic to a product X X { discrete set 1, then the sheaf is really the constant sheaf associated to a discrete Abelian group. Example 7.5.7 (The sheaf associated to a space of germs). Consider a space of germs a l> X over the topological space X. For each open subset U C X, we denote by a (U) the _ space of continuous sections U -> 8. The correspondence U »-> 8(U) clearly a sheaf. 8 is called the sheaf associated to the space of germs.

Proposition 7.5.8. (a) Let 8 l> X be a space of germs. Then

3 = 8. _

(b) A presheafS over the topological space X is a sheaf zuzana' only 1f3

S.

Exercise 7.5.9. Prove the above proposition. Definition 7.5.10. If S is a presheaf over the topological space X , then the sheaf 3 is called the sheaf associated to S , or the sheafification of S.

Definition 7.S.11. (a) Let A morphism between the (pre)sheaves of Abelian groups (modules etc.) S and S over the topological space X is a collection of morphisms of Abelian groups (modules etc.) he : S(U) -> SIU), one for each open set U C X, such that, when V C U, we have he o TuV = ,=Up o he. Above, TuV denotes the restriction morphisms of S, while the ,:UV denotes the restriction morphisms of 3. A morphism h is said to be injective if each he is injective. (b) Let S be a presheaf over the topological space X. A sub-presheaf of S is a pair (T, 2), where 'J' is a presheaf over X , and 'L : 'J' -> S is an injective morphism. The morphism 'L is called the canonical inclusion of the sub-presheaf.

316

Lectures on the Geometry

of Manzfolds

Let h : S -> 'J' be a morphism of presheaves. The correspondence

U »-> kerhu C 5(U>, defines a presheaf called the kernel of the morphism h. It is a sub-presheaf of S.

Proposition 7.5.12. Let h : S -> 'J' be a morphism of presheaves. If both S and 'J' are sheaves, then so is the kernel of h. The proof of this proposition is left to the reader as an exercise.

Definition 7.5.13. (a) Let S 1 X (i = 0, 1) be two spaces of germs over the same topological space X. A morphism of spaces of germs is a continuous map h : 80 -> 81 satisfying the following conditions. (al) tf1 o h = 7r0, i.e., h(tr0 1(x)) C Tr1 1(a:), Va: 6 X. (a2) For any so 6 X, the induced map ha, : 7r0 1(:z:) -> w1 1(:I:) is a morphism of Abelian groups (modules etc.).

The morphism h is called injective if each

hrr:

is injective.

(b) Let 8 l> X be a space of germs. A subspace of germs is a pair (313), where 3' is a space of germs over X and j : 3' -> 8 is an injective morphism.

Proposition 7.5.14. (a) Let h : 80 -> 81 be a morphism between two spaces of germs over X . Then Algol 3 X is a space of germs over X called the image of h. It is denoted by lm h, and it is a subspace of 81. Exercise 7.5.15. Prove the above proposition. Lemma 7.5.16. Consider Iwo sheaves S and 'J', and let h : S -> 'T be a rnorpnism. Then h induces a morphism between the associated spaces of germs h : S -> 'J'. The definition of /E should be obvious. If f 6 So, and x 6 U, then he) = [h(fl]m, where h ( f ) is now an element of 'J'(U). We let the reader check that l is independent of the various choices, and that it is a continuous map S -> 'ii with respect to the topologies described in Example 7.5.6. The sheaf associated to the space of germs lm /E is a subsheaf of 'T called the image of h, and denoted by lm h.

Exercise 7.5.17. Consider a morphism of sheaves over X, h : S -> 'J'. Let U C X be an open set. Show that a section g 6 'J'(U) belongs to ( l m h)(U) if an only if, for every as 6 X, there exists an open neighborhood Vx C U, such that g I v h ( f ) , for some

f 6 S(V).

317

Cohomology

" D u e to Proposition 7.5.8(a), in the sequel we will make no distinction between sheaves and spaces of germs.

Definition 7.5.18. (a) A sequence of sheaves and morphisms of sheaves,

-> sn 3

Sn-I-1

in

41

1

Sn-l-2

->

7

is said to be exact if lm hn - her hn+1, 'v'n. (b) Consider a sheaf S over the space X. A resolution of S is a long exact sequence 0 +5+80@>51@>~-~->s"s8H+1->

Exercise 7.5.19. Consider a short exact sequence of sheaves 0 - > S _ 1 ->S0->S1 ->0.

For each open set U define S(U) = S0(U)/S_1(U). (a) Prove that U +-> S(U) is a presheaf. (b) Prove that 81 = S = the sheaf associated to the presheaf S.

~

Example 7.5.20 (The DeRham resolution). Let M be a smooth n-dimensional manifold. Using the Poincaré lemma and the Exercise 7.5.17 we deduce immediately that the se-

quence 0

d d RMQ0M->Q}W->

mM denotes the sheaf of k-forms on M while d

is a resolution of the constant sheaf Be, denotes the exterior differentiation.

7.5.2

d

->Q"M-> 0

beef eohomology

Let U »-> S(U) be a presheaf of Abelian groups over a topological space X. Consider an open cover u = ( U S A o f X . A simplex associated to 'LL is a nonempty subset A = { O d 0 , . . . , C ¥ q C A , such that q

UOH ¢ (D.

II

UA

0

We define d m A to be one less the cardinality of A, dim A q-dimensional simplices is denoted by Nq(U). Their union,

III

1. The set of all

U NG (U) q

is denoted by N(U), and it is called the nerve of the cover. For every nonnegative integer q we set Aq := {0, . . . , q}, and we define an ordered q-simplex to be a map O' : Aq -> A with the property that o(Aq) is a simplex of N(U), possibly of dimension < q. We will denote by (U) the set of ordered q-simplices, and

31

318

Lectures on the Geometry

we will use the symbol 0(k) Oak. Define

Q07

• • •

705q

Cq(S, U)

of Manzfolds

to denote an ordered q-simplex 0 : Aq -> .A such that

So, H ueJ Iv,

(x O 5(c),0) q

q+1

Z(

(-1)J' j

0

j

q

(-1l1"(0» (As,

(-1>i j

l )'"(c» 9é"+1h,(@)) Iv,

k

0

: j a

w\j,llj,

,llql)

Iv,

0

q

l-1)Z_.l_1(C, (Au,

-|-

w\j»llj,

»lik,

7

Nql) Iv

€=j q

(-w j

(-1l/°(0» (As, k

0

: p a

7 / \ j 7 / j ?

)la»ql) Iv,

0

q

l-1l@-}-1(07 I Q ,

-I-

7 ) \ j 7 / / j 7

»lik,

:Aj-lafUja

),Uql)

7

e=j+1 q

.

(-1)"{

-|-

ac, ( m

•

j=0

+(c, ( m . . .

7

* j » I'Lj-!r-17

7

llql)

Iv

IV

/Jql)

Iv

321

Cohomology

The last term is a telescopic sum which is equal to

(C, (}\0 v

• •

-,As Iv

-(C, ILLQ, . . 7 l L q l ) •

lo= (Q*c,0)

(1°*c,0l.

If we change the order of summation in the first two term we recover the term (-5Xe,o). Part (c) is left to the reader as an exercise. We now have a collection of graded groups { H°(U,S) ; U - open cover of X }, and morphisms

,Qi . H,lib 5) _>

V) 5%

H. I

•

u '< V

7

such that, u Zu

W II and Zu

W 3 ? / V

'V

0?/LL,

whenever U -< V -< W. We can thus define the inductive (direct) limit

H°(x, go

:

lim H°('L(, S).

u

The group H' (X, S) is called the beef cosmology of the space X with coefficients in the pre-sheaf S. Let us briefly recall the definition of the direct limit. One defines an equivalence relation on the disjoint union

H I-I'(U,S), 'II

by

H.

(U)9f

g 6 I-I'(\7)

is

> 2w>'L1,v= @uWf=@9.

We denote the equivalence class of f by j. Then

iium H' (up

r

Note that we have canonical morphisms, nu : H' (U,

so -> H. (x,s).

Example 7.5.26. Let S be a sheaf over the space X. For any open cover 'LL = (Up)» a 0-cycle subordinated to U is a collection of sections f 6 S(Ua) such that, every time Ua re UB go (Z), we have JU0

IU0BI

f 5IUo0

According to the properties of a sheaf, such a collection defines a unique global section f 6 S(X). Hence, H0(X,S) = S(X).

~

322

Lectures on the Geometry

of Manzfolds

Proposition 7.5.27. Any morphism of pre-sheaves h : S0 -> 51 over X induces a morphis in cosmology

h* Sketch of proof.

H.

(X>50)

-)

H' (X, 81).

Let U be an open cover of X . Define

h : cq -> c'1 H°(U, 81) which commutes with the refinements limits.

eX.

The proposition follows by passing to direct

Theorem 7.5.28. Let p 0 - > 8 _ 1 l>S0->S1 ->0,

be an exact sequence of sheaves over a paracompact space X . Then there exists a natural long exact sequence J*

6* ->H H co(u,s) -> 0.

We obtain a long exact sequence in cosmology

->

H Hq(u,50) -> H'1('u,5> -> H+11(u,x> ->

Passing to direct limits we get a long exact sequence

-> Hq(X,S_1) -> Hq(X,S0) -> Hq(X,S) -> Hq-l-1(x,5_1) -> To conclude the proof of the proposition we invoke the following technical result. Its proof can be found in [ l 19].

Lemma 7.5.29. If two pre-sheaves S, S' over a paracompact topological space X have isomorphic associated sheaves then

H.

(x,5) 'E H ' ( x , s'

Definition 7.5.30. A sheaf S is said to be fine if, for any locally finite open cover U (UQMQA there exist morphisms ha : S -> S with the following properties.

323

Cohomology

(a) For any $ 6 .A there exists a closed set CO( C UQ, and ha(sas) = 0 for ac Q Co, where So denotes the stalk of S at cz: 6 X . (b) Za ho is. This sum is well defined since the cover U is locally finite.

Example 7.5.31. Let X be a smooth manifold. Using partitions of unity we deduce that the sheaf of of smooth k-forms is fine. More generally, if E is a smooth vector bundle over X then the space Q; of E-valued k-forms is fine.

Proposition 7.5.32. Let S be a fine sheaf over a paracompact space X . Then HQ (X, S) Oforq > 1.

r\/

Proof. Because X is paracompact, any open cover admits a locally finite refinement. Thus, it suffices to show that, for each locally finite open cover U = l U a l a € A » the cohomology groups HQ (U, S) are trivial for q > 1. We will achieve this by showing that the identity map C'q(U, S) -> C"I(U, S) is cochain homotopic with the trivial map. We thus need to produce a map Xi :

0 -> cQ-1(u,8l,

such that XC]-I-15q

-|-

5q-1Xq

= JL

(7.5.l)

Consider the morphisms ho, : S -> S associated to the cover U postulated by the definition of a fine sheaf. For every a 6 A, O' 6 Nqwl, and every f 6 C'q(U,S), we construct Ito, ( f ) , of) 6 sl(/'O'l as follows. Consider the open cover of Uo-

==

{ V = U , m Uo, w uO' \ Co, }, (suppla C ca). lvfww= 0 and, according to the axioms of a sheaf, the section ha(f IV) can

Note that haf be extended by zero to a section (t0,(f), 0) 6 s(UO'). Now, for every f 6 C'q(U, S), define

xqf 6 C'q-1(U,S) by

..II

(Xqlfl) 0)

RMQ0Mi>Q}W->

Exercise 75.36. Describe explicitly the isomorphisms

HW)

~=

H1(m,Bm) and H2(M)

2

H2(m,Bm).

Remark 7.5.37. The above corollary has a surprising implication. Since the Cech cohomology is obviously a topological invariant, so must be the DeRham cosmology which is defined in terms of a smooth structure. Hence if two smooth manifolds are homeomorphic they must have isomorphic DeRham groups, even if the manifolds may not be diffeomorphic. Such exotic situations do exist. In a celebrated paper [93], John Milnor has constructed a family of nondiffeomorphic manifolds all homeomorphic to the sphere S7_ More recently, 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. (This is possible only for 4-dimensional vector spaces!) These three mathematicians, J. Milnor, S. Donaldson and M. Freedman were awarded Fields medals for their contributions.

325

Cohomology

Theorem 7.5.38 (Leray). Let M be a smooth manifold and U

ofM,

IUoQQA

a good cover

i.e., dim M O'

Then

Ho (U, Be )

f\J

H.

(m).

Proof. Let Zn denote the sheaf of closed k-forms on M. Using the Poincaré lemma we deduce 2'"(u;) = dak'-1lUO'). We thus have a short exact sequence

0 -> cqlu, zkl -> Cow, Qk-1)

c'1(u, z'*) -> 0.

Using the associated long exact sequence and the fact that Qk-l is a fine sheaf, we deduce as in the proof of the abstract DeRham theorem that

iv

H1(u, z, 1 ,

Hence, for any good cover U

H.(m,Qm)

H.

Remark 7.5.40. A combinatorial simplicial complex, or a simplicial scheme, is a collection UC of nonempty, finite subsets of a set V with the property that any nonempty subset B of a subset A 6 fK must also belong to the collection UC, i.e.,

A

ex,

B CA, B ;é(D-» B

ex.

The set V is called the vertex set of UC, while the subsets in U( are called the (open) faces) of the simplicial scheme. The nerve of an open cover is an example of simplicial scheme. To any simplicial scheme UC, with vertex set V, we can associate a topological space l5 R with the property that f (Fu) 0, for all but finitely many 'u's. In other words,

RV

R. 'UGV

326

Lectures on the Geometry

of Manzfolds

Note that RV has a canonical basis given by the Dirac functions 61, : V -> R, 5U,u)

1

u

'U

0 u ¢

'U.

We apologize RV by declaring a subset C 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. For any face F C UC we denote by As C RV the convex hull of the set {6'U§ '06 F }. Note that As is a simplex of dimension IFI - 1. Now set

l5 G. The Lie group G operates on its Lie algebra g via the adjoint action

Ad : G -> GL(g),

g »-> Ad(g) E GL(g),

where Ad(glx :: gxg

1 7

VX E g, g E G.

We denote by Ad(P) the vector bundle with standard fiber g associated to P via the adjoint representation. In other words, Ad(P) is the vector bundle defined by the open cover (Ua), and gluing cocycle Adlgog) : UQ5 -> GLIEI. 329

330

Lectures on the Geometry

of Manzfolds

The bracket operation in the fibers of Ad(P) induces a bilinear map

1-, .1 : Q'*(Ad(p) )

X

Q'(Ad(p>) -> Qk+"(Ad(p) )

7

defined by k

(wk A s ) (X) [X,y], > V @ (X) Y] Y 7 for all wk Q Qklml, 77 Q Q0( Ad(p)). and X Q* Exercise 8.1.1. Prove that for any inv dlpl) the following hold. w

(X)

"( 2Q Q

7

[(»J,,,] =

[ 7]7] -

[[W77

(8.l.1)

lA l[77

I.l?7 (-llloJ

-Q

[w, (I5],77] -|- l - l

-

I I Ad(P) ), [ , ] ) is a super

In other words, Q'

)|w|

.

(8.l.2)

"*J],

|¢|[

We

je

[WI 1.

(8.l.3)

algebra.

Using Proposition 3.3.5 as inspiration we introduce the following fundamental concept.

Definition 8.1.2. (a) A connection on the principal bundle P defined by the open cover "LL l e a ) a € A 9 and the gluing cocycle .Quo : Uog -> G is a collection 2

AA 6 Q1(U> (X)

Q,

satisfying the transition rules Aglilil

g 5(w)d g@(w ) + g ; ( x ) A ( w ) g @ ( x )

- ( dg@a(:u) log 1 (so) + g@a(2¢)A0(w)g5 1 (Lil, Vcc E Uog. We will denote by A(U, 9-») the set of connections defined by the open cover U and the gluing cocycle g., U.. -> G. (b) The curvature of a connection A 6 All, 9--) is defined as the collection Fa E Q2(Ua) (X) g where

Fa

¢ZAo,

l

-2[A O A o ,]

Remark 8.1.3. Given an open cover (Ua) of M, then two gluing cocycles g,B0 a h

e : U0g

G,

define isomorphic principal bundles if and only if there exists smooth maps Ta:Uo->G,

such that h@o(¢ul

Z

T@(t)9@0(@@)T0(1'l'1> ,'v'0,B, Van E Uo0.

If this happens, we say that the cocycles g., and h., are cohomologous. Suppose (gag), (hog) are two such cohomologous cocycles. Then the maps To induce a well defined correspondence

..

'T : A('L(, g

->A('L(,h

..

331

Characteristic Classes

given by

A("L(, 9--)

EIAaI

I

u'

> ( Bo

== - ( d T ) T

1

+ TAOlTa

1) E Am, h.. .

The correspondence T is a bisection. Suppose 'Ll = (U0,)o€A and V = (Vz l i € I are open covers, and QBQ : Uog -> G is a gluing cocycle. Suppose that the open cover l v i l i € I is finer that the cover q U a ) a € A 9 i.e., there exists a map (p : I -> A such that C

U(f)(i))

W E I.

We obtain a new gluing cocycle Q; : Vi -> G given by ac'

This new cocycle defines a principal bundle isomorphic to the principal bundle defined by the cocycle (gg0). If (Aa) is a connection defined by the open cover (UOf) and the cocycle lg5al» then it induces a connection

Ag'

Ago(i)

Iv

defined by the open cover (W), and the cocycle gin. We say that the connection Aw is the V-refinement of the connection A. The correspondence

'Tau :A(U»g--) -> A(\7,gCD so

7

A+->ALf'

is also a bijection. Finally, given two pairs (open cover, gluing cocycle), (U, g,.) and (V, h,,), both describing the principal bundle P -> M, there exists an open cover W, finer that both U and W, and cohomologous gluing cocycles in, B.. : W., ->G refining g.. and respectively h.,, such that the induced maps

Aw, 9--)

>A(W, §..)

>A(W,l

..


9_1 + Q go

QBQ

_1

QQ

-1

QQBCZQQB -|- QQBBQQQ5'

333

Characteristic Classes

This shows that the collection (Bo) defines a connection on P. (b) We need to check that the forms Fo satisfy the gluing rules

_

FB - 9

where g

-1

For

7

We have

QoB = 9 g 1 -

1

FB

= CMB -|- 51AB» AB]

= d(9-1d9 + Q'1AQ9) Set w

1 + §[9-ldg + g-1AQ9, 9-1d9 + 9-1Aog1-

g-1dg. Using (8.1.2) we get 1

FB

= do + §[W7 W] + d(g

-1

Aug) + [mg

-1

1

-1

Aw] + 519 A y

-1

(8.l.5)

Aw]-

We will check two things. A. The Maurer-Cartan structural equations. 1

dw-|- §[w,w] = 0.

B.

d(gllAag) + [w, 9'1Ao9l

= g`1(dAo)g-

Proof of A. Let us first introduce a new operation. Let M, K) denote the associative algebra of K-valued n X n matrices. There exists a natural operation /\ :

Q'"(u> ®

m, K) x M U )

m, K) -> Qk+'lUo,) ®

®

Q , K),

uniquely defined by (wk®A) A (ne®B)

=(

J A ) ®(A-Bl,

(81.6)

where wk' 6 Q":(U0,), 17e E Qt((/o) and A, B 6

(n,Kl (see also Example 3.3.l2). The space M, K) is naturally a Lie algebra with respect to the commutator of two matrices. This structure induces a bracket K 2 k+ (X) (X) l7 (n,Kl al® )X ) Q) Q

.. 1.§M(u

QIIlf '

SEIne

,

In

defined as in (8.1.1). A very simple computation yields the following identity. w / \ 77

l

= 5lW»*7l We

E

QUO)

(X)

QW Kl.

The Lie group lies inside GL(n, K), so that its Lie algebra g lies inside think of the map QoB as a matrix valued map, so that we have do = d(glldg) = (d9'1) /\ do = -(9-1 -do g-1)d9 = -(9-1d9) /\ (g-1d9) 1 w A w(8.1.7) -

§[W»W]-

(8.1.7)

, K). We can

334

Lectures on the Geometry

of Manzfolds

Proof of B. We compute d(g'lAo9)

= (d9llAo) ~g + 9-1ldAa¢l9 + g'1Aod9 = -9-1 - do - 9-1 /\ As ' Q + g`1(dAo)g + (g'1Ao9) /\ g'ldg = - w A g - y + g - M g A w + 9-1ldAa¢l9 (8.1.7)

=

1

1

- §[W»9 Mag] + §[W»91Ao9] + 9 1(dAo)9

(81.2) - [w',9'1Ao9] -|- g"(dAo)9-

Part (b) of the proposition now follows from A, B and (8.1.5). (c) First, we let the reader check the following identity (8.l.8)

d[(»Jn7] = [dim] + (-1) '"'[w» dt1]» where w,17 E Q*(UO)

(X)

d(Fa)

g. Using the above equality we get 1 lg.1.2l[dAa As] i

§{[dA0,AO,]

Z

[FoHAQI -

[A c g ( h ) :z 9119.

Q

G.

335

Characteristic Classes

The set of gauge transformations forms a group with respect to the operation (SO) ' (Ta) :z

ISQTQI-

..

this group. One can verify that if P is described the (open cover, We denote by §(U, g gluing cocycle)-pair (V, h ».) , then the groups §("Ll, g..I and 907% h ».) are isomoiphic. We denote by glpl the isomorphism class of all these groups. The group §('L[, g acts on A('Ll, g according to

..

9(p)

X

..

A(p) 9 (T, A) »-> TAT-1 ¢ = ( - ldT)Ta 1 -|- T A a T

1

) Q A(Pl.

The group glpl also acts on the vector spaces Q°(Ad(p)) and, for any A 6 A(P), T E glpl we have FTAT-1

= TFAT-1

We say that two connections A0, A1 E A(P ) are gauge equivalent if there exists T 6 9(P) such that A1 = TA0T-l. 8.1.2 G-vector bundles

Definition 8.1.7. Let G be a Lie group, and E -> M a vector bundle with standard fiber a vector space V . A G-structure on E is defined by the following collection of data. (a) A representation p : G -> GL(V). (b) A principal G-bundle P over M such that E is associated to P via p. In other words,

there exists an open cover (Ua) of M, and a gluing cocycle gal? : Uog -> G, such that the vector bundle E can be defined by the cocycle

p(9o@l : Uow -> G L W ) . We denote a G-structure by the pair (P, p). Two G-structures (Pi, p on E, i = l , 2, are said to be isomorphic, if the representations Pt are isomorphic, and the principal G-bundles P are isomorphic. Example 8.1.8. Let E -> M be a rank T' real vector bundle over a smooth manifold M. A metric on E allows us to talk about orthonormal moving frames. They are easily produced from arbitrary ones via the Gramm-Schimdt orthonormalization technique. In particular, two different orthonormal local trivializations are related by a transition map valued in the orthogonal group OU), 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(rl. In other words, a metric on a bundle induces an O(7") structure. The representation p is in this case the natural injection O(r) GL(1°, R). Conversely, an O(7"l structure on a rank r real vector bundle is tantamount to choosing a metric on that bundle. Similarly, a Hermitian metric on a rank k complex vector bundle defines an um)structure on that bundle.

336

Lectures on the Geometry

of Manzfolds

Let E = (P, p, V) be a G-vector bundle. Assume P is defined by an open cover ( U I , and gluing cocycle QoB

: U0g -> G.

If the collection { A a E Q1 ( U ) (X) g} defines a connection on the principal bundle P, then the collection p*(AO) defines a connection on the vector bundle E. Above, p* : g -> End(V) denotes the derivative of p at 1 E G. A connection of E obtained in this manner is said to be compatible with the G-structure. Note that if F(A0,) is the curvature of the connection on P, then the collection p* (F(AO)) coincides with the curvature F(p* (Ao)) of the connection p* (Aa). For example, a connection compatible with some metric on a vector bundle is compatible with the orthogonal/unitary structure of that bundle. The curvature of such a connection is skew-symmetric which shows the infinitesimal holonomy is an infinitesimal orthogonal/unitary transformation of a given fiber. 8.1.3 Invariant polynomials Let V be a vector space over K

R, C. Consider the symmetric power sklv*l C (v*)®'" 7

which consists of symmetric, multilinear maps

9o:Vx--»xV->K. Note that any SO E 5I M be a principal G-bundle over the smooth manifold M . Assume P is defined by an open cover (UQ) and a gluing cocycle QoB

: U0g -> G.

Pick A E A(P) defined by the collection As, E Q1(Ua) (X) g. Its curvature is then defined by the collection

Fa

dAa +

I

1 [AORTAOf]. 2

Given (Is E MG), we can define as in the previous section (with A P(Fal :

(z5(Fo 7

7

Qeven

(UP)»

V

9)

F

-(/)(lA )Ft])Ft9 . . . , F t , C ' 0 ) - - - - - ( 2 5 ( F t , . . . , F t , [ A t , F t ] , C ' O , ) (8.1.10) i

¢(Ft»..- ,Ft,dC'O,+[At,Co]) = ¢ ( d C a + [ A t , C a ] , F t , . . . , F t ) .

Hence 1

¢(F1) - N, and any principal G-

bundle P -> N.

343

Characteristic Classes

Hence, the Chern-Weil construction is just a method of producing G-characteristic classes valued in the DeRham cosmology.

Remark 8.1.20. (a) We see that each characteristic class provides a way of measuring the nontriviality of a principal G-bundle. (b) A very legitimate question arises. Do there exist characteristic classes (in the DeRham cosmology) not obtainable via the Chern-Weil construction? The answer is negative, but the proof requires an elaborate topological technology which is beyond the reach of this course. The interested reader can find the details in the monograph [98] which is the ultimate reference on the subject of characteristic classes. (b) There exist characteristic classes valued in contravariant functors other then the DeRham cohomology. E.g., for each Abelian group A, the tech cosmology with coefficients in the constant sheaf _A defines a contravariant functor H°(-, A), and using topological techniques, one can produce H' (-, A)-valued characteristic classes. For details we refer to [98], or the classical [124]. 8.2

Important examples

We devote this section to the description of some of the most important examples of characteristic classes. In the process we will describe the invariants of some commonly encountered Lie groups.

8.2.1 The invariants of the torus T"

--

The n-dimensional torus T" = U(l) X - X U(l) is an Abelian Lie group, so that the adjoint action on its Lie algebra t" is trivial. Hence

I.

(T*) = 5°((i")*)_ In practice one uses a more explicit description obtained as follows. Pick angular coordinates 0 < H'L < 2tr, l < i < n, and set 1 . do] to 2tr'i •

The acts form a basis of (tn * and now we can identify /

7

I . lTN)£R[$1,

7

fn]-

8.2.2 Chem classes

Let E be a rank T' complex vector bundle over the smooth manifold M. We have seen that a Hermitian metric on E induces an U (T)-structure (P, p), where p is the tautological representation p : U(1°) GL(1°,lC).

Exercise 8.2.1. Prove that different Hermitian metrics on E define isomorphic U(7°)structures.

344

Lectures on the Geometry

of Manzfolds

Thus, we can identify such a bundle with the tautological principal U (T)-bundle of unitary frames. A connection on this U (T)-bundle is then equivalent with a linear connection V on E compatible with a Hermitian metric ( , ) , i.e.,

..

A{(VxU,U)

Vi (Au, u)

-|-

l'u,, V X v } } ,

VA 6 (C, 'u,,v 6 COO(E), X 6 Vect ( M ) . The characteristic classes of E are by definition the characteristic classes of the tautological principal U(1°)-bundle. To describe these characteristic classes we need to elucidate the structure of the ring of invariants I°(U(r)). The ring I'(U(r)) consists of symmetric, 1°-linear maps TXT-1 E QU), T E

up.

It is convenient to identify such a map with its polynomial form

P¢(X) = d ) ( X , . . . , X ) . The Lie algebra _u(1°) consists of T' X T' complex skew-hermitian matrices. Classical results of linear algebra show that, for any X 6 _u(r), there exists T 6 U(r), such that TXT-1 is diagonal

TXT-1

idiag(/\1, . . . , /\v*).

The set of diagonal matrices in Q ) is called the Carton algebra of_u(r), and we will denote it by €2(1,°). It is a (maximal) Abelian Lie subalgebra of QU). Consider the stabilizer

..II

SU(7`)

{ T 6 U(1~) ; TXT-1 = x, VX

e @(.~)}»

and the normalizer NU(r)

:=

{T E UITI;

T€!(1°)T-1 C @Q(1°)}°

The stabilizer SU(1°) is a normal subgroup of NU(,.°, so we can form the quotient

Wu(¢~)

2 = Nu(1~)/51/('~l

l I

17

II

..

called the Weyl group o f U ( r l . As in Subsection 7.4.4, we see that the Weyl group is isomorphic with the symmetric group S because two diagonal skew-Hermitian matrices are unitarily equivalent if and only if they have the same eigenvalues, including multiplicities. We see that P is Ad-invariant if and only if its restriction to the Cartan algebra is invariant under the action of the Weyl group. The Carman algebra is the Lie algebra of the (maximal) torus T" consisting of diagonal unitary matrices. As in the previous subsection we introduce the variables 2'/Ti

dO j .

The restriction of P to €_u(#°) is a polynomial in the variables all, . . . , a s . The Weyl group S permutes these variables, so that P¢ is Ad-invariant if and only if PIbléli1, . . . , SET is a

345

Characteristic Classes

symmetric polynomial in its variables. According to the fundamental theorem of symmetric polynomials, the ring of these polynomials is generated (as an R-algebra) by the elementary

ones

Cr

£111

337r

Thus

I.

(UW)

R[C1, CQ,

c

1.

In terms of matrices X 6 QU) we have Co (Xltk

det

]l-

k

,X 2'rrz

6 I . (U(7°))[¢].

The above polynomial is known as the universal rank T' Chem polynomial, and its c o e f f i cients are called the universal, rank 7" Chem classes. Returning to our rank T' vector bundle E, we obtain the Chem classes of E

== co(F(Vll G H2"=(m),

bE) and the Chem polynomial of E

(MIM-

Ct

..

Above, V denotes a connection compatible with a Hermitian metric (

7

) on E, while

F(V) denotes its curvature. Remark 8.2.2. The Chern classes produced via the Chern-Weil method capture only a part of what topologists usually refer to characteristic classes of complex bundles. 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. The following facts are essentially due to Kodaira and Spencer [81], see also [55] for a nice presentation. Let L -> M be a smooth complex Hermitian line bundle over the smooth manifold M. Upon choosing a good open cover (Ua) of M we can describe L by a collection of smooth maps zag : Uog -> U(1) = S1 satisfying the cocycle condition

~

II W # -

ZQBZBWZWQ

(8.2. 1)

If we denote by COO(-, S1) the sheaf of multiplicative groups of smooth S1-valued functions, we see that the family of complex line bundles on M can be identified with the Czech group H1 (M, COO(-, g11l. This group is called the smooth Picard group of M. The group multiplication is precisely the tensor product of two line bundles. We will denote it by PicOO ( M ) . If we write zag = exp(2'rri9o@) (93O = -Hog 6 COO(Uo5,]R)) we deduce from (8.2. 1) that VUQB7 ¢ (Z) Hog

-|-

HM

-|-

H

= "QM E Z.

346

Lectures on the Geometry

It is not difficult to see that, VU

of Manzfolds

¢ QL we have

"B76 - "OMS -|- "0M36

- "QM

i

0.

In other words, the collection "QM defines a Czech 2-cocycle of the constant sheaf Z. On a more formal level, we can capture the above cocycle starting from the exact sequence of sheaves 0 -> Z--> C°°(-JR)

exp(2'rr"i~)

>

COO 0.

The middle sheaf is a fine sheaf so its cosmology vanishes in positive dimensions. The long exact sequence in cohomology then gives 0 -> Pic" (m) i> H2(m, z ) -> 0.

The cocycle ( "QM ) represents precisely the class 6(L). The class 6(L), L 6 Pic°° (M) is called the topological first Chern class and is denoted by etlop (L). This terminology is motivated by the following result of Kodaira and Spencer, [81 ] : The image of Diop ( L ) in the DeRham chorology via the natural morphism

~ HERIMI

H*(M,Z) -> H*(M,lR) =

coincides with the ]inst Chem class obtained via the Chem-Weil procedure. The Chern-Weil construction misses precisely the torsion elements in H2(M, Z). For example, if a line bundle admits a flat connection then its first Chern class is trivial. This may not be the case with the topological one, because line bundle may not be topologically trivial.

8.2.3 Ponhyagin classes Let E be a rank T' real vector bundle over the smooth manifold M. An Euclidean metric on E induces an O(r) structure (P, p). The representation p is the tautological one p : O(1°) GL(7°,R).

Exercise 8.2.3. Prove that two metrics on E induce isomorphic OU) -structures. Hence, exactly as in the complex case, we can naturally identify the rank 1°-real vector bundles equipped with metric with principal O(r)-bundles. A connection on the principal bundle can be viewed as a metric compatible connection in the associated vector bundle. To describe the various characteristic classes we need to understand the ring of invariants

I.

(O(1"llAs usual, we will identify the elements of In(O(r)) with the degree k, Ad-invariant polynomials on the Lie algebra Q(1°) consisting of skew-symmetric T' X T' real matrices. Fix P 6 Ik:(O(1°)). Set m = [T/2], and denote by J the 2 X 2 matrix II

..

J

347

Characteristic Classes

Consider the Carton algebra €Q(r)

{A1J® ® AmJ 6 QU); *j 6 ]R}, {A1J® ® AmJ ® 0 6 QU); *j 6 R},

7°

2m

7°

2m

-|-

l

The Cartan algebra 6900 is the Lie algebra of the (maximal) torus RE1 ® ---®R@m 6 O(1°), RE1 ® ® Ram ® HR E OITI, where for each H E [0, 27r] we denoted by RE the 2 .. II

2m

7°

2m -I- l

7

2 rotation

_ ' Q

- cost

RE

X

7°

;;M

sin H

As in Subsection 8.2.1 we introduce the variables 1 27r

»-doj .

"la

Using standard results concerning the normal Jordan form of a skew-symmetric matrix, we deduce that, for every X 6 Q(1°), there exists T 6 O(1") such that TXT-1 G €9(r)° Consequently, any Ad-invariant polynomial on Q(1") is uniquely defined by its restriction to the Caftan algebra. Following the approach in the complex case, we consider

{ T e of); TXT-1 = x,

SO(1~)

Nom

{T

G

Olrl;

VX e

@Q(»~)

7

T € 9 ( t ~ ) T - 1 C @9(»,~)}-

The stabilizer SO('r') is a normal subgroup in N(O(r)), so we can form the Weyl group WO(T)

..

NO(r)/SO(r)°

Exercise 8.2.4. Prove that We(Tl is the subgroup of GL(m, R) generated by the involusons (7

j

(X17'°'7Xi7"°7Xj7"'7

go:

(Xl

7

,.ZUlu,...,

to)

am)

»-> (-17

|

(-17

7 X j 7 ° ' ° 7 X i 7 ' ° ' 7 aim

,-$j,...,

as

.

The restriction of P 6 Ik (O(1°)) to €9(',°) is a degree k homogeneous polynomial in the variables 5131, . . . , :am invariant under the action of the Weyl group. Using the above exercise we deduce that P must be a symmetric polynomial P = P(x1, . . . m), separately even in each variable. Invoking once again the fundamental theorem of symmetric polynomials we conclude that P must be a polynomial in the elementary symmetric ones PI P2 Pm

= Z

Z j"2

2 i I°(50(¢~)) is an isomorphism.

Exercise 8.2.6. Prove the above lemma. The situation is different when 7° is even, T' Qm. To describe the ring of invariants I . (50(2ml), we need to study in greater detail the Cartan algebra €Q(2m)

= { AIJ ®

® AmJ 6

Q }

and the corresponding Weyl group action. The Weyl group Wso(2m)» defined as usual as the quotient Wso(2m) = Nso(2m)l5so(2m) 7

is isomorphic to the subgroup of GL((3Q(2,,,,)) generated by the involutions Oiij

(Al,..,A¢,...,Ai,...,ATTIl»->(A1,...,Aj,...,A,...,ATTI),

349

Characteristic Classes

and 5 :

( u ...

7

All P-> l€1A17

. . swAm), 7

---

where 61, . . . ,am = :l:l, and 6 1 am = l . (Check thi5!) Set as usual Xi := -Ai/27r. The Pontryagin O(2m)-invariants Pj(SU1,

7

:am

.CU

Maj

)2 7

1 pa, Y »-> A is the ideal (YQ - Zm).

Proof of Step 1. Note that W S o ( 2 m ) has index 2 as a subgroup in W 0 ( 2 m l . Thus W s 0 ( 2 m l is a normal subgroup, and

9 The group 9 = {]1, e} acts on I°(SO(2m)) by l€F)l£U1,£C2,

...736i)

I

Fl-£C1,£U2,

am

...

= Ker(]l -

2).

7

and moreover,

I°lo(2mll

-F(-17§627

7

mm

7

350

Lectures on the Geometry

of Manzfolds

For each F 6 I ° ( S O ( 2 m ) ) we define F+ := (II ker(l - 2). Hence F+

P(p1, . . . , p T T I ) .

On the other hand, the polynomial F' Indeed, F


0(2)

+

(8.2.3)

We will discuss these two situations separately.

The complex case.

The sequence in (8.2.2) induces a projective sequence of rings

R 1° (H)-

The elements of her go* C I°(G) are called universal identities. The following result is immediate.

Proposition 8.3.2. Let P be a principal G-bundle which can be reduced to a principal H-bundle Q. Then for every 17 6 her go* we have

17(P) = 0 in H *.

Proof. Denote by LG (respectively L H) the Lie algebra of G (respectively H), and by 90* the differential of up at l 6 H , J H £ / G .

359

Characteristic Classes

Pick a connection ( A l on Q, and denote by (Fa) its curvature. Then the collection go* (Aa) defines a connection on P with curvature up* (Fa). Now v((p*(Fal)= ((/)*t1)(Fo)

0.

The above result should be seen as a guiding principle in proving identities between characteristic classes, rather than a rigid result. What is important about this result is the simple argument used to prove it. We conclude this subsection with some simple, but very important applications of the above principle.

Example 8.3.3. Let E and F be two complex vector bundles over the same smooth manifold M of ranks r and respectively S. Then the Chem polynomials of E, F and E ® F are related by the identity CtlE

®

Fl =

CE)

(8.3.1)

• CtlFl»

where the "-" denotes the A-multiplication in Heven ( M ) . Equivalently, this means et(E€BF)=

C¢(El'CJ(Fli+j=k

To check this, pick a Hermitian metric g on E and a Hermitian metric h on F. g ® h is a Hermitian metric on E ® F. Hence, E ® F has an U (r -|- 5) structure reducible to an U(r) X U(s) structure. The Lie algebra of U(r) X U(s) is the direct sum u(r) Q; _u(s). Any element X in this algebra has a block decomposition XE II

XT' GX87

where X (respectively XS) is an T' X T' (respectively S X s) complex, skew-hermitian matrix. Let z denote the natural inclusion QU) QBQISI _u(1°+s) and denote by 6 I* (I/(V1) [t] the Chem polynomial. We have

ii"

*

z (c

1°-I-8

) )(XT-

® X8)

I

( (

det ]11°+s det

11 T

t

. X ® XI

2tr'I

.X

2Trz

w(

) 118

t

.Xs

2tr'I

cy)

(s)

Ct

(8).

The equality (83.1) now follows using the argument in the proof of Proposition 8.3.2.

Remark 8.3.4. Consider the Grassmannian Grk((C") of complex k-dimensional subspaces in (Cn. The universal complex vector bundle U p n -> Gr k( Grk((CTL). The trivial bundle Cn is equipped with a canonical Hermitian metric. We denote by Q k , n the orthogonal complement of U p n in Q" so that we have an isomorphism

_

Qn

Uknggkn-

360

Lectures on the Geometry

of Manzfolds

From the above example we deduce that Ct

) lUk," )@( Qw

C t (Q"

I

)

l.

Denote by Up the j-th Chem class of U p T L and by 'Up the 6-th Chem class of k Ct

Then

n-k

I Uk,n )

Cal

ujtj, j

Qkyn-

Q/c ,"

)

0

ugh.

e=o

I

One can then prove that the cosmology ring H' Gr 1 _u(")We obtain a morphism Z*

I.

(I/(n)) ->

i* (C21

=

6(k )

(X)

6(6)

LetX = X1 €(k)

(X)

€(£)(Xk ®

Xi).

The above example has an interesting consequence.

Proposition 8.3.14. Let E and F be two real, oriented vector bundle of even ranks over the same manifold M . Then

-

e ( E ® F ) = e ( E ) e(F),

-

where denotes the A-multiplication in

Heven ( M ) .

oWe are not won'ied about convergence issues because the matrices for which we intend to apply the formula have nilpotent entries.

363

Characteristic Classes

Example 8.3.15. Let E be a rank 2h real, oriented vector bundle over the smooth manifold M. We claim that Q' E admits a nowhere vanishing section 5 then e ( E ) = 0. To see this, fix an Euclidean metric on E so that E is now endowed with an SO(2h)structure. Denote by L the real line subbundle of E generated by the section g. Clearly, L is a trivial line bundle, and E splits as an orthogonal sum

E

L ® Ll.

The orientation on E, and the orientation on L defined by f induce an orientation on L_L so that LJ- has an SO(2k - 1)-structure. In other words, the SO(2k) structure of E can be reduced to an SO(l) X SO(2k-1) SO(2k - 1)-structure. Denote by 2* the inclusion induced morphism

7

r\/

I*(50 I* E is the zero section. The geometric Euler class

.

1 eger

(El

(g71ry

if rank (E) is even if rank (El is odd

pf (-F(V)) 0

7

where V is a connection on E compatible with some metric and 21° = rank (E). The next result, which generalizes the Gauss-Bonnet theorem, will show that these two notions of Euler class coincide. 'ii

Theorem 8.3.17 (Gauss-Bonnet-Chern). Let E -> M be a real, oriented vector bundle over the compact oriented manifold M . Then

et0plEl

egeomlE)~

364

Lectures on the Geometry

of Manzfolds

Proof. We will distinguish two cases. A. rank (E) is odd. Consider the automorphism of E i : E - > E ur->-'u,'v"u€E. Since the fibers of E are odd dimensional, we deduce that i reverses the orientation in the fibers. In particular, this implies or*i*TE =

= tr* (-TEL

-tl'*TE

where or* denotes the integration along fibers. Since phis theorem), we deduce i*TE

TE

7T*

is an isomoiphism (Thom isomor-

.

Hence etoplE)

I

(8.3.4)

-C0i*7-E.

On the other hand, notice that Cui*

C0-

Indeed,

C0 i*

= (i§0)*

(-Co)* = (§)* (Co =

Z

-Col-

The equality stop = e g e r now follows from (8.3.4). B. rank (E) = Qk. We will use a variation of the original argument of Chern [29]. Let V denote a connection on E compatible with a metric g. The strategy of proof is very simple. We will explicitly construct a closed form w 6 Q2';¢(E> such that (i) Tr*w (ii) ggw

1G

QOIMI.

€(Vl

Z

(2%)

P f (-F(v)).

The Thom isomorphism theorem coupled with (i) implies that w represents the Thom class in H2p"'t(E). The condition (ii) simply states the sought for equality stop = e g e r . Denote by S(E) the unit sphere bundle of E, S(E)

IUI9

{'u, 6 E ,

Z

1}.

Then S ( E ) is a compact manifold, and

dimS(E) = d i m M + 2 k - 1 . Denote by fro the natural projection S(E) -> M, and by fro (E) -> S (E) the pullback of E to S ( E ) via the map two. The vector bundle w0 (E) has an SO(2k)-structure, and moreover, it admits a tautological, nowhere vanishing section

T SlEx)9€*->€€Ex

(7f0(E)x)e (co E M ) -

Thus, according to Example 8.3.15 we must have egeom (770

(E) )

0 E H2'"(s(E)),

365

Characteristic Classes

where

geom (fro E)

denotes the differential form geom (fro

Hence there must exist

l (27f)k P f (-F(710V))-

V)

w 6 Q21 3/4. Finally, define

- p(~)#(e(v))

The differential form w is well defined since p' (T) E 0 near the zero section. Obviously w has compact support on E, and satisfies the condition (ii) since Cow

Z

-,0(0lCo7T*€(V) = 6(V)-

From the equality E/M

p(t~we(v) = 0,

we deduce IE/m

IE/m

p'(1°)d1° /\ W)

I°° 0

0'(1°ld7"

/

S(E)/M

\11

(V)

366

Lectures on the Geometry

of Manzfolds

\I/(V)

==-(p(l)-~p(0l)

(8.3.6)

l.

S(E)/M

To complete the program outlined at the beginning of the proof we need to show that Ld is closed. do (

p'(r)d7° A d o ) - 0'(t") A or*elVl

I

¢

.

8 3 5

)

*

Tt*6(V)}-

p'(f~)dti A mew)

The above form is identically zero since ¢r0e(V) = ¢r*e(V) on the support of p' . Thus w is closed and the theorem is proved.

Proof of Lemma 8.3.18. We denote by V the pullback of V to fro E. The tautological section T s(E) -> tr0E can be used to produce an orthogonal splitting

L@Ll

%E

7

where L is the real line bundle spanned by T, while L_L is its orthogonal complement in 7r0 E with respect to the pullback metric g. Denote by

P

-> 7r0 E

7r0 E

the orthogonal projection onto LJ-_ Using P, we can produce a new metric compatible connection V on 7r0 E by

= (trivial connection on L ) ® POP. We have an equality of differential forms 1r0*e(V) Since the curvature of

1

V

::

Z

p f (-Fw.

splits as a direct sum FW)

0 Q FW),

I

(Vl denotes the curvature of IL we deduce P f I F I I = 0. We denote by Vt the connection V -|- t(V - e) so that V0 = V, and V1 where F'

7

V If Ft is

the curvature of Vt, we deduce from the transgression formula (S.1.12) that 111111»

7r0

e(V) = e(V) - e(V) = d

Pf(V

v, Ft

7

We claim that the form k

W)

1

P f (V

k 0

satisfies all the conditions in Lemma 83.18.

FR

,F")dt

,Ftldt

367

Characteristic Classes

By construction,

div) = 7V0@(V)7 so all that we need to prove is

S(E)/M

m y ) = -1 6 Q°(M)~

It suffices to show that for each fiber Eas of E we have

\I/(V)

l.

Et

Along this fiber or0 E is naturally isomorphic with a trivial bundle 7I*o

u _ )9¢1 /\ (En-I-1 the polynomial P defines a complex linear map L -> (C, and hence an element of L*, which we denote by P | L . We thus have a well defined map 9

CIP" 9 L »-> PILe L* = U* IL, and the reader can check easily that this is a smooth section of II*, which we denote by [P] . Consider the special case P0 = Zn. The zero set of the section lP0l is precisely the image of [H1. We let the reader keep track of all the orientation conventions in Exercise 83.21, and conclude that C1 (ml is indeed the Poincaré dual of [H] .

372

Lectures on the Geometry

of Manzfolds

~

Exercise 8.3.23. Let Us = S2 \ {north pole}, and Un = S2 \ {south pole} = (C. The overlap Un O Us is diffeomorphic with the punctured plane CC* = (I \ {0}. For each map g : C* -> U(1) = S1 denote by Lg the line bundle over S2 defined by the gluing map

~

QNS :

Un re Us

U(1),

9NS(Zl

= 9(2)-

Show that $2

L 01( g)

=d t , e 9

where deg g denotes the degree of the smooth map g |S1 be* -> S1.

Chapter 9

Classical Integral Geometry

Ultimately, mathematics is about solving problems, and we can trace the roots of most remarkable achievements in mathematics to attempts of solving concrete problems which, for various reasons, were deemed very interesting by the mathematical community. In this chapter, we will present some very beautiful applications of the techniques developed so far. In a sense, we are going against the natural course of things, 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. Integral geometry, also known as geometric probability, is a mixed breed subject. The question it addresses have purely geometric formulations, but the solutions borrow ideas from many mathematical areas, such as representation theory and probability. This chapter is intended to wet the reader's appetite for unusual questions, and make him/her appreciate the power of the technology developed in the previous chapters.

9.1 9.1.1

The integral geometry of real Grassmannians Co-area formulae

As Gelfand and his school pointed out, the main trick of classical integral geometry is a very elementary one, namely the change of order of summation. Let us explain the bare bones version of this trick, unencumbered by various technical assumptions. Consider a "roof"

X A

B

B

B where X -> A and X -> B are maps between finite sets. One should think of a and 5 as defining two different fibrations with the same total space X . Suppose we are given a function f : X -> R. We define its "average" over X as the "integral" a

(f)x

f(33)a:6X

373

374

Lectures on the Geometry of Manu'olds

We can compute the average ( f }X in two different ways, either summing first over the fibers of a, or summing first over the fibers of 5, i.e.,

(

2 2

a A

f(33l

0(:1:)=a

)

(f)x

(

2 2

boB

f(f/3)

5(2vl=b

)

(9.l.1)

We can reformulate the above equality in a more conceptual way by introducing the functions a * l f l : A - M C , 0,*(f)(0)=

f(f/3)» a(:13)=a

and

I5'*(f) B ->(C, B*(f)(b)=

f( A(€1 A

•

•

A en),

where (el, . . . , en) is the canonical basis of R" . If V0 and VI are vector spaces of the same dimension n, and g : V0 -> VI is a linear isomorphism then we get a linear map

g* : IAls(V1) -> Ills(%l» IAls(V1) 9 A +-> g*A, where

( Q ) (/WMI

I

Al/\z'lg'l)¢l

)»

\V/U17

7

vn€%~

If V0 = VI = V so that g 6 Aut(V), then ldetglSA.

9*>\

h*g*9 and thus we have a left action of Aut(V)

For every g, h 6 Aut(V) we gave (oh)*

on IAIS(V) Aut(V)

Aut(V)

X

IAIS(V) -> IAIS(V)7

X

IAls(V) 9 (9, A) »-> QM = (9`1)*% = ldetgl'sA.

We have bilinear maps

IAI5(V) (X) III¢(V)

*

|A|s-I-tMv (Ml)

*-

A'

,U-

To any short exact sequence of vector spaces

0

U i v 15w a d l l l. u

Ma

d i V =

Ma

dimW=p

we can associate maps

lAI+(u)

I=IAls(V)

X

X

IAls(V) -> IAIS(w),

III+(w) -> Ills(Ul»

and X

IAls(U)

X

IAls(W) -> IAls(V)

as follows.

.

Lets, 6 III+(U), A 6 IAIS(V)9 and suppose lifts 'Up 6 V o f

we, such that Blvj)

lwj)1SjSp

We, and a basis

a(u1), . . . ,a(um71), U1

7 • •

is a basis of W. Now choose of U such that

l'UȢ)1gi$1m,

¢

7

'Up 7

is a basis of V. We set (MW

( Ajwj)

*( (/\¢O¢(U¢l ) A ( / \ j ' U j l m( Am )

It is easily seen that the above definition is independent of the choices of v's and 'u,'s.

376

.

Lectures on the Geometry

Let A 6 IAIS(V) and u 6 LAI+(W). Given basis l'u»i)1gi$vm of U, extend the linearly independent set ( ) C V to basis {a(u1), . . . , a(um),01, :Van of V and now define

(Aw) ( I\i'LLi ) :z

.

of Manzfolds

*( (/\@O¢(U¢)) /\ (/\j'Uj) ) . lal I\i5('Vi) )

(91.2)

Again it is easily verified that the above definition is independent of the various choices. Let I.L 6 IAIS(U) and u 6 IAIS(W). To define I.L x u : det V -> R it suffices to indicate its value on a single nonzero vector of the line det V. Fix a basis (Uillgfzgm of U and a basis l w j l l é j é p of W. Choose lifts (vi) of we to V. Then we set

Ill

X

v)( (A¢@(U51) /\ lAJwj)l

i

,LLI/\¢Uil1/I/\jUjl.

Again one can check that this is independent of the various bases (UU and (We Exercise 9.1.1. Prove that the above constructions are indeed independent of the various choices of bases.

Remark 9.1.2. The constructions \, /, and spaces

X

associated to a short exact sequence of vector

0 -> U > V B> w -> 0, do depend on the maps a and B! For example if we replace a by B7' = TB, t,T > 0, and if l~* G IAI5(U)> A G I A l5(V)» I/ G IAI3(W), a

Ut

to and

5 by

then

\0¢»5-r A

(t/T)S#\o,@ A, A/Omg-r I/ (t/7')SA/01,5 VI A x O r t 5 V = (t/7')-sp X05 u. The next example illustrates this. i

Example 9.1.3. Consider the short exact sequence

0->U=R

a

>

v=R2

5

>W=]R->0

given by @(s) = (45, 105), l3(w,y) = 5:15 - 23/Denote by e the canonical basis of U, by (el, et) the canonical basis of V and by f the canonical basis of W. We obtain canonical densities ,\U on U, AV on V and AW on W given by /\Ul€l I\V(€1 /\ 6 2 ) = w ( f ) 1. We would like to describe the density >V/5*,\W on V. Set I

I

.ii 0(e) = (4, 10). We choose f 2 6 V such that B(f 2) = f , for example, f2 = (1, 2). Then 4 1 A v / 5 * w ( € l = Av(f1 /\ f 2)/>\W(f ) 10 2 Hence Av/@*Aw = ZAp.

ldetl H

II N

I

377

Classical Integral Geometry

Suppose now that E -> M is a real vector bundle of rank n over the smooth manifold M. Assume that it is given by the open cover (Ua) and gluing cocycle QBQ :

U0g -> Aut(V),

where V is a fixed real vector space of dimension n. Then the bundle of s-densities associated to E is the real line bundle IAISE given by the open cover (Ua) and gluing cocycle

ldetg50|-3 : U0;-3 -> Aut( IAIS(V)) 2 R* We denote by cOO (IAISE) the space of smooth sections of IAISE. Such a section is given by a collection of smooth functions Aa : Ua -> IAIS(V) satisfying the gluing conditions A;-xlflil

1

(DB 1)*>\o(g3)

ldet

1

g5ol-sAol£ul,

Va,B,

at

6 UQ;-3.

Let us point out that if V = R", then we have a canonical identification IAIS(]R") -> R and in this case a density can be regarded as a collection of smooth functions AO! : Ua -> R satisfying the above gluing conditions. An 5-density A E C'°°(IAIS E) is called positive if for every : B E M we have My) E

III+(E,,,l. If QUO : N -> M is a smooth map, and E -> M is a smooth real vector bundle, then we obtain the pullback bundle 1r*E -> N. We have canonical isomorphisms

IAI

S

E

§

tr*

IAISE,

and a natural pullback map

~

¢* =0*°°(IA15 E) -> C'°°(tr* Ills E) = 0°°(IA1 tr*E). Given a short exact sequence of vector bundles 0->E0->E1->E2->0

over M , we obtain maps \¢C(>°(IAI+ EQ)

I:COO>(IA1S El)

X

X

6®(IA18 El) -> C7°°(IAI5 ET)»

0'>°(IA1+ EQ) ->

COOllAI5 Eol,

C°°(IAlsE2) ->

C`°°(IAI5 E1)-

and X

:COO(l AIS Et)

X

Observe that for every positive smooth function f : M -> (0700) we have

(fml\A

i

(f-1(/\A\, ,v(fy)

i

(f"1)(/1/M

In the sequel we will almost exclusively use a special case of the above construction, when E is the tangent bundle of the smooth manifold M. We will denote by IAIS(M) the line bundle IAIS(TM), and we will refer to its sections as (smooth) s-densities on M. When s = 1, we will use the simpler notation IAIN to denote IAI1(M).

378

Lectures on the Geometry

of Manzfolds

As explained in Subsection 3.4.1, an s-density on M is described by a coordinate atlas l, and smooth functions Ao : Uo, -> R satisfying the conditions IUQ, /

AB

Ida I

-SAG

where d

)

= det

1

ii

\

,

n

dim M.

(9.l.3)

1 N is a diffeomoiphism and ldpl = (UQ, Po, Idyll is a density on N, then we define the pullback of ld,0l by (I) to be the density go* ldpl on M defined by

d)*ldf0l =

(¢>-1(uo),pldyl)

7

y;

fzzgogb.

The classical change in variables formula now takes the form

/

N

/

ldpl

M

(25*ldpl.

Example 9.1.5. Suppose (b : M -> N is a diffeomorphism between two smooth L 0 6 Qm*(M), and lwl is the associated density. Then

dimensional manifolds,

¢*lw l

Z

|¢*w|-

Suppose a Lie group G acts smoothly on M. Then for every g 6 G, and any density

ldpl, we get a new density g* ldpl. The density ldpl is called G-invariant if g*ldpl =

Idpl,

Vg € G.

Note that a density is G-invariant if and only if the associated Borel measure is G-invariant. A positive density is invariant if the Jacobian g* ldpl is identically equal to 1,'v'g E G. Id/>l

Proposition 9.1.6. Suppose ldpl and ld7'l are two G-invariant positive densities. Then the Idol Jacobian J Idol is a G-invariant smooth, positive function on G. Proof. Let :BE M and g a G . Then for every open neighborhood U o f x we have

Idol U

/

g(U)

ldpl,

/

U

we

Idol 9(U)

\.

In add

1

ft] do

I

Idol f g ( U ) Idol I(U)

7

and then letting U -> {:z:} we deduce J(w)=J(9f11),

V:1;eM,geG.

Corollary 9.1.7. If G acts smoothly and transitively on the smooth many'old M then, up to a positive multiplicative constant there exists at most one invariant positive density.

380

Lectures on the Geometry

of Manzfolds

Suppose : M -> B is a submersion. The kernels of the differentials of form a vector subbundle TVM -> TM consisting of the planes tangent to the fibers of ®. We will refer to it as the vertical tangent bundle. Since is a submersion, we have a short exact sequence of bundles over M. 0 -> TVM TM

*TB -> 0.

Observe that any (positive) density ldul on B defines by pullback a (positive) density *ld1/l associated to the bundle ®*TB -> M. If A is a density associated to the vertical tangent bundle TVM, then we obtain a density A X * ld1/l on M. Suppose ldlnl is a density on M such that is proper on the support of ldnl. Set k := dim B, T' := d i M - dim B. We would like to describe a density ®*ld/nl on B called the pusnforward of ldll by . Intuitively, ®* id/nl is the unique density on B such that for any open subset U C B we have

Idul-

®*ld,LLI

U

-1(U)

Proposition 9.1.8 (The pushforward of a density). There exists a smooth density * ld,a | on B uniquely characterized by the following condition. For every density ld1/l on B we have

'1>*ld#l = V,,ld1/l, where VV G C°°(B) is given by

ld~l/*ld1/I.

II

..

v,(b>

-1(b)

Proof. Fix a positive density ld1/l on B. Along every fiber Mb = Tm1, -> (Tm)lm,,

Mi (*TB)IMb -> 0.

To understand this density fix at 6 Mb- Then we can find local coordinates (3/i)1$J$I< near b 6 B, and smooth functions (f13')1ld/>l = V,,ld1/l. In other words, the density VVld1/l on B is independent on I/. It depends only on

Idul-

Using partitions of unity and the classical Fubini theorem we obtain the coareaformula for densities.

Theorem 9.1.9 (Coarea formula). Suppose the : M -> B is a submersion, ldvI is a density on B and Idol is a density on M such that is proper on the support of ldlul.

/

M

Idol

/

B

I(/

®*ldpl

B

l/*ld1/I Id1/l~

(9.l.4)

-1(b)

Remark 9.1.10. Very often the submersion B satisfies the following condition. For every point on the base b e B there exist an open neighborhood U of b in B, a nowhere vanishing form w G Qk' (U), a nowhere vanishing form Q € Q't.

383

Classical Integral Geometry

P

1 9

9

t

1

O

Fig. 9.1

p'

t

>

I I

Slicing a sphere by hyperplanes.

The map 'ii is a submersion on the complement of the poles, M = S" \ {Pi }, and 7r(M) = (-1, 1). We want to compute or* ldvv,l. Observe that 7/"1 (t) is the (in - 1)-dimensional sphere of radius (1 - t 2 l 1 / 2 . To find the gradient V'/r observe that for every p e Sn the tangent vector V7r(p) is the projection of the vector it on the tangent space TpS", because 3,5 is the gradient with respect to the Euclidean metric on Rn-l-1 of the linear function 'or : Rn-1-1 -> R tr(t, x ) = t. We denote by 9 the angle between it and TpS"', set p' = 7r(p), and by t the coordinate of p' (see Figure 9.1). Then 9 is equal to the angle at p between the radius [0, p] and the segment [p, pI]. We deduce

cos 9 = length [p,p/]

= (1 - t2)1/2

Hence IVy(p)l = (1 - t2)1/2 7 and consequently,

1

T-1(¢l

= -1/2

=

0,,_1 0.

(9.l.8)

It is known (see [134], Section $2.4l) that up

OO

Bop, Q) 0

(1

_|_ £I3lp+q

do: QUO

F(P)F(ql

F(P

-|-

q)

7

(9.l.9)

384

Lectures on the Geometry

of Manzfolds

where I`(;z;) denotes Euler's Gamma function F(fv)

€":t513-1dt. 0

We deduce 2F()H+1 O'

(9.l.10)

r(ngl)

n -

The equality (9.1.7) can be further generalized. We start with a simple linear algebra result. Lemma 9.1.12. Suppose that U and V are Euclidean spaces of dimensions n + k and respectively k, n 2 0, and A : U -> V is a surjective linear map. Then there exist Euclidean coordinates bel, . . . ,;13""-k on U, Euclidean coordinates y l , . . . ,gk on V and positive numbers ,u1, . . . , ,up such that, in these coordinates the operator A is described by

ye The numbers #21 . . . , p V

-> V

l.LjfDj

J

1§

7

S If.

are the eigenvalues of the positive symmetric operator AA*

so that P'1°'°llk

= \» detAin*

Proof. Let W denote the orthogonal complement of kerA in U. Denote by A0 the restriction of A to W so that A0 : W -> V is a linear isomorphism. Note that W coincides with the range of the adjoint operator A* : V -> U so that AoA;

AA*.

Z

We want to find a linear isometric R : V ->W

B

such that the operator

AoR : V -> V

>=

is symmetric. Note that since R is an isometric we have R-1 commutative diagram

w

Ao

V

I I

R

V Note that AoA*

R* Moreover we have a

\

1

iv

V

V -> V is positive and symmetric. We define R := A0(A0,40)

1/2

.• V -> W.

Let us show that R is indeed an isometric. Indeed, for any 'U 6 V we have

( R e , R v ) = (A0(A0A0)-1/2/0, A0(A0A0)'1/2u ) Z

=

((A@A0)-1/2), A@A0(A0A0)-Mv) l l A 0 A 0 ) - 1 / 2 U 7 I A 0 A 0 I 1 / 2 ' U l = (m).

385

Classical Integral Geometry

Clearly AoR = A0A0(A0A0)-1/2 = (A0A0)112 is symmetric. Now choose an orthonormal basis that diagonalizes B. Transport it via R to an orthonormal basis of W. With respect to these bases of W and V the operator A is described by a diagonal matrix with entries consisting of the eigenvalues of AoR = (A0A0)1/2.

Suppose now that A : U1 -> U2, dim U1 = n -|- lc, dim U2 = Is is a surjective linear map between Euclidean spaces. Let lit 6 III(U¢) denote the Euclidean density on U p , i = 1, 2. Set U0 := her A and denote by Mo the Euclidean density on U0 . Using Lemma 9.1.12 we can find Euclidean coordinates 513l, . . . ,:1:n+'° on U1, Euclidean coordinates y1, . . . , yn on U2, and positive numbers up, . . . , ,on such that A is described the equations, Yj=lJ,

j=l>-.~,H-

Then /Io = id:v"+l A

•

-

•

/\ dx"+'"l, M1 = ld:1:1 A

• •

- /\ dm"+'"l,

1

# 2 - I d y A»-~Ady"l.

Consider now the short exact sequence of vector spaces ii

A

0 -> U0 U1

> $2 -> 0,

where i is the canonical inclusion. Using (9.l.2) we deduce M2/A*M1 =

l I.L1 .

•

a

,Un

ldxn+1 /\

A do:"`*'" |

l

JA M

0.

(9.l.11)

Suppose now that (M, gel, (B, gel are Riemannian manifolds, : M -> B is a submersion and f : M -> R is a continuous function such that is proper on the support of f. Denote by ldVml the metric volume density on M and by dVB. We define its Jacobian to be the function J

: M -> R, ](P)

where D(p) that

\ det(D(a:))*D(p

7

is the differential of at p. From (9.1.11) we deduce 1

ldVml/*ldVNI l>-1(5)

JI, ldV-1(b)l»

where ldVq>-1 lb) | denotes the metric volume density along the fiber ®'1 (6), b 6 B. From the coarea formula (9. 1.4) we deduce that metric coarea formula

M

fldvml

W B

_1(5)

JI,

b

B

(9.l.12)

386

9.1.2

Lectures on the Geometry

of Manzfolds

Invariant measures on linear Grassmannians

.

Suppose that V is an Euclidean real vector space of dimension n. We denote by the inner product on V. For every subspace U C V we denote by PU the orthogonal projection onto U. We would like to investigate a certain natural density on Gr IUt at t 1

he, Uo) If we write Pt

I

.2

0, then

.

d dt

|

PHS? t

0

PHS we deduce from (1 .2.5) that Pt

(HL -I-t2S*S)-1 tS(]1L +t 2S *S )-1

t(11L + t2s*S)-15* t2s(]1L +t2s*s>-1s*

Hence

pt SO

S*PLJ_ -I- SPL,

0

that

h(U0, Uo) We can be even more concrete.

1

2(tr(SS*)

+ t1~(5*5l) =

tr(SS*).

(9.1.13)

387

Classical Integral Geometry

_ _ of L, and an orthonormal basis Let us choose an orthonormal basis l€-f2l1 = 12 tr(P0, P0) = tIllXL6L,L0XLO_L,L0)

I 'éavl I 2

(9.l.l5)

a,i

We want to interpret this in the language of moving frames. Suppose M is a smooth m-dimensional manifold and L : M -> G r k ( V ) is a smooth map. Fix a point Po 6 M and local coordinates ( U l 1 V, f A(Sl

fA

TAB

h=

.

TAB

Z= S B A ,

d f 8 . The above metric metric has the

® 9AB-

A>B

The associated volume density is

'd'Yn I

TAB

A>B

Step 2. Fix an orthonormal frame ( A l of V such that L0 = span (et ; 1 < i < of). We can identify V with R", O(V) with O(nl, and L0 with the subspace Rh ® On-k C R".

An orthogonal n the equalities

X

n matrix T is uniquely determined by the orthonormal frame

TAB

pA

.

TE8.

Define p : O (n)-> Gr klfnl, P ( T l = T ( L 0 ) -

More explicitly, we have p(T)

-

span l T € i ) 1 g i $ k -

lT€Al via

392

Lectures on the Geometry of Manu'olds

We will prove that we have a principal libration

o O(nl

of the map p : O(n) -> Grk,(R"). The section can be identified with a smooth family of orthonormal frames l¢A(Ll, L E U ) 1 5 A g n o R " , such that 1 $ i < If).

= span (¢¢(L);

L

To such a frame we associate the orthogonal matrix ¢(L) E O(n) which maps the fixed frame ) to the frame (¢8 I, It is a given by a matrix with entries

leA

(l( /\ He

15

/\

/\

Q>,B

i>j

0,'é

)

is the pullback of a nowhere vanishing form defined in a neighbor-

hood of L0 in Grk (RTL), whose associated density is ld.*yTLkl. We now find ourselves in the situation described in Remark 9.1 10. We deduce C

" P 1(Lo)

(f

O(k)

I D ) /\ Qu

A

ldVkl

H

a>,5'

i>j

)(/

O(n-k)

ldVn-kl

u

cucak.

394

Lectures on the Geometry

of Manzfolds

Hence Cn CkCn-k

Cn,k

Step 3. Fix an orthonormal basis {eA} of V, and denote by S n+ S n+

{6'€ V; III

l

1,

i

'U

.

1

the open hemisphere

> 0 }.

81

Note that Gr1(V) 2- RIP7n-1 is the Grassmannian of lines in V. The set of lines that do not intersect S n+ 1 is a smooth hypersurface of GII1 (V) diffeomorphic to ]R]Pm-2 and thus has kinematic measure zero. We denote by Gr 1(Vl* the open subset consisting of lines intersecting S n+ 1 . We thus have a map

w

Gr (V) -> S n-|-

1

6+-MO S n+

7

1

This map is a diffeomorphism, and we have Cn,1 Gr1(vl

Now observe that

ld'Yn,1l

Gr{(v)

ld'Yn,ll

s+

(¢'1)*|dvn,1|-

w is in fact an isometric, and thus we deduce 1

Cn,1

Un

= §area(S" 1l =

__

1

2

non 2 .

Hence Cn-I-1

=

CnClCn+1,1

=

UHCH 7

which implies inductively that n-1 Cn--IO'n_1°°°U'2C2=2

H

'n-1

H

Of

k=1

U3-

j=0

In particular, we deduce the following result.

Proposition 9.1.14. For every 1 < k < n we have Cn,k

H H

)~( H

H

UP

we)

Following [79], we set

[to]

l Un 1 2 Wn-1

H n k

n

Town 7

20.1n-1

k

[nli- k

l[

ll[ /€][

H la]

[up n

]!>

win!

in

7

1

()k w

Un

(9.1.20)

395

Classical Integral Geometry

Denote by

d]/7L,k

the unique invariant density on Gr

n such that

(9.1.21)

We have d]/n,k

["]

I

Cr"kk ld'Ynk

Example 9.1.15. Using the computation in Example 9.1.13 we deduce 0S9
> 2oh

22h7II-hhI!

(2h)!

(9.23)

Hence l (mTTI

Sm O' 'rn

1,

lThe orientability assumption is superfluous. 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, and in particular, it is orientable.

402

Lectures on the Geometry

of Manzfolds

so that 1

§*M mm

llllllllllllll

Um

M

deg

am.

Denote by g the induced metric on M, and by R the curvature of g. We would like to prove that the integrand §MdAm has the form

rdAm

P(Rm)dVm,

where dVm denotes the metric volume form on M and P(Rm) is a universal polynomial m of degree in the curvature R of M. 2 Fix a positively oriented orthonormal frame é' = (el, . . . , et) of TM defined on some open set U C M, and denote by 6 (0 1, . . . , 0 ml the dual coframe. Observe that 01 A

(We

We set SU

Sm(€i» o i l

-

.

Na

A 0 'rn

Rfzjke

o l e , R ( e k , , eelej

Theorem 9.2.2 implies that

Observe that Rtjke ;£ 0

>'¢A¢ 512

S¢I'ik 512

Sn

>'1l
0.

Hence deggp = degX-,

for any admissible vector field X. Suppose X is a nondegenerate admissible vector field. This means that X has a finite number of stationary points, Zx = {I>1, •

•

-

7PV}7

X(p¢)

0,

and all of them are nondegenerate, i.e., for any p G Z x , the linear map AX,p

:

TpRm+1

-> TpRm+1,

TpRm+1 9 U


D

uXIIPI

407

Classical Integral Geometry

is invertible. Define 6X

: Zx -> {$1}»

€(p) = sign det

App.

For any 5 > 0 sufficiently small the closed balls of radius 5 centered at the points in Zx are disjoint. Set

U

D

DO

BE (p)

-

PQZx

The vector field X does not vanish on D69 and we obtain a map

X:D€->SM

17

1

X

IX I

X.

Set

l

Q

X*dAm.

0' 'rn

Observe that

do =

1

x*d(dAm) = 0 on D .

O' m

Stokes' theorem then implies that

do /3D

0 -:'» deg

lD€

aD

/

an

Q=

2/

PGZx

Q, 3B@(1>)

where the spheres 335 (p) are oriented as boundaries of the balls BAP) If we let e -> 0 we deduce deg

aD

€x(P)»

(9.2.8)

PQZx

for any nondegenerate admissible vector field X. To give an interpretation of the right-hand side of the above equality, consider the double olD. This is the smooth manifold 15 obtained by gluing D along 3D to a copy of itself equipped with the opposite orientation, Uan (-D) ~ The manifold without boundary 15 is equipped with an orientation reversing involution go : D -> D whose fixed point set is AD. In particular, along AD C D we have a cpinvariant decomposition

Tl5l6D = T619 @ L, where L is a real line bundle along which the differential of go acts as -II L . The normal vector field n defines a basis of L. If X is a vector field on D which is equal to n along 31), then we obtain a vector field X on D by setting II

(>
Tq15,

A

'u

+->

is an isomorphism. This map is independent of the connection V, and we denote by @;( S " , (BD 9 p »-> 'n(pl = unit outer normal, and by SD the secondfundamentalform of 31),

SD ( X , Y)

TL

.

(D

X

Y), VX, Y 6 Vect ( u p ) ,

Then 1 0' TTY

an

d€tl-sDldv3D

: :

deg §0

x(Dl-

Classical Integral Geometry

409

9.3 Curvature measures In this section we introduce the main characters of classical integral geometry, the so called curvature measures, and we describe several probabilistic interpretations of them known under the common name of Crofton formulae. We will introduce these quantities trough the back door, via the beautiful tube formula of Weyl, 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. Surprisingly, these coefficients depend only on the Riemann curvature of the induced metric on the submanifold. The Crofton Formulae state that these curvature measures can be computed by slicing the submanifold with affine subspaces of various codimensions, and averaging the Euler characteristics of such slices. To keep the geometric ideas as transparent as possible, and the analytical machinery to a minimum, we chose to describe these facts in the slightly more restrictive context of tame geometry. 9.3.1

Tame geometry

The category of tame spaces and tame maps is sufficiently large to include all the compact triangulable spaces, yet sufficiently restrictive to rule out pathological situations such as Cantor sets, Hawaiian rings, or nasty functions such as sin(1/t). We believe that the subject of tame geometry is one mathematical gem which should be familiar to a larger geometric audience, but since this is not yet the case, we devote this section to a brief introduction to this topic. Unavoidably, we will have to omit many interesting details and contributions, but we refer to [31 , 36, 37] for more systematic presentations. For every set X we will denote by IP(X) the collection of all subsets of X . An R-structure2 is a collection S = {8"}n219 5" C IP(]R"), with the following properties. Ei- Sn contains all the real algebraic subsets of R", i.e., the subsets described by finitely many polynomial equations. EQ. Sn contains all the closed affine half-spaces of R" P1. Sn is closed under boolean operations, U, (W and complement. PQ. If A E S o , and B Q S", then A X B E SM-V". P3. If A E So, and T : RM -> R" is an affine map, then T(A) E S" Example 9.3.1 (Semialgebraic sets). Denote by 521g the collection of semialgebraic subsets of R", i.e., the subsets S I R " which are finite unions

S = S1 U USu, where each 5 is described by finitely many polynomial inequalities.

The Tarski-

Seidenberg theorem (see [20]) states that S a l g is a structure. 2This is a highly condensed and special version of the traditional definition of structure. The model theoretic definition allows for ordered fields, other than R such as extensions of R by "infinitesimals". This can come in handy even if one is interested only in the field R

410

Lectures on the Geometry

of Manzfolds

To appreciate the strength of this theorem we want to discuss one of its many consequences. Consider the real algebraic set • •

Zn

l£I',(I,0,...,(I,n_1l

Q R " +1.7 a0-i-a1a3+--a,,_1a:" - 1

-I-.Clin

= 0 }.

The Tarski-Seidenberg theorem implies that the projection of Zn on the subspace with coordinates ( G i l is a semialgebraic set • •

in

6

007

7

an-ll

E Rn;

Ha: E R, PA l$l3l

0 }7

where P5(x)

=

Ag -|- @158

+ -

&n_1$n

-1

ax"

-|-

In other words, the polynomial PA has a real root if and only if the coefficients co' satisfy at least one system of polynomial inequalities from a /nite, universal, collection of systems of polynomial inequalities. For example, when n = 2, the resolvent set 322 is described by the well known inequality - 4a0 2 0. When n = 3, we can obtain a similar conclusion using Cardano's formulae. For n > 5, we know that there cannot exist any algebraic formulae describing the roots of a degree n polynomial, yet we can find algebraic inequalities which can decide if such a polynomial has at least one real root.

as

Suppose 8 is a structure. We say that a set is S-definable if it belongs to one of the S"'s. If A, B are S-definable then a function f : A -> B is called S-de_lQnable if its graph Ff := {(a,b) E A

X

B; b = f(a)

}

is 8-deiinable. The reason these sets are called definable has to do with mathematical logic. A formula3 is a property defining a certain set. For example, the two different looking formulas {;z3€R; a 2 2 0 } , {a;'€]R; 2 3 / E R :

LU

2

y }

describe the same set, [0, ool. Given a collection of formulas, we can obtain new formulas, using the logical operations A, v, -, and quantifiers EI, V. If we start with a collection of formulas, each describing an S-definable set, then any formula obtained from them by applying the above logical transformations will describe a definable set. To see this, observe that the operators As V, correspond to the boolean operations, O, U, and taking the complement. The existential quantifier corresponds to taking a projection. For example, suppose we are given a formula (l5(a, b), (a, b) 6 A X B, A, B definable, describing a definable set C C A X B. Then the formula -l

{ s e A ; Elb E B :

¢(a,b)}

describes the image of the subset C C A X B via the canonical projection A X B -> A. If A C RT", B C R", then the projection A X B -> A is the restriction to A X B of the linear oWe are deliberately vague on the meaning of formula.

411

Classical Integral Geometry

projection RM X R" -> RM and P3 implies that the image of C is also definable. Observe that the universal quantifier can be replaced with the operator -E-\. Example 9.3.2. (a) The composition of two definable functions A i> B i> C is a definable function because Fg0f=

1(a,c)€

A

X

C;§lb€ B : (a,b) € Ilf, ( b , c ) € I ` g } .

Note that any polynomial with real coefficients is a definable function. (b) The image and the preimage of a definable set via a definable function is a definable set. (c) Observe that 'Salg is contained in any structure S. In particular, the Euclidean norm

I

.

1/2

II]Rn->]R,

l(3317

7 son)

2

I

3% 'é 1

is 8-definable. Observe that any Grassmannian Gr k(R"l is a semialgebraic subset of the Euclidean space End+ (R) of symmetric operators R" -> R" because it is defined by the system of algebraic (in)equalities Gr 1 R with the property that there exists a real analytic function JF defined in an open neighborhood U of the cube Cn := [-1, 1]"' such that

f(:v)

.f(w) :u E co, x€R"\C'n 0

we denote by S a n the structure obtained from Sals by adjoining the graphs of all the restricted real analytic functions. For example, all the compact, real analytic submanifolds of R" belong to S a n . The structure 'San is tame. (c) (Walkie, van den Dries, Macinlyre, Marker) The structure obtained by adjoining to San the graph of the exponential function R -> R t »-> e*, is a tame structure. (d) (Khovanski-Speissegger) There exists a tame structure S' with the following properties (dl) San C 8/.

(do) If U C R " is open, connected and S'-definable, Fl, . . . , Fn . U definable and C1, and f : U -> R is a C1 function satisfying

8

a f = H n »

VQCQR,

7

QQ

l,...,n,

X

R -> R are S'-

(9.3.1)

then f is S'-definable. The smallest structure satisfying the above two properties, is called the pfajian closure4 of S a n , and we will denote it by San . 4 Our definition of pfaMan closure is more restrictive than the original one in [78, 122], but it suffices for many geometric applications.

413

Classical Integral Geometry

Observe that if f five F : ( a , b ) - > ] R

(a, b) -> R i s 01,

3

-definable, and kg 6 (a,b) then the antideriva-

f (t)dt ,

F(23)

1

(a, b),

000

is also

-definable.

The definable sets and functions of a tame structure have rather remarkable tame behavior which prohibits many pathologies. It is perhaps instructive to give an example of function which is not definable in any tame structure. For example, the function ac +-> sin as is not definable in a tame structure because the intersection of its graph with the horizontal axis is the countable set we which violates the 0-minimality condition T. We will list below some of the nice properties of the sets and function definable in a tame structure S. Their proofs can be found in [3l,36].

.

(Piecewise smoothness of tame funetions and sets.) Suppose A is an S-definable set, p is a positive integer, and f : A -> R is a definable function. Then A can be partitioned into finitely many 8 definable sets 51, . . . , So, such that each S is a Co-manifold, and each of the restrictions f | s. is a Co-function. The dimension of A is then defined as max dim S . (Dimension of the boundary.) If A is an S-definable set, then dim(el(A) \ A) < dim A. (Closed graph theorem.) Suppose X is a tame set and f : X -> R" is a tame bounded

.. . .

function. Then f is continuous if and only if its graph is closed in X X R" . (Curve selection.) If A is an 8-definable set, h > 0 an integer, and :Hz 6 el(A) \ A, then there exists an 8-definable Ck-map *y : (0, 1) -> A such that :u = limt_,0 'y(t). (Triangulability) For every compact definable set A, and any finite collection of definable subsets {517 . . . , S;.,}, there exists a compact simplicial complex K, and a definable homeomorphism K -> A

..

such that all the sets -1 (Si) are unions of relative interiors of faces of K. Any definable set has finitely many connected components, and each of them is definable. (Definable selection.) Suppose A, A are S-definable. Then a definable family of subsets of A parameterized by A is a subset

SEA

X

A.

We set

{ a A ; (a,A>€s}, and we denote by As the projection of S on A. Then there exists a definable function S : AS -> S such that

.

s(A) Q S , VA Q As. (Dejinability of dimension.) If (SA) AGA is a definable family of definable sets, then the

function

A 9 A »-> dim SA Q R is definable. In particular, its range must be a _finite subset of Z.

414

Lectures on the Geometry

of Manzfolds

.

(De/nabilily of Euler characteristic.) Suppose (SA) AeA is a definable family of compact tame sets. Then the map

A

the Euler characteristic of SA 6 Z

9 A +-> X(SAl

. .

is definable. In particular, the set { x(5A); A Q A, } C Z is finite. (Scissor equivalence.) If A and B are two compact definable sets, then there exists a definable bijection go : A -> B if and only if A and B have the same dimensions and the same Euler characteristics. (The map go need not be continuous.) (Dejinable triviality of tame maps.) We say that a tame map Q) : X -> S is dejinably trivial if there exists a definable set F, and a definable homeomorphism T : X -> F X S such that the diagram below is commutative T

X

>

S

X

F

its

S

If \I/ : X -> Y is a definable map, and p is a positive integer, then there exists a partition of Y into definable Co-manifolds Y1, . . . , Ye such that each the restrictions

if : \P-1(Y1 ye is definably trivial.

Definition 9.3.5. A subset A of some Euclidean space Rn is called tame if it is definable within a tame structure S. Exercise 9.3.6. Suppose S is a tame structure and up : [0, l] -> R2 is an S-definable map. (a) Prove that up is differentiable on the complement of a finite subset of [0, l] . (b) Prove that the set

{ (t, m) G [0, l]

X

R 2, . go is differentiable at t and m

(p'(tl }

is 8-definable. (c) Prove that the curve

{(t(0(ill;

C=

t€[0,1]}

has finitely many components, and each of them has finite length. 9.3.2 Invariants of the orthogonal group

In the proof of the tube formula we will need to use H. Weyl's characterization of polynomials invariant under the orthogonal group. Suppose V is a finite dimensional Euclidean space with metric (-, -) We denote by ( - 7 - ) the canonical pairing

(-,-)¢v*

X

V-HR, (

7

u)

(U), V u Q V , ATV* = Hom(V, R).

We denote by O(V) the group of orthogonal transformations of the Euclidean space V.

415

Classical Integral Geometry

0% where E is a finite dimensional real vector

Recall that an O(V)-module is a pair (E, space, while p is a group morphism

p : O(V) -> Aut(E), g +-> p(g).

A morphism of O(V)-modules (E¢»P¢)» ¢ = 0, 1, is a linear map A E0->E1 such that for every g Q O(V) the diagram below is commutative E0 90(9)

A>E1

l

ET

l

p1(9)

A

>

E1

We will denote by Hom0(v) (EO, El) the spaces of morphisms of O(V)-modules. The vector space V has a tautological structure of O(V)-module given by

= go, Vg 6 O(V), 'u 6 It induces a structure of O(V)-module on V* = Hom(V, R) given by T

: O(V) -> Aut(V), T(g)v

V

Pt : UW) -> Aut(V*l, 9 »-> pt(9l» where

(Mg) A u) Equivalently, 0+(9) action on (V*)®n,

(A ,9-1v),v A

Q

V*,v 6 M

p(g-1)t, where p(h)t denotes the transpose of p(h). We obtain an

( p ) ® " : olv) -> Auto (v*)®"

),

9 »-> /)1(9)®"-

We denote by (V*)§FV> the subspace consisting of invariant tensors, w6

(v*)§W>
R Hermann Weyl has produced in his classic monograph [133] an explicit description of (V*)V> . We would like to present here, without proof, this beautiful result of Weyl since it will play an important role in the future. We follow the elegant and more modern presentation in Appendix I of [8] to which we refer for proofs. Observe first that the metric duality defines a natural isomorphism of vector spaces D:V->V*

7

v+->vT 7

defined by

(UT, u

( p a ) , iv"u,,v 6 V

416

Lectures on the Geometry

of Manzfolds

This isomorphism induces an isomer rphism of O(V)-modules

D : (VZ al -> (V*»p1l~ We conclude that for ever nonnegative integers r, S we have isomorphisms of G-modules IV-*l®(v"-I-s) (V*®1") V®8 %" Hom(V®", v ® s > . (X)

In particular, ( (v*)®""+8" )0(v> 4 lHom(V ®'r 7 v ® s > low)

HO1T1o(v)(V®» v ® s > .

I

Let us observe that if we denote by S the group of permutations of {l, . . . ,t°}» then for every up Q S we obtain a morphism of O(V)-modules T¢>

6 HOM0(v)(V

®'r 7

v®*>,

Tpl'U1 (X)

(X)

U R,

(of)

P(/)

E

T'

E Z>_0,

8,.>,

defined by

Pgo tut . . . 7 U r 7 7

V17

7

v)

(u1»v®' {l,27...7 d}

llifhk

such that i

(2m)!

m

define

P- (Tl

T,,(1)...u(h) ' ' • T;1(hk-h+1)...p(hk)

I

-

116Cv"-

The collection

{P~§

iv

is a matching of Ihk }

is a basis of the space of degree k invariant polynomials v®/* -> R For example, the space of degree l invariant polynomials in tensors of order 4 has dimension 2 2 1 ) = 6. Each polynomial in a basis constructed as above is a sum of d2 terms. We see that things are getting "hairy" pretty fast. 9.3.3

The tube formula and curvature measures

Suppose that M is an m-dimensional submanifold of R". We set C

:= codi

M

n

m.

In this section we will assume that M is compact and without boundary, but we will not assume that it i5 orientable. For T' > 0 we define the tube of radius T' around M to be the closed set 1ll,.(M) := {:u

Q

M; dist (;u,Ml
0 such that, for every T' G (0, to), the exponential map IEm induces a diffeomorphism

EM : n,.(m) -> Mm). If we denote by

ldvn I the Euclidean volume density on R"

v(m,¢~)

vol ('l',.(Ml

) 1r,(m>

we deduce

Adm

]P3MdVn|N,~(M)

If 77 N,.(M) -> M denotes the canonical projection, then we deduce from Fubini's theorem that

V(M,rl

1r*IEMldV,,l.

(9.3.2)

M

D;

. 0

We want to give a more explicit description of the density Tr* IBM ldV,,l. We will continue to use the indexing conventions we have used in Subsection 9.2. 1. Fix a local orthonormal frame leA) of (T]R")l M defined in a neighborhood U C M of a point Po 6 M such that for all l < i g m vector field ii is tangent to U. We define t

D

tal

ltm-f-1

I

7

we

,t") € Re;

2

N,°(U)

I

N~(M)lu,

22)

*

(tO'ea(a:),a:) E NrlM)a

and thus we can identify D; X U with the open subset 71' -1(U) C N,°(M), and we can use as Q M and Fe D; as local coordinates on 7r"1(U). Define 1r,,(U)

¢=

IEm(N»~(U)) C R"

7

420

Lectures on the Geometry

of Manzfolds

and

IU) -> R" by éA(;13-I-taeO) = et(ac).

pA

We have thus extended in a special way the local frame (6A) of (TO-")lu to a local frame of lTR")l1r,.(u) so that

Do pA = 0, 'v'a,A.

(9.3.3)

We denote by (0 A) the coframe of IU) dual to pA. Over D; X U we have a local frame later, , e ) with dual coframe (¢>A) defined by bi To* 0 dtO! . v (Ba I

Z

Consider the 1-forms Q1(1r,(u)) associated to the Levi-Civita connection D by the frame (pA) on or(II), and set (9 A B 6

A

@C B

ac J

D

=

A Q BE W,

so that ec 6 B

A ®CBeA.

Using (93.3) we deduce A 9 aB

Finally set

'Pg

=

)

AIM r*(®B

7 The equalities (9.3.4) imply

Q

QUO

AB

)

Rm

)1S©,JS'"1' We deduce that

(-1)"p(the sum of all the V

am-i-el

of Manzfolds

minors of

.F)

tr AVUIU

7

al

up?) symmetric with respect to the diagonal.

.

These minors are parametrized by the subsets I C {l, . . . , m } of cardinality #I = V. For every w Q Re we denote by of( uij w ) the corresponding minor of U(w), and by EI its average, #Mum

0 .

F] (Um )

Se-1

.

w)dw.

Note that HI is an O(c)-invariant polynomial in the variables {'~*@j } i , j € I ~ Let

I

{1 Iim_2(M, g) is precisely the Hilbert-Einsteinfunctional we discussed in Subsection 4.2.6, Definition 4.2.33. To find the const we compute lim-QIM) when M = So. Using (93.13) we deduce

20' m O' 2

m 2

()

1,

M

where Sfround denotes the scalar curvature of the round metric on the unit sphere. Observe that the Grassmannian of oriented codimension one subspaces of lR'"+1 can be identified with the unit sphere So. Hence, the oriented Gauss map of the unit sphere Sm is the identity. In particular, the shape operator of Sm Rm-l-1 is the identity operator. From Theorem Egregium we deduce that all the sectional curvatures of S" are equal to one. Using the equalities (4.2.3) we deduce Sround

2

2() m

1

Rijij

2

M

"id

Hence

-

2o m U2

const

(e) The polynomial

QUO

=

1

m 2

C)

m 2

()

COI1Stm

1

1

llll...l.lllll

o2

Zo m

411'

> Um-2lM)9)

I

471'

M

S9|dV9|-

still has a "reasonable form"

Q2(Rl=

QUO #I=4

Then

/

#5, - 6 and QI

€(07»S0)Ru102(p1¢2R0304(/)3(04

I

o,(p€S

R 0 1 0 2 U 1 02 R U 3 0 4 0 3 0 4

u'S]

-|-

€(07,

§0)Ru102(P1(P2R0304(P3(P4°

o;écp€S

The first sum has only three different monomials, each of them appearing twice is them sum. The second sum has different monomials (corresponding to subsets of cardinality 2 of SQ) and each of them appears twice.

Definition 9.3.13. If (M, g) is a closed, compact, oriented, Riemann manifold, m dim M, and w is nonnegative integer. If m - w is odd we set

up(ml

0.

428

Lectures on the Geometry of Manu'olds

If m - w is an even, nonnegative integer, m w

211, then we set

1

uw(M,9)

l27Tlhh!

Qh ( R )| dVm I .

M

We will say that P*wlM) g) is the weight w curvature measure of (M, g). We set ld.»Uw

1

I :=

(2.rl"h! Qh ( R )|dVm I ,

and we will refer to it as the (weight w ) curvature density.

Remark 9.3.14. (a) Let us observe that for any Riemann manifold M, orientable or not, the quantities Iduwl are indeed well defined, i.e., independent of the choice of local frames used in their definition. 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. For submanifolds of R", 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, for any finite set I, the relative signature e(0, LPI of two permutations up, O' : I -> I is defined by choosing a linear ordering on I , but it is independent of this cnoiee. (b) The weight of the curvature density has a very intuitive meaning. Namely, we should think of l w ( M , g) as a quantity measured in meter" . Proof of Lemma 9.3.10. Consider the integral

2

We have e

6-lt°l2 C

I(h1, . . . , hc) =

H (/

j=1

(u

SO

that C

OO

AC

6-8252h1 ds

J

-oo

OO

H (/ 1

e

52

0

32) C

OO

H (/

j=1

e'"'th'j'1/2du

0 •

)p

On the other hand, using spherical C00td1 yes ' N3 $1 -|-|- he, we deduce that

--

_

C

HF(

2hi-I-1\

j 1

laI»

lwel2hcdw

I(h1,

7

1

F t; and recalling

w

) (/

oo

e

-p 2

p2h+

0

(u

52hj ds

2

p ) OO

1

lwll2h1

2

SC-1

1 F 2

e+2h 2

-~~

lwel2h"doJ

e'"u 0

) (/

Sc-1

I$1 I 2h1

. . . ac Zhedw

I

c-l-2h 2

I

.

'ldu

C_

1

dp

that h

9.3.4 Tube formula

-

429

Classical Integral Geometry

Gauss-Bonnet formula for arbitrary submamfolds

of

an Euclidean space

Suppose Mm C R" is a closed, compact submanifold of Rn. We do not assume that M is orientable. As usual, set C = n - m, and we denote by g the induced metric on M. For every sufficiently small positive real number 1° we set M := {;z: E RTL; dist (at, Ml = T'} = 3'lll,.(M).

The closed set M is a compact hypersurface of R", and we denote by gr the induced metric. Observe that for r and 5 sufficiently small we have

MMfr) = 'r'I"+(M) - 'r,,-(M) so that,

V(M,.,e) = V(M,v° + 5) - V ( M , 7° - 5), which implies that 1+2h W1+2h(°$ Mn-1-2h(Mr»

g)

h>_ 0

(T + 6)

c-l-2k

c

+2

k

-(t~-6)

C

+2

k

}un-¢-2k(M,9)-

k>0

We deduce 2 Iin-1-2hlMragrl

I

U1 2h

+

k>0

w

c+2k

(

e-I-2k - 1 2k-2h 'ac + Hn-e-2klMag)° l+2h

)

We make a change in variables. We set p : = n - l -2h, w

n-c-2k=m-Zk.

T h e n c + 2 k = ' n , - w , l + 2 h = n - p , e - | - 2 k - l - 2 h = p - w , so that we can rewrite

the above formula as m UplMr»grl

I

2 w=O

n

W

( )

Un

MM Mm Q)

: :

r p ' w p w (M,

9)-

(9.3.14)

71-17

In the above equality it is understood that uw(m) deduce that If the codimension of M i5 odd then

1

w

0 if m

2u»p(M>9)> V0 S P < m

w is odd. In particular, we

dim M.

(9.3.15)

If in the formula (9.3.14) we assume that the manifold M is a point, then we deduce that M is the (n - 1)-dimensional sphere of radius r, M = sp-1, and we conclude that n-1

l"LPIST where

[ p]

(9.3.13).

n _ )- 2 p

()

can wn_p

H

n up, n - p = 1 mod 2, We p

up: 2

(9.3.16)

is defined by (9.1.20). The last equality agrees with our previous computation

430

Lectures on the Geometry

of Manzfolds

If in the formula (93.14) we let p = 0 we deduce i0lMr)grl

2/i0(M,gl, V0
(v, 'U + L ) € U .

We obtain a double libration j

V

Graff

C

and we set

:1 :=

£-1(M) = { (v,S) 6 V

X

Graff e . u € S V W M } . 7

Since dime = d i V + d i m GrC = n-I-c(n - c ) we deduce

dimU(M) =

n-|-

c(n - c) -codimM =

Again we have a diagram

U(M) e

M

Graff C

11+

e(n - c).

435

Classical Integral Geometry

The map T' need not be a submersion. Fortunately, Sard's theorem shows that 1° fails to be a surjection on a rather thin set. Denote by Graff ° ( m ) the set of codimension C affine planes which intersect M transversally. The set GraHI°(M) is an open subset of GraffC, and Sard's theorem implies that its complement has measure zero. We set U(M)* :=

7°"1

l Graff eIM>).

The set U(M)* is an open subset of U(M), and we obtain a doublejbration

U(M)* (9.3.23)

Graff °(Ml

M o n

L VI M which is the boundary of the The fiber of T' over L 6 Graffe is the slice ML domain DL :: IL O D) C L. The vertical bundle of the libration T' : 3*(M) -> Graff ° ( m ) is equipped with a natural density given along a fiber L O M by the curvature density id,uk1Ll | of the domain DL . We will denote this density by id;i,l. As explained in Subsection 9.1. 1, using the pullback r* ldel we obtain a density

luAu

dl~*pLI X r*ldCI

on U*(M) satisfying,

lAI :1*(m)

/

GraHI°(M)

(f

LOM

)

lduLI ld.%(L)l P

,lGraH'c

/Jpl-L nD)ld2;¢(L)l_

To complete Step 1 in our strategy it suffices to prove that there exists a constant f , depending only on m and C such that €*ldAI

1

old,Ucl,

where the curvature density is described in Definition 9.3.13. Set h := ( m - C The points in U(M) are pairs (x, L), where at 6 M, and L is an affine plane of dimension h -|- l. Suppose (3120, Lo) E 3*(M). Then we can parametrize a small open neighborhood of (330, Lo) in U* (M) by a family •

(16, € 0 ( S l »

ellSla

' ' •

7

Ehlsl, € h + 1 l S l » '

7

em(s)),

where :u runs in a small neighborhood of :UQ 6 M, S runs in a small neighborhood U0 of lLol in Gr C so that the following hold for every S.

.

The collection {€00§)»€19)»

..

,oh(Sl, €h+1(S)»

is an orthonormal frame of R" The collection {en, 81, . 7oh} is a basis of S. The collection span {$1, . . .,e;,} is basis ofTH30M

re S.

»am($)}

436

of Manzfolds

Lectures on the Geometry

.

The above condition imply that 80 is a normal vector to the boundary of the domain D re L C L. We require that 80 is the outer normal.

A neighborhood of

(5130,

Lo) in U is parametrized by the family

(vi, e0(s), 81lSl, . . . , € h l $ l 7 € h + 1 l $ ) 7 . . . 7 €,,,,(5)

),

where F' runs in a neighborhood of 5130 in the ambient space V. We denote by SD, the co-oriented second fundamental form of D, by SL the cooriented second fundamental form of DL C L, and by ldVLmMI the metric volume density o n L y . Then, i f w e sets = d i m L - p = m - c - p , w e d e d u c e 1 ld.,LLpLI

k

I

tfkl-sLlldvLqml.

O'

In the sequel we will use the following conventions. '£,j,k denote indices running in the set {0, . . . , h}.

0¢,B,'y denote indices running in the set {h + 1, . 7 m . A, B, C' denote indices running in the set {0, 1, . . 7 m • We denote by (HA) the dual coframe of (cAl» and set eA

TAB

.

(D

dBl.

Then, the volume density of the natural metric on GrC is

/\$0i

ld'Ycl

0,i

Then

am I

D Foea

X

ld'Ycl

a

AS

X

ld'Yc I 7

a

and

lAI

L

X

l U' k

d€tl-sLmmlldvLmml X I\@ 1, and it is a disk of dimension (m + 1 - C = dimL and radius (1 - r2)1/2 if T' < 1. We conclude that 'r1.

We set llwmzv

__ NplMm

_

+1_

C

C) - We

' -

P

_ -

m

(-U'rn-I-1-c (.0m-}-1-c-p

+l

C

P

Using Theorem 9. 1.16 we deduce #oIL

n ]n>m+1)ld/7CI(L)

(f L (f [LP /*p

llln,c,p

lDm-I-1 m

x €[L ] l ,

(az -|- [L])) ldV[L]; l(22l

Id1/@l([L])

P /2 (1 - | II 2) ld.[L];l(x)

x

ld1/Cll[L]l

I l X ( L f W M ) € Z

442

Lectures on the Geometry of Manu'olds

is definable. Its range is a definable subset of Z, i.e., a finite set. To see that it has compact support it suffices to observe that since M is compact, an affine plane which is too far from the origin cannot intersect M.

Definition 9.3.25. For any compact tame subset S C R", and for every integer 0 < p < 7 1 we define

x(L

II

a,,(s)

raff P(R")

D) ld;;1(Ll.

The proof of Proposition 9.3.24 shows that the integral in the above definition is well defined and finite. This proposition also shows that

aplDl

: :

MpIDI

if D C R" is a tame domain with smooth boundary. The above definition has a "problem": a priori, the quantity /i ,,(s) depends on the dimension of the ambient space R" D S. The next exercise asks the reader to prove that this is not the case. Exercise 9.3.26. Fix tame structure 5, and suppose S C R" is a compact, S-definable set. Let N > n and suppose z' : R" -> RN is an amine, isometric embedding. (a) Prove that GraH7'(R")X

1/ (Lm5)ld-I(L)

GraH7 '(RN)X l

U O z(' s l ) l of1/I( U )»

so that [Lp(S) is independent of the dimension of the ambient space. (b) Prove that for every real number A we have ii pa/\Sl:

IAINa,,($)-

(c) Suppose 51, S2 C R" are two compact S-definable sets. Prove that

ii p(51) + l

ii pIS1 u S2)

2lS2l

ii pIS n S ) .

(d) Suppose T C R'n is a triangle, i.e., the convex hull of three non collinear points. Compute Ii p(T), Vp > 0.

Proposition 9.3.27. Fix a tame structure 8. Suppose K C R" $5 a compact, S-definable set, A $5 an arbitrary S-definable set, and A C K X A i5 a closed .S-definable subset of K X A. For every A 6 A we set A Then the function A 9 A I

II

>fpl)'l

{& E K; (a7>\) E A}_ p P (AAA 6 R is bounded and Bore] measurable. A

Proof. Denote by Graff pIK) the set of affine planes of codimension p in IRE." which intersect K. This is a compact, definable subset of Graff". Define F :A

X

Graff "(K) ->

z,

(A, L) »-> X 0 we set Ny0(r) := fig re D , see Figure 9.2. The collection (No lye L0rwm forms a vector subbundle N0 -> L0 O M of I T R H I I L o V - M ' The exponential map on TR" restricts to a map EL 0oM I

NO * Lo.

446

Lectures on the Geometry

I

I

\

of Manzfolds

M

\ /

l |

I I l I |

r l I l I l I l

Not)

/

•y |

m

LE

/ \

l l

1 I I

I I I I I I I

Dr

p * " - - 1 - - " § q

-' I

¢ "

C * *

I l

- -

*-.

~* `

I

5

11

I

1

1

\

l

Fig. 9.2

Slicing the tube Dr around the submanifold M by a plane L0 .

Denote by 6 the pullback to N0 of the distance function :u »-> d(:z;) = dist (az, M),

AL0 = d O IEL0nm : N0 ->]R. For every y 6 L0 ii M, the restriction to fig of the Hessian of d at y is positive definite. Hence, there exists p = p(L0) > 0 such that the map EL0riM : {a: G NO; 5(;z3) g

p} - > L 0 O D p

is a diffeomorphism. We deduce that we have a natural projection

L0 VI Dp -> L0 M M, which is continuous and defines a locally trivial libration with fibers fig (p) . For every y 6 L0 O M, the fiber Ny°(0) is homeomorphic to a disk of dimension k. Thus, L0 O Dp is homeomorphic to a tube in L0 around L0 (W M C LQ, so that

xIL0 M pp.

oIL() VW

I

The downward gradient flow of the restriction to L0 O Dp of the distance function d(a2) produces diffeomorphisms of manifolds with boundary L0V1Dp%L0MD,., 'v'r€(0,p).

Hence

xIL0 M D ) xIL0 M pp) xIL0 M ml, VT E Since the restriction to L0 re Dp of the distance function d(ac) has no critical points other than the minima y 6 L0 re M, we deduce that L0 is transversal to the level sets I

I

{d(;13) = 1°} = M , VT 6 (0,p]. This proves L0 6 Xp.

447

Classical Integral Geometry

Corollary 9.3.32. Suppose C C R2 i5 a smooth, closed, compact tame curve. For every line L 6 Gr 1(]R2) : Gr 1inv) we set 7"Lc(Ll Z = # I L VW cl. Then the function L »-> n g ( L ) belongs to L°°(Gr1(]R2), ld17l21), has compact support and

length (C)

NclLl\d12,1

I (L)-

Gp1lR2)

Remark 9.3.33. Theorem 9.3.29 offers a strange interpretation to the Hilbert-Einstein functional. Suppose M is a tame, compact, orientable submanifold of the Euclidean space R". Let m = dim M, and denote by h the induced metric on M. Then l €M(h), #vi-zIM) HI = -27r where €m( h) is the Hilbert-Einstein functional

am(h)

M

s(hldvm(h).

Denote by Graff m-2(M) = Graff ( M ) the set of codimension - 2) affine planes in Rn which intersect M along a nonempty subset, and by Graff*m 2lml C Graff m-2(M) the subset consisting of those planes intersecting M transversally. For every L E Graff;n-2 ( M ) , the intersection LOM is a, possible disconnected, smooth, orientable Riemann surface. By Sard's theorem the complement of Graff;n-2(]l/I) in Graff 1"-2(M)

has zero measure. For every R > 0 we denote by Graff M-2(R) the set of codimension (m - 2) planes in R" which intersect the disk IDS of radius R and centered at 0. Note that since M is compact, there exists R0 > 0 such that Graff '*"°'2(M) C Graff rz-2(R), VR > Ro. We set

cIR)

ld.;;(L)l, c(Ml

Q;

GraH"m'2(R)

GraH'm'2(Ml

ld;;(/;l1,

Using Crofton's formula in Proposition 9.3.24 and the simple observation that X(LVW]D)"R)=1,

VL Q Graff M-2(Rl»

we deduce that

C(Rl

Z

Lim

#fn-2(D"R)

QRm

2

Observe now that cllm ld;7(/;l1 is a probability measure on Graff '"'2(R), and we can regard the correspondence

Graff m-2(R) 9 L I > x ( L r 1 M ) e Z as a random variable OR whose expectation is 1 m-2 M = 1 8 h. (oR)

c(R)/LL

(

)

2'rrc(R) m( )

We deduce

amah) =

Q

'Wm-2RM-2('lR) =

lim RM-2oo

27r I

I

448

Lectures on the Geometry

of Manzfolds

Theorem 9.3.34 (Crofton Formula. Part 2). Denote by Graff °(M) the subset of Graff C consisting of amine planes which intersect M transversally. Then, for every 0 < p < m-ewehave

19+ C

[ ] p

P¢p+clM)

/

I

Graff° ( M )

lplLm M)ld;;I(Ll

/

I

a,,(Lim)lw>(L)l.

Graff"

Proof. We continue to use the same notations we used in the proof of Theorem 9.3.29. From (93.15) we deduce lip+elMl :

12%

»Up+clDv°l-

The set D is a tame domain with smooth boundary, and Corollary 9.3.28 implies p-I-c C

]

»U»p+clDv°)

/

=

lJfpll. m D~)ld'17l(L)

=

Graff"

lim

/

np(LeD,,)l¢{;l(Ll.

GraffC

Thus it suffices to show that 1*->0

/

a,,(L re D,,,)ld;7(L)l =

Graff"

/

a,,(L m M)ld'7(L)l.

Graff

C

For r E 10,R) we define gr

Graff C -> R, 9,.(L> = a,,(L m D,.>.

From Proposition 9.3.27 we deduce that EIC' > 0 :

lg»,~(Lll < C,

VL E Graff

C 7

1°

E [0,R).

(9.3.25)

Lemma 9.3.3S. 1-

gT`lL)

:

90(L)

for all L 6 Graff °(MlProof. Let L 6 Graff °(Ml and set S := L lim

?°\,0

apluS""l

nD. :

We have to prove that

ii pls0l'

Denote by Graff "(SO) the set of codimension p affine planes in R" which intersect SO transversally. The set Graff" (50) is an open subset of Graffp, and its complement has measure zero. Then,

9'rlLl

ii (50

x(U VW $.~)|d'17"(Ul|-

X(U O S»~ll c°°(F>. The space Op ( E , El is an associative K-algebra. The spaces of smooth sections cOO (E) and cOO (F) are more than just K-vector spaces. They are modules over the ring of smooth functions cOO ( M ) . The partial differential operators are elements of Op which interact in a special way with the above C'°°(M)-module structures. First define

PDO 0(E,F) := Hom(E,F). Given T E PDO0, u E C'°°(E), and f e Cvoo(M), we have T(fu) - f ( T u ) terms of commutators,

IT, flu =

0 or, in (10.l.1)

T(fu) - f ( T u ) = 0.

Each f E CV0°(M) defines map

ad(f) U p (E,F) -> 019 (E,F),

by

ad(f)T

== T

of -f

O

T

= IT, fl,

W 6 Op(E, F).

Above, f denotes the C'°°(M)-module multiplication by f. We can rephrase the equality (10. 1. 1) as

PDO " L E , F ) = { T e U p ( E J ) , ad(f)T

0 W' E C°°(M)}

her ad .

Define PDO (fol) (E,

Fl :=

her adm+1

{T E Op (E,F); T E kerad(f0)ad(f1)

-

.ad(fml, Vfi E Cvoo(M)} .

The elements of PDO (fol) are called partial deferential operators of order § m. We set

PDO (E, F)

U

==

PDO

(m) ( E ,

F).

7m>0

Remark 10.1.1. Note that we could have defined PDO lm) inductively as

PDO( m )

I

{T

Q Op

[T, f]

Q

PDO (m-1) ,

This point of view is especially useful in induction proofs.

of

Q

c°°}.

453

Elliptic Equations on Manifolds

Example 10.1.2. Denote by R the trivial line bundle over RN . The sections of K are precisely the real functions on RN . We want to analyze PDO (1) = PDO( 1 ) ( m ) . Let L E PDO ( l ) , u,f E cOO(nN). Then [L,f]u = o(f) u, for some smooth function o(f) E cOO (RN). On the other hand, for any f , g E cOO (RN)

-

0(fg)u = [L, flu = [L, f](gu)

-|-

-

f([L» 9110 = 0 ( f ) 9 u

-|-

f0(9)

'u-

Hence 0(fg) = o(f)g -|- fo(g). In other words, the map f »-> o(f) is a derivation of c°°(RN), and consequently (see Exercise 3.1.9) there exists a smooth vector field X on RN such that (t(f) Let ,u := L(1) E

x-f8

c°°

Fac

is a linear isomorphism. The following exercises provide a complete explicit description of the p.d.o.'s on RN.

Exercise 10.1.14. Consider the scalar p.d.o. on RN described in Corollary 10.1.5 L

Clolilflga

loll

Show that c°°(RN),

a

2

a.

Let f E CooeRN). Then

ad(f) (be)

ad(f)(A)

2 Z

-§j{ad(f)( 9°'1(M)»

6 where

* is the Hodge *-operator *

QUO

(M) -> Q"'° (M).

The operator 6 is a first order p.d.o., and moreover 01 (6) : *01ld)* : *e(§) * This description can be further simplified. Fix 5 E Tas M, and denote by g* 6 TxM its metric dual. For simplicity we assume l§l = l€*l = 1. Include 5 in an oriented orthonormal basis (g1, . . . ,§") of Tas M, £1 = §, and denote by ii the dual basis of T x M . Consider a monomial Ld = £] 6 A'*Tac M, where I = denotes as usual an ordered multi-index. Note that if 1 Q I, then U1(€1)

0(d)(€) + M be a smooth vector bundle over the Riemann manifold ( M ) - A second order p.d.o. L : c°° -> c°° is called a generalized Laplacian i f

02

(Lllgl

151

" O b s e r v e that all the generalized Laplac ians are elliptic operators.

459

Elliptic Equations on Manifolds

10.1.3

Formal adjoints

For simplicity, all the vector bundles in this subsection will be assumed complex, unless otherwise indicated. Let El, E2 -> M be vector bundles over a smooth oriented manifold. Fix a Riemann metric g on M and Hermitian metrics ¢ on E , i = 1, 2. We denote by dvg - *l the volume form on M defined by the metric g. Finally, c5° (E,-) denotes the space of smooth, compactly supported sections o f E, . 'Q

-

Definition 10.1.24. Let P E PDO (El, EQ). The operator Q E PDO (EQ, El) is said to be aformal adjoint of P if Vu E C'§°(E1), and Vu E 7

I P , v)2dVg

(;5°lE2) we have

(Fu, Qu)1dVg.

M

M

Lemma 10.1.25. Any P E PDO (El, EQ) admits at most one formal adjoint. Proof. Let 621,

QUO

be two formal adjoints of P. Then, Vu E C5° (EQ), we have M

Lu, (621 - Q2lu)1dvg = 0 Vu E CO lEll.

This implies (Q1 - Q2)v = 0 Vu E Cf-§°(E2). If now U E COO(E2) is not necessarily compactly supported then, choosing a partition of unity (Q) C Cgo(M), we conclude using the locality of Q = QUO - QUO that

Qu

Q(avl = 0.

The formal adjoint of a p.d.o. P 6 PDO lEt) EQ), whose existence is not yet guaranteed, is denoted by P*. It is worth emphasizing that P* depends on the choices of g and

(~, Proposition 10.1.26. (a) Let L0 E PDO (Et, El) and L1 E PDO (El, E2) admitformal adjoint L11 E PDO (E'i-I-17 E t ) ( i = 0 , 1), with respect to a metric g on the base, and

metrics (», -la on Et, j = 0 , 1 , 2. Then L1L0 admits a formal adjoint, and

in

PDO

1fL Q

(WED, El), then L* Q

Proof. (a) For any Up 6

PDO

(WE1, EQ).

c5°(Et) we have (LQUo, L1u0)1dVg

(L1L0'LL0, U2)2dvg

M

( U Q , L0L1'LL2)0d'/9

M

.

M

(b) Let f Q C°°(M). Then

lad(fl/;l*

I

(L o f - f

O

L)*

I

-[Li fi

I

- ad(f)L*.

Thus

adlf0l adlfll .

a

. ad(ji,,,)L*

I

(-1)"'+1(ad(f0)ad(f1)

ad(fm)Ll*

0.

The above computation yields the following result.

Corollary 10.1.27. IfP E PDO ( m ) admits a formal adjoint, then o,,,(P*)

where the

I

(-1)""0m(Pl*»

* in the right-hand-side denotes the conjugate transpose of a linear map.

460

Lectures on the Geometry

of Manzfolds

Let E be a Hermitian vector bundle over the oriented Riemann manifold ( M g ) -

Definition 10.1.28. A p.d.o. L E PDO ( M ) is said to be formally self-adjoint L L*

if

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*

Q'

( M ) - > Q' (M), on an oriented Riemann manifold (M, g) is formally self-adjoint 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 ( E , (~, .>,), ii = 1, 2, be two arbitrary Hermitian vector bundle go. Then any L E PDO (El, EQ) admits at least

over the oriented Riemann manifold ( M ,

(and hence exactly) one formal adjoint L*.

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 = Qp and E2 = Qq. By choosing orthonormal moving frames, we can assume that the metrics on Et are the Euclidean ones. According to the exercises at the end of Subsection 10.1.1, any L E PDO(m) (El, ET) has the form

AO,(a3)3°'

L

7

Ialém

where Aa 6 0°°(RN,Mqxp((C)). Clearly, the formal adjoint of A transpose

A;

is the conjugate

EQ.

To prove the proposition it suffices to show that each of the operators C°°(E), A = V*V. To justify the attribute Laplacian we will show that A is indeed a generalized Laplacian. Using (10.1.8) we deduce that over U (chosen as above) we have

A

{2 (v + a {2

log

v

Riga)

ock

k

Vi

-|-

3m1.

log

m)

O

H

dzcj

o

(X)

Vi

j

l9k"~Val

k

U

o

9 k y V k -|- Qnk 9ky -|- g k j a m k

lm

gkjvkvj

kJ

-I-

/ a w e )

a

Vi

Vi

}

463

Elliptic Equations on Manifolds

The symbol of A can be read easily from the last equality. More precisely, Ct2(A)(€)

-gjk§yfk

I

I

-151g2'

Hence A is indeed a generalized aplacian. In the following exercise we use the notations in the previous example.

Exercise 10.1.33. (a) Show that 9k£

'II

Mg

Fee

),

where Pm denote the Christoffel symbols of the Levi-Civita connection associated to the metric g. (b) Show that to;

As

WT* M®EvEl

where VT*M®E is the connection on T*M (X) E obtained by tensoring the Levi-Civita connection on T*M and the connection VE on E, while

tr; : c°°(T*m®2

(X)

E) -> C°°(E)

denotes the double contraction by g,

tfglszj up = (X)

gjsiju.

Proposition 10.1.34. Let E and M as above, and suppose that L E PDO2(E) is a generalized Lap lacian. Then, there exists a unique metric connection V on E and 52

IR(/2) 6 End ( E ) such that

L

I

v*v + 32.

The endomorphism 32 is known as the Weitzenbéck remainder of the Laplacian L.

Exercise 10.1.35. Prove the above proposition. Hint. Try V defined by Vfgrad(h)u

f = -{(Ag/Uu 2

(ad(hlLlu} f , h 6 c°°(m>, u

Q

c°°(E).

Exercise 10.1.36 (General Green's formula). Consider a compact Riemannian manifold (M, go with boundary coM. Denote by 18 the unit outer normal along M be Hermitian vector bundles over M and suppose L E PDO k ( E , F). Set 90 = g Iam, E0 = E lam and F0 = F lam. The Green formula states that there exists a sesquilinear map BLICOOIEI

X

C°°(F)->C°°(c9M),

such that M

(Lu/u)d'u(9)

8MBL("~'»='UldU(90)+

M

(u»L*v)dv(9)~

464

Lectures on the Geometry

of Manzfolds

Prove the following. (a) If L is a zeroth order operator, then BL 0. (b) If L1 E PDO (F, G), and L2 E PDO ( E , F ) , then I

BL1L2 (u,vl

= BL1(L2u, Fu)

-|-

BL2 (u, LIE).

(c) BE*(v,u) = -BE(u,v).

(d) Suppose V is a Hermitian connection on E, and X E Vect ( M ) . Then B o X ( u , v ) = (u,v)g(X,15,'), Be(u,u) = (u,in~u)E,

where in denotes the contraction by 18. (e) Denote by 17 the section of T*M I am g-dual to ii. Suppose L is a first order p.d.o., and

set J := oL(17). Then BL(7u,,'u} = (.]u,v)p.

(10.l.9)

(D Using (a)-(e) show that, for all u E COO(E), 'u E COO(F), and any £1:0 E AM, the quantity BL(U, v)(:c0) depends only on the jets of u, 'U at £130 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, u) = 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 [23], or the very comprehensive [40, 137].

10.2.1 Sobolev spaces in RN Let D denote an open subset of RN . We denote by LgOClpl the space of locally integrable real functions on D, i.e., Lebesgue measurable functions f : RN -> R such that, Va E 05° (RN), the function of is Lebesgue integrable, of E L1 (RN). Definition 10.2.1. Let f E L M D L and 1 < lc < N. A function g E LgOClpl is said to be the weak k-th partial derivative of f , and we write this g = (in, f weakly, if gcpdzv = 'D

Lemma 10.2.2. Any

f

féikcpdac,

We 6 C0 (D).

'D

6 Ll1OC(Dl admits at most one weak partial derivative.

465

Elliptic Equations on Manifolds

The proof of this lemma is left to the reader. " N o t all locally integrable functions admit weak derivatives.

Exercise 10.2.3. Let f E C°°(D), and 1 < lc < N. Prove that the classical partial derivative as, f is also its weak k-th derivative.

Exercise 10.2.4. Let H 6 LgOClkl denote the Heaviside function, H(t) H(t) E 0 for t < 0. Prove that H is not weakly differentiable. Exercise 10.2.5. 6 _|. 92 weakly. L e t 1f , f2 e L 1loel D ). I f kfz 91

. = 92

E

1

LlOC

1, for t > 0,

weakly, then 8k(f1

-|-

f2l

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 E LgOClp). Then we say that Lf = g weakly i f

990 d o ;

in

)

aL*wdo:

W

Q

COoo( D).

Above, L* denotes the formal adjoint of L with respect to the Euclidean metric on RN

Exercise 10.2.6. Let f , 9 6 C°°(D)~ Prove that

Lf = g

Lf = g

classically ~I '

weakly.

Definition 10.2.7. Let k E Z+, and p E l1, oo]. The Sobolev space L f w l p l consists of all the functions f E Lplpl such that, for any multi-index a satisfying lal < k, the mixed partial derivative c9"'f exists weakly, and moreover, c9°"f E Lp(D), For every f E I;I we set 1/19

llfllk,p

llfll/ 0, and moreover

UR

•

f

L'°p

>f

as

R ->

>

0 a smooth function UR 6

lac°l > R + 1, and ldtIR(a))l < 2

oo.

Proof. We consider only the case k = 1. The general situation can be proved by induction. We first prove that m

follows easily by induction over k. We let the reader fill in the details.

Theorem 10.2.24 (Rellich-Kondrachov). Let (k,pl, (m, q) E Z+

X

[l, oo) such that

k > m and 0 > can(k,p) > 0N(m,q). Then any bounded sequence (url C L'

0 has a

I.

subsequence strongly convergent in LM»q(1R(N

Proof. We discuss only the case k = 1, so that the condition 0N(1, p) < 0 imposes l < p < N. We follow the elegant presentation in [23, Chap. IX]. The proof will be carried out in several steps. Step 1. For every h 6 RN we define the translation operator Th,

: LQIRNI -> Lq(R"),

Thf(1u)

= f(w + h),

of

Q

Lq(1Re.N), :c

Q

RN.

Fix Q G [1»p*)» P* = N p / ( N - p) and aradius R > 0. Let a E (0, l] be defined by l sq, S > 0 and the fact that P6 (y)dy is a probability measure we deduce q

B

IT-yu(wl - u(t)l¢>6(:u)dy

IT-yu(wl - u(w)l"pa(y)dy

< B6

we deduce

106 * u(a:)

_ u(@)lq
2'(R") is a bounded of functions with supports contained in BR. We need to prove that for any e > 0 we can find a finite cover of 'Ll with Le(BRl-balls of radius 5. To prove this, we set U0:

N B1~(f12)

7'

ld.(y)lqdy S 017'-

'VQ/JQRA

0 0. Thus 1x1

aw)

- u(0)l

C'

t'*'1dt

all"

0

The energy estimate is a very useful tool in the study of nonlinear elliptic equations. The Morley embedding theorem can be complemented by a compactness result. Let (k,p) 6 2+ X [l, oo] and (m, a) 6 Z+ X (0, l ) such that

0N(k,p) > 07(m,0) k > m.

480

Lectures on the Geometry

of Manzfolds

Then a simple application of the Arzela-Ascoli theorem shows that any bounded sequence in L"'=P(RN ) admits a subsequence which converges in the cYTrn,a norm on any bounded open subset of RN. The last results we want to discuss are the interpolation inequalities. They play an important part in applications, but we chose not to include their long but elementary proofs since they do not use any concept we will need later. The interested reader may consult [4] or [15] for details.

Theorem 10.2.29 (Interpolation inequalities). For each R > 0 choose a smooth, c u t # function UR 6 03° (RN) such that 'UR

if lac° < R

7

'UR EO ifl5UI>R-i-1, and

ld17R(:c)l

[l,ool, and (ha) 6 Z+

Fix (m,pl 6

L

(a) For every 0

< 7°
0 and

X

0, and for all 'U 6 C'k"* ( R N ) , we have

j < k,

HHURUHHMN -< 0ll»Rull,,Rn 0e-j45)

do,

WE E CO

Note the following obvious inclusion {Lp - weak solutions} D {LP - strong solutions}.

The principal result of this subsection states that when L is elliptic the above inclusion is an equality.

Theorem 10.3.6. Letp 6 (1, oo) and suppose that L : COO(E) -> cOO (E) is an elliptic operator of order m, with smooth coejicients. If u 6 L O C and Lu 6 L then u 6 LloePlEl°

490

Lectures on the Geometry of Manz]'old5

Remark 10.3.7. Loosely speaking the above theorem says that if a "clever" (i.e., elliptic) combination of mixed partial derivatives can be defined weakly, then any mixed paltial derivative (up to a certain order) can be weakly defined as well.

Proof. 2 We prove only two special special cases of the theorem. Namely we assume that L has either constant coefficients or it is a first order operator. The essential ingredient in the proof is the technique of mollification. For each 5 > 0 we denote by TO the operator of convolution with p6. Fix a radius R > 0 and a partial differential operator S of order m > 0. We first prove that there exists a constant C' > 0 depending only on R, N, p, S such that

II [S,T6]u HpaBR+6

_< CIIUIIM-1,P»BR+1) Vu E

00

(l0.3.7)

(8R+1)-

Using the equality

[s1s2, $51 = [So T5]S2 + SO $s2, TO] we deduce that if (10.3.7) holds for the operators $1 and $2 of positive orders then it also holds for their product $152- Clearly (10.3.7) is trivially satisfied for operators S with constant coefficients because in this case [S, TO] = 0. Thus, it suffices to verify (10.3.7) for first order operators of the form a(:c)3,~ -|- b(;I:). Denote by Mb : 05° -> 03° the operator of multiplication of by the smooth bundle morphism b(a2). Then [Mb) T6]u(a:)

Rn P6(3U - y)( b(:z:l - 6(9) Lu(y)dy,

and the smoothness of b implies that there exists a constant C' > 0 independent of 5 > 0 such that

II [Mb,T6] lIP»BR+6 < CIIUIIP,BR+1 v

Vu G CO (BR+1)-

Next observe that

[

T6]u(w)

Z

(@w(H 1 -

y)

/

a(;u)3¢

=

1 6n+1

Rn

/

Rn

M23 - you(uldy

a

0

+6 1+1

/

-8y~0(;v

- y))

Rn

I )

8yi p

.CU` y

ac - y

5

MY

6n+1

lx-yl 0 to a section U0 6 L"""'(BRl. On the other hand, from the Rellich-Kondrachov compactness theorem we deduce that Thru converges in the LLp-norm to up. This implies that U0 = u so that u E Lloep' Remark 10.3.8. The proof of the general case of the regularity theorem requires considerable more analytic work than we are willing to include in a geometry book. However, the general idea is easy to describe. Using the theory of pseudo-differential operators, one can construct a parametrix for any elliptic operator L on RN T. This is an integral operator T : C8° (RN ) -> C°°(RN ) which serves as a sort of inverse for L, i.e., the operators S = LT - II and Se = T L - II are smoothing operators. This means that they map locally integrable functions (and more generally, distributions) to smooth functions. The convolution with PA is a prime example of smoothing operator, while the convolution with the function lacl2'N is the parametrix of the (geometers') Laplacian on RN . If Lu = 'U weakly then Tv=TL'u,=u+Sgu.

Thus To - 'LL is a smooth function, and thus it suffices to prove that To 6 LlOCp . This is achieved, again using the theory of pseudo-differential operators, the Calderon-Zygmund inequality, and the special case of Theorem 10.3.6 when L is a constant coefficients operator. For details we refer to [126].

492

Lectures on the Geometry

Corollary 10.3.9. [few 6 U

LCIEI i

|

of Manzfolds

of Lu = u, 1 < < 1° < R we have

weak LLp-solution

6 L I E ) , then u 6 L?O*""°"(E), and for every 0

p


>7 via 6 00

.

To put it differently, the section Qu is a LLp-weak solution the equation Ll8z"LLl

=

6i1)

-

[g L]u.

Observe that [a L] is a p.d.o. of order g m which means that the right-hand side of the above equality is in since the inductive assumption implies that u 6 LiO;1+M'P. . . . . . k-1 . . Invoking the inductive assumption again we deduce that 3-7u, 6 Lloc --M'P, 'v'z, 1.e., 'U 6 Le+m,p

L5;;1"'

loe

•

To prove the elliptic estimate pick a sequence un 6 C3°(E) such that u n - > u strongly in Lzko` m'p(El-

Then lc m, Lun -> Lu strongly in Ll0+ PIEI,

and

llunllm+k,p,8, < C(llunll0,p,8R + ||L'w-||k,p,8R)The desired estimate is obtained by letting n -> oo in the above inequality.

Corollary 10.3.10 (Weyl Lemma). If 'LL E and 'U i5 smooth, then u must be smooth.

LQDOCIEI

is an LLp-weak solution of Lu

k Vi E ZJ Hence u 6 Proof. Since 'U is smooth we deduce U E LlOP Morley embedding theorem we deduce that u 6 ClOCO Wm 0.

LlkO+m'p

U,

Wk. Using

The results in this, and the previous subsection are local, and so extend to the more general case of p.d.o. on manifolds. They take a particularly nice form for operators on compact manifolds.

493

Elliptic Equations on Manifolds

Let (M, g) be a compact, oriented Riemann manifold and E, F -> M be two metric vector bundles with compatible connections. Denote by L E PDO"'(E, F) an elliptic operator of order m.

Theorem 10.3.11. (a) Lets 6 I;1>(E) and

M

Then

'LL

(u, Q5)Fd'U9

6 Lfwlpl, 1

0.

If un ->u strongly in L1'2(M) we deduce using the Fatou lemma that

M

F


0 such that

ldul2dvg 2 C M

lull2dV9, Vu G Ll 2(M). M

Proof. We argue by contradiction. Assume that for any 6 > 0 there exists u5 E L1,2(M), such that IU_LI2dVg

= l,

ldu5l2dvg

and

M


quence

11,

L1,2lM

). Since

weakly in L1»2(M).

L2lml is compact, (Rellich-Kondrachov) so that, on a subseb J_E - > v strongly in L2lml.

This implies 'U 7é 0, since M

Wdvg

lull2dvg = 1.

lim 5

M

On the other hand,

duE

du -> 0, strongly in L2.

500

Lectures on the Geometry

of Manzfolds

We conclude

(do, d¢) dog

WE E L1)2(M).

M

In particular, (do, do) dog = 0, M

so that do = 0. Since M is connected, we deduce 'U lim

'v(w)dVg(sl

C

M

C

const. Moreover,

uJ_dvg

0.

M

5

This contradicts the fact that 'U ¢ 0. The Poincaré inequality is proved.

Ll,2lml such that

We can now establish the coercivity of I(u). Let I 0, and u 6

lV M

2 ldu(H>)l2 + F(u(w))

s(:L°)u(:z:)

dvglxl 1. Invoking elliptic regularity again we deduce that u 6 I;4» 1. (At this point it is convenient to work with f , rather than with p which was only 02.) Feed this back in h(a:), and regularity theory improves the regularity of 'U two orders at a time. In view of Morrey embedding theorem the conclusion is clear: u 6 COO(M). The proof of Theorem 10.3.15 is complete. From the theorem we have just proved we deduce immediately the following consequence.

Corollary 10.3.24. Let ( M , g) be a compact, connected, oriented Riemann manifold and s(;c) 6 C ° ° ( M ) . Assume volg(M) = l. Then the following two conditions are equivalent. m l ; = [M s(a3)dVg > 0 ( b ) For every A > 0 there exists a unique u = UA 6

cOO

( M l such that

Au -|- Ae" = s(:z:). Proof. (a) => (b) follows from Theorem 10.3.15. (b) => (a) follows by multiplying (10.3. 15) with 'u(:c) that 5

A

(10.3.15)

1, and then integrating by parts so

e"'(as)dVg(a:) > 0. M

502

Lectures on the Geometry of Manz]'old5

Although the above corollary may look like a purely academic result, it has a very nice geometrical application. We will use it to prove a special case of the celebrated uniformization theorem. Definition 10.3.25. Let M be a smooth manifold. Two Riemann metrics 91 and 92 are said to be conformal if there exists f 6 C°°(M), such that 92 = 6f91. Exercise 10.3.26. Let ( M , g ) be an oriented Riemann manifold of dimension N and f E COO(M). Denote by g the conformal metric g = egg. If s(ac) is the scalar curvature of g and § is the scalar curvature of § show that e -f

5(a3

>+ (n

- II Agf

1> 2 is classically called a uniformizing parameter. It is very similar to the angular coordinate 9 on the circle S1. This is a uniformizing parameter on the circle, which is an "inverse" of the universal covering maple 9 9 +-> et 6 S1. In the following exercises (M,g) denotes a compact, oriented Riemann manifold without boundary.

Exercise 10.3.31. Fix e > 0. Show that for every f 6 I;2(M) the equation Au -|- cu = f has an unique solution 'U G L2,2lMl_ Exercise 10.3.32. Consider a smooth function f : M X R -> R such that for every :IJ 6 M the function u I-> f(H1» u) is increasing. Assume that the equation Age = f(:1:,u) (10.3.17) admits a pair of comparable sub/super-solutions, i.e., there exist functions up, U0 6 L1'2(M) VW LOO(M) such that U0(;z:) > u0(;z:), a.e. on M, and _ Ague, weakly in L1,2(Ml. Ag(/'0 > f(w7 U0(H:)) > f(a:,u0(;z:)) > Fix C > 0, and define IU/'nln21-C L2,2(Ml inductively as the unique solution of the equation Au,,,(ac°) -|- cun(.§c) = cu,,_1(:13) -|- f(ac°, 'uun_1(ac)).

(a) Show that u0(:c) < u1(:z:) _< u2(.5c) < < uTL(.5c) < < U0(x) Va: 6 M. (b) Show that uTL converges uniformly on M to a solution 'LL E COO(M) of (10.3.17) satis-

lying U0

< 'LL
Y be a linear operator, not necessarily continuous, defined on the linear subspace D ( T ) C X . The operator T is said to be densely deaned if its domain D ( T ) is dense in X . The operator T is said to be closed if its graph, II

..

FT

{(@,To) G DID)

X

YCX

X

Y }7

is a closed subspace in X X Y . (b) Let T : D ( T ) C X -> Y be a closed, densely defined linear operator. The adjoint of T is the operator T* : D(T*) C Y -> X defined by its graph

FT*

{(y*,;I:*) Q Y x X ; (:z:*,;/:l = (y*,T:z:) Va: 6 D(T),

.>

where (-, : Z X Z -> K denotes the inner product3 in a Hilbert space Z. (c) A closed, densely defined operator T D ( T ) C X -> X is said to be self-adjoint if

T

I

T*.

Remark 10.4.2. (a) In more concrete terms, the operator T : D ( T ) C X -> Y is closed if, for any sequence (izinl C D ( T ) , such that (in,Txn) -> (:z:,y), it follows that (i) as 6 D(T), and (ii) y = T:1:. (b) If T : X -> Y is a closed operator, then T is bounded (closed graph theorem). Also note that if T : D(T) C X -> Y is a closed, densely defined operator, then kerT is a closed subspace of X . (c) One can show that the adjoint of any closed, densely defined operator, is a closed, densely defined operator. (d) The closed, densely defined operator T : D ( T ) C X -> X is self-adjoint if the following two conditions hold. 3 When K

i.e., (2:1,A2:2)

(C, _we use the convention that the inner product ( Alzl, Z2), VA E C, 2 1 , 2 2 E Z.

- > is conjugate linear in the second variable,

505

Elliptic Equations on Manifolds

(Taz, y) =

D(T)

(QC,

{y

I

T y ) for all :1:, y 6 D ( T ) and

E X;

EIC > 0, l(T11,y)l _
L2(E), D(La) = L'2(E),

given by u »-> Lu for all u 6 L"'=2(E).

Proposition 10.4.4. (a) The analytical realization La of L is a closed, densely a'e]?ned linear operator. (b) IfL* : COO ( F ) -> cOO ( E ) is the formal adjoint o f L then

(Fla

I

(LJ

In particular, the analytical realization La is self-adjoint if and only if L is formally selfadjoint.

Proof. (a) Since c°

Un

Fu,

strongly in L2(E), and Lun -> 'U strongly in L2(F).

From the elliptic estimates we deduce IUn -

'Uflnllk S C( HI/Un - Lumll +

-

umm)

-)

0

51S

mm, ->oo.

Hence (un) is a Cauchy sequence in Lk,2(E), so that un -> u in Lk,2(E). It is now clear that

'U

= Lu.

(b) From the equality

(u, L*v)dVg , Vu G L'°»2lE),

(Lu, v)dVg

'u Q

L/"»2(F>,

M

M

we deduce *

_

*

D( (La) ) 3 D ( ( L )a) - L

k,2

(F),

and (La)* = (L*)a on L'*'2(F). To prove that

D( (La>*) C D ( I m p ) = LWF) we need to show that if 'U 6 L2 (F) is such that EIC > 0 with the property that (Lu,v)dVg _< M

CIIuII

7

Vu 6 L'°'2(E),

506

Lectures on the Geometry

of Manzfolds

then 'u 6 L":=2(F). Indeed, the above inequality shows that the functional

(Lu, v)dVg

u »-> M

extends to a continuous linear functional on I;2(E), Hence, there exists 6 L2(E) such that M

(Fu, ¢)dvg Vu E Lk»2(El.

( m m , 'U )dVg

M

In other words, 'U is a L2-weak solution of the elliptic equation L*'u regularity theory we deduce 'U 6 L'"'2(M). he proposition is proved.

Using elliptic

Following the above result we will not make any notational distinction between an elliptic operator (on a compact manifold) and its analytical realization.

Definition 10.4.5. (a) Let X and Y be two Hilbert spaces over K = R, (C and suppose that T : D ( T l C X - > Y is a closed, densely defined linear operator. The operator T is said to be semi-Fredholm if the following hold. (i) dim kerT < oo and (ii) The range R (T) of T is closed. (b) The operator T is called Fredholm if both T and T* are semi-Fredholm. In this case, the integer

ind T :=

diIIl]K

her T - dim her T*

is called the Fredholm index o f T . Remark 10.4.6. The above terminology has its origin in the work of Ivar Fredholm at the twentieth century. His result, later considerably generalized by F. Riesz, states that if K : H -> H is a compact operator from a Hilbert space to itself, then HH -|- K is a Fredholm operator of index 0.

Consider again the elliptic operator L of Proposition 10.4.4. Theorem 10.4.7. The operator La : D(Lal C L2(El ->

L2lFl is Fredholm.

Proof. The Fredholm property is a consequence of the following compactness result.

Lemma 10.4.8. Any sequence (Unln20 C

Lk,2lEl such that { flu,,ll

-I-

n>0

i5

bounded contains a subsequence strongly convergent in L2 (El.

Proof. Using elliptic estimates we deduce that

Ilunllk

_< C(llLunll + mun) _< const.

Hence (un) is also bounded in L k , 2 (E). On the other hand, since M is compact, the space Ll*'2(E) embeds compactly in L2(E). The lemma is proved.

507

Elliptic Equations on Manifolds

We will first show that dim her L < oo. In the proof we will rely on the classical result of F. Riesz which states that a Banach space is finite dimensional if and only if its bounded subsets are precompact (see [23], Chap. VI). Note first that, according to Weyl's lemma her L C C°°(E). Next, notice that her L is a Banach space with respect to the L2-norm since according to Remark 10.4.2 (a) her L is closed in L2lEl. We will show that any sequence (uTL) C kerL which is also bounded in the L2-norm contains a subsequence convergent in L2. This follows immediately from Lemma 10.4.8 since ll'u.nll -|- ||Lu,,|| = ll'u.,,ll is bounded. To prove that the range R ( T ) is closed, we will rely on the following very useful inequality, a special case of which we have seen at work in Subsection 9.3.3. Lemma 10.4.9 (Poincaré inequality). There exists C > 0 such that

IIuII 0

as m,n

-> oo.

in L k , 2 (E). Clearly Lu

"U,

508

Lectures on the Geometry of ManQ'olds

We have so far proved that her L is finite dimensional, and R (L) is closed, i.e., La is semi-Fredholm. Since (La)* = (L*)0, and L* is also an elliptic operator, we deduce (La)* is also semi-Fredholm. This completes the proof of Theorem 10.4.7. Using the closed range theorem of functional analysis we deduce the following important consequence. Corollary 10.4.10 (Abstract Hodge decomposition). Any k-zh order elliptic operator L : cOO ( E ) -> cOO ( F ) over the compact manifold M defines natural orthogonal decompositions ofL2(E) and L2lFl,

L2(E) = her L QUO R (L*) and

L2lFl =

kerL* ® R(L).

.

Corollary 10 4.11. If her L* = 0, then for every 'U Q L2 ( F ) the partial a'Qj'erential equation Lu = 'U admits at least one weak L2-solution u 6 L2 (E). Corollary 10.4.12. Suppose that L is additionally formally self-adjoint. If k e r L = 0, then for any j E Z20, the induced linear map Lk-I-j,2lEl -> LJ»2(F) is a topological isomorphism. The last corollary is really unusual. It states the equation Lu = 'U has a solution provided the dual equation L*v = 0 has no nontrivial solution. A nonexistence hypothesis implies an existence result! This partially explains the importance of the vanishing results in geometry, i.e., the results to the effect that her L* = 0. With an existence result in our hands presumably we are more capable of producing geometric objects. In the next chapter we will describe one powerful technique of producing vanishing theorems based on the so called Weitzenbéck identities.

Corollary 10.4.13. Over a compact manifold her L = her L*L and her L* Proof. Clearly her L C her L*L. Conversely, let W

lIe¢ll

2

M

(Lab/l¢)dVg

Q

her LL*.

C°°(E) such that L*L1b = 0. Then

I/;*L(p,zpldvg = 0. M

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. Unfortunately, this would require a lot more extra work to include it here. However, in the remaining part of this subsection we will try to unveil some of the natural beauty of elliptic operators. We will show that the index of an elliptic operator has many of the attributes of a topological invariant. We stick to the notations used so far. Denote by Ell klEvFl the space of elliptic operators cOO (E) -> cOO (F) of order k. By using the attribute space when referring to Elli we implicitly suggested that it carries some structure. It is not a vector space, it is not an affine space, it is not even a convex set. It is only a cone in the linear space PDO( m ) but it carries a natural topology which we now proceed to describe.

509

Elliptic Equations on Manifolds

L€tL1, L2 Q Ell k(E7 F ) . We set 6(L1, LQ) = sup{llLlu -

L2'UII3 IIuII k

1}

Define

dll

1,

LQ)

I

1'Il3JX

{(S(L1, LQ), 6(L1, L2l} .

We let the reader check that ( Ell k , d) is indeed a metric space. A continuous family of elliptic operators ILAIAQA, where A is a topological space, is then a continuous map

A E } I - > L ) € Ell k Roughly speaking, this means that the coefficients of LA, and their derivatives up to order /Q depend continuously upon A.

Theorem 10.4.14. The index map ind : Ell 1Z,

L +-> ind (L)

is continuous.

The proof relies on a very simple algebraic trick which requires some analytical foundation. Let X, Y be two Hilbert spaces. For any Fredholm operator L : D(L) C X -> Y denote by 'LL : kerL -> X (respectively by PL : X -> her L) the natural inclusion kerL --> X (respectively the orthogonal projection X -> her L). If L : D(LZ~) C X -> Y (i = 0, 1) are two Fredholm operators define

72Lo(L1) : D(L1) ® kerL0 C X ® kerL0 -> Y ® kerL0 by

72L0(L1)(u, (be = (LW

+ ZL5€25,PL0Ul,

u Q D(L1l, QUO Q her L0-

In other words, the operator RL0 (Ll) is given by the block decomposition /1

RIG (Ll)

ZL5

o_fL1

at Lo. The operator L0 is called the center We will call RL0 (Ll) is the regularization of the regularization. For simplicity, when L0 = L1 = L, we write

RL

RLILI.

The result below lists the main properties of the regularization.

Lemma 10.4.15. (a) RL0 (Ll) is a Fredholm operator: (b) R20(1;1) RL8 (LT)(c) RL0 is invertible (with bounded inverse). I

Exercise 10.4.16. Prove the above lemma.

510

Lectures on the Geometry

of Manzfolds

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. It is a very good "routine booster".

Proof of Theorem 10.4.14. Let Lo E Ell VL Q Ell k(E, F) satisfying d(L0, L) < r, we have

We have to find 1 ° > 0 such that,

ind (L) = ind (LQ). We will achieve this in two steps. Step 1. We will find 1°> 0 such that, for any L satisfying d(L0,L) < T , the regularization of L at L0 is invertible, with bounded inverse. Step 2. We will conclude that if d(L, Lo) < r, where r > 0 is determined at Step 1, then ind aLl = ind (Lo). Step 1. Since RL0 (L) is Fredholm, it suffices to show that both RLoILI, and 'R,Lo*lL*l are injective, if L is sufficiently close to Lo. We will do this only for RL0 (L), since the remaining case is entirely similar. We argue by contradiction. Assume that there exists a sequence (un, (in) C L""2(E) X her Lo, and a sequence (Ln) C Ell k(E, F) such that

Ilunllk + llcbnll Rho (Le)(un» (in)

=

1,

(10.4.1)

10701,

(10.4.2)

and (10.4.3)

From (l0.4.l) we deduce that l¢nl is a bounded sequence in thejinite dimensional space her Lo- Hence it contains a subsequence strongly convergent in L2, and in fact in any Sobolev norm. Set $5 := lim (in. Note that llg2§ll = limn II¢nll. Using (l0.4.2) we deduce Lour

-0%7

i.e., the sequence (Lour) is strongly convergent to now gives

QUO in I;2(F). The condition (10.4.3)

lIL0un1 L'nunll < 1/n, i.e., lim Liun = lim Loun TL

= - OO. ilk strongly converges in L k , 2 to some u. Moreover,

Uvrl

lim Ilunllk TL

I

Ilullk

and

Ilullk -|- ||¢||

I

1.

511

Elliptic Equations on Manifolds

Putting all the above together we conclude that there exists a pair (we) E her L0» such that

Ilullk + II¢ll Lou

Lk,2lEl

X

1,

I

and u J_ her L.

(10.4.4)

This contradicts the abstract Hodge decomposition which coupled with (10.4.4) implies 'LL = 0 and = 0. Step 1 is completed.

Step 2. Let T' > 0 as determined at Step 1, and L

Q

Ell ilE,Fl. Hence

RIG (L) is invertible. We will use the convertibility of this operator to produce an injective operator

her L* ® her Lo --> kerL ® her L0. This implies dim her L* -|- dim her L0 < dim her L + dim her Lo, i.e., ind (Low < ind(L). A dual argument, with L replaced by L*, and L0 replaced by Lo, will produce the opposite inequality, and thus finish the proof of Theorem 10.4. 14. Now let us provide the details. First, we orthogonally decompose

IPIEI = (her L ) l

® her L and

HIFI = (her L*>i ® her L*.

Set U := her L ®ker Lo, and V := her L* éBker L0. We will regard RL0 (L) as an operator

RL0 (L) : (her L)l ® U -> (her L*l1 ® V As such, it has a block decomposition

72L0(L) =

A C'

7

where T : Lk'2(E) O (ketL)l C (kerL)

-> (her L*l-L = Range (L)

denotes the restriction of L to (her L)_L. The operator T is invertible and its inverse is bounded. Since 72L0(L) is invertible, for any 'U E V we can find a unique pair (aW) Q (kerlll ® U, such that

H

RLoIL) 'LL = u(v) is injective.

(z5('vl and

'LL

= u(v). We claim

512

Lectures on the Geometry

of Manzfolds

Indeed, if u(v) = 0 for some "u, then Tqzé = 0, and since T is injective U. Theorem 10.4.14 is proved.

The theorem we have just proved has many topological consequences. We mention only one of them.

Corollary 10.4.17. Let Lo, L1 ind lLll.

Q

Ell k ( E , F ) Q" 0k1.r(L0) = 0k(L1) then ind(L0)

Proof. For every t 6 [0, 1] Lt (1 - t)L0 + tL1 is a k-th order elliptic operator depending continuously on t. (Look at the symbols.) Thus ind (Lt) is an integer depending continuously on t so it must be independent of t. This corollary allows us to interpret the index as as a continuous map from the elliptic symbols to the integers. The analysis has vanished! This is (almost) a purely algebraictopologic object. There is one (major) difficulty. These symbols are "polynomials with coefficients in some spaces of endomolphism". The deformation invariance of the index provides a very powerful method for computing it by deforming a "complicated" situation to a "simpler" one. Unfortunately, our deformation freedom is severely limited by the "polynomial" character of the symbols. There aren't that many polynomials around. 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) . At this point one should return to analysis, and try to conceive some operators that behave very much like elliptic p.d.o. and have more general symbols. Such objects exist, and are called pseudo-dy erential operators. We refer to [83, 121] for a very efficient presentation of this subject. We will not follow this path, but we believe the reader who reached this point can complete this journey alone. Exercise 10.4.18. Let L Q Ell k(E, F). A finite dimensional subspace V C I;2(Fl is called a stabilizer of L if the operator SL , V :Lk'2(E) ® V -> L2(F)

SL,V(U

® u) = Lu

+ 'U

is surjective. (a) Show that any subspace V C L2(F) containing her L* is a stabilizer of L. More generally, any finite dimensional subspace of I;2(F) containing a stabilizer is itself a stabilizer. Conclude that if V is a stabilizer, then

ind L

dim

k€I` S L y

dim V.

Exercise 10.4.19. Consider a compact manifold A and L : A -> Ell ME, F) a continuous family of elliptic operators. (a) Show that the family L admits an unQ'orm stabilizer, i.e., there exists a finite dimensional subspace V C L2 (F), such that V is a stabilizer of each operator LA in the family L.

513

Elliptic Equations on Manifolds

(b) Show that if V is an uniform stabilizer of the family L, then the family of subspaces k€IISL ,V defines a vector bundle over A. (c) Show that if VI and VI are two uniform stabilizers of the family L, then we have a natural isomorphism vector bundles her SLy1 ®

r\/

_2

her $L,v2 G E .

In particular, we have an isomoiphism of line bundles

det her SL »VI

(X)

Vi* 2

det

det her SL,v2

(X)

det

v;

Thus the line bundle det her SL,v (X) det V* -> A is independent of the uniform stabilizer V. It is called the determinant line bundle ofthefamily L and is denoted by det ind (Ll-

10.4.2 Spectral theory We mentioned at the beginning of this section that the elliptic operators on compact manifold behave very much like matrices. Perhaps nothing illustrates this feature better than their remarkable spectral properties. This is the subject we want to address in this subsection. Consider as usual, a compact, oriented Riemann manifold (M, g), and a complex vector bundle E -> M endowed with a Hermitian metric ( , ) and compatible connection. Throughout this subsection L will denote a k-th order, formally self-adjoint elliptic operator L : COO(E) -> C°°(E). Its analytical realization

..

La : Lk,2lE) C L2(El -> L2(E), is a self-adjoint, elliptic operator, so its spectrum spec(L) is an unbounded closed subset of R Note that for any A 6 R the operator A - La is the analytical realization of the elliptic p.d.o. All E - L, and in particular, the operator A - La is a Fredholm, so that A G spec:(L)

I

I \

\ : )

L)¢0.

ker(A

Thus, the spectrum of L consists only of eigenvalues of finite multiplicities. The main result of this subsection states that one can find an orthonormal basis of L2(El which

diagonalizes La.

Theorem 10.4.20. Let L 6 E11k ( E I be a formally self-adjoint elliptic operator: Then the following are true. (a) The spectrum spec(L) is real, spec(Ll C R, and for each A E spec(Ll, the subspace ker(A - Ll is jinite dimensional and consists of smooth sections. (b) The spectrum spec(Ll is a closed, countable, discrete, unbounded set. (c) There exists an orthogonal decomposition

L2(El

ker(A - L) .

I

A€spec(L)

(d) Denote by P the orthogonal projection onto ker(A - L). Then

Lk,2(E)

D(La)

A2llp¢ll

w€L2(E), A

2

-> 0°°(s1,(c>. The operator L is clearly a formally self-adjoint elliptic p.d.o. The eigenvalues and the eigenvectors of L are determined from the periodic boundary value problem

Au, 7u,(0) = u(27r), which implies

u(0l = C'exp(iA0) and exp(21rAil = 1.

Hence spec(L) = Z and ker(n - L)

span@{exp(in9)}.

The orthogonal decomposition

LQISII

ker(n - L) n

is the usual Fourier decomposition of periodic functions. Note that 71(0)

Un

€Xp(in9) 6 L1)2(51)

TL

if and only if

(1 -|- n2)|un|2 < oo. 1162

516

Lectures on the Geometry of ManQ'olds

The following exercises provide a variational description of the eigenvalues of a formally self-adjoint elliptic operator L Q E11k (E) which is bounded from below, i.e., inf

{/

M

(Lu,u)d»g; u Q L E ) ,

Hun

1

Exercise 10.4.22. Let V 6 LL¢»2(E) be a finite dimensional invariant subspace of L, i.e., L(V) C V. Show that (a) The space V consists only of smooth sections. (b) The quantity

Aly_Ll

inf

{/

(L'u,, u)dtug; u 6 Lk,2lE) re M

v

Hun

1

is an eigenvalue of L. (V denotes the orthogonal complement of V in L2lE)

Exercise 10.4.23. Set V0 = 0, and denote A0 = ,\(V0J_). According to the previous exercise A0 is an eigenvalue of L. Pick (be an eigenvector corresponding to A0 such that lld)0 I = 1 and form VI = Vi GB span {gz50}. Set $1 = Alvl_Ll and iterate the procedure. After m steps we have produced m -|- l vectors QSo, QS1, . . . ,(/)m corresponding to m -|- l eigenvalues A0,/\1 S ' • ' Am ofL. Set Vm+1

and Am+1

== A(v£+1) etc.

spanc{g250, (Zh, •

• • 7 (1)m}

(a) Prove that the set

{¢1,I.I,¢m,---} is a Hilbert basis o f L2(18), and _ Am . . . } . spec(L) ={/\1 S ° ° °
IR is an eigenfunction of ASn,-1 if and only if there exists a homogeneous harmonic polynomial p such that go = p, (c) Set As, := n(n -|- lc - 2). Show that

dim her (As

A Sn

1

k-I-n n

'l

(

k+n 3 n 2

(d) Prove Weyl's asymptotic estimate (10.4.5) in the special case of the operator A Sn

1.

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Lectures on the Geometry

10.4.3

of Manzfolds

Hodge theory

We now have enough theoretical background to discuss the celebrated Hodge theorem. It is convenient to work in a slightly more general context than Hodge's original theorem. Definition 10.4.28. Let ( M , g ) be an oriented Riemann manifold. An elliptic complex is a sequence of first order p.d.o.'s 0 -> 0(E0)

D

c°°(El> 9%

Dm

1

-> c°°(E,,,> -> 0

satisfying the following conditions.

(i) (C°°(E), DZ-) is a cochain complex, i.e., D¢D¢-1 = 0, VI < i < m . (ii) For each (co, go Q T*M \ {0} the sequence of principal symbols

0-> (Eolx

o(

0(D0)(> ,€)

D

'rn_

1

3(x,€) (Et

-> 0

is exact. Example 10.4.29. The DeRham complex (Q* ( M ) , d) is an elliptic complex. In this case, the associated sequence of principal symbols is (e(§) = exterior multiplication by go 0 -> R eos T;M

e(q

eos set(T;m) -> 0

is the Koszul complex of Exercise 7.1.23 of Subsection 7.1.3 where it is shown to be exact. Hence, the DeRham complex is elliptic. We will have the occasion to discuss another famous elliptic complex in the next chapter. Consider an elliptic complex (COO (E, D. )) over a compact oriented Riemann manifold (M, g). Denote its cohomology by H' ( E , D.l. A priori these may be infinite dimensional spaces. We will see that the combination ellipticity + compactness prevents this from happening. Endow each E with a metric and compatible connection. We can now talk about Sobolev spaces and formal adjoints D: . Form the operators AS

D - D + D¢_1D-_1 : COOlEd) -> COOlE@l.

We can now state and prove the celebrated Hodge theorem. Theorem 10.4.30 (Hodge). Assume that M is compact and oriented. Then the following are true. (i) H{(E.,D.) E kerr C C°°(E¢)! Vi. (ii) dim H¢(E., D.) < oo (iii) ( Hodge decomposition). There exists an orthogonal decomposition

w.

L2(Eil = ke1~A,-

Q9

R (DH) ® R

ID. I,

where we view both D1-1, and Di as bounded operators L1,2 -> L2.

519

Elliptic Equations on Manifolds

Proof. Set E

® D?7

65E2 7 D

*7 A = ®

D*

Thus D, D*, A E PDO ( E , E). Since D D _ 1 implies

A

I

,D=D+D*

QUO Di

D*D+DD*

I

0, we deduce D2

(D +D *)2

I

(D*l2 = 0 which

DQ.

We now invoke the following elementary algebraic fact which is a consequence of the exactness of the symbol sequence.

Exercise 10.4.31. The operators D and A are elliptic, formally self-adjoint p.d.o. (Hint: Use Exercise 7. 1.22 in Subsection 7.1.3.) Note that according to Corollary 10.4.13 we hav her A orthogonal decomposition

L2 her At. Set

r u e c°°(E); D o , = 0 } and 1% L

Di-llCOOlEi-lllv

so that

Hi(E., D ) = Z / B We claim that the map Pi : Zi -> her As descends to an isomorphism H E . , D . ) -> her As. This will complete the proof of Hodge theorem. The above claim is a consequence of several simple facts. Fact 1. her At C Zn'_ This follows from the equality her A = her D.

Fact 2. If u GZz- then u -

Pll,

6 B . Indeed, using the decomposition (l0.4.6) we have

p,-fu, -|-

u

Etp to 6

LE.

Since u - Pll, 6 C'°°(E), we deduce from Weyl's lemma that in 6 C°°(El. Thus, there exist 'U 6 COO(EZ~_1) and w 6 C°°(E¢) such that = 'U ® w and

w

u =

Pn

-|- Di_1'U -|-

D- w.

Applying Di on both sides of this equality we get 0 Since her D'i

D* w

I

her Dz~D, , the above equalities imply 'LL -

PL

I

Di_1'U E

Bi

We conclude that Pi descends to a linear map her As -> H'i (E, D. ) Thus Bi C

RID) =

(ketA)l,

520

Lectures on the Geometry

of Manzfolds

and we deduce that no two distinct elements in her As are cohomologous since otherwise their difference would have been orthogonal to her As. Hence, the induced linear map Pz- : ker As -> H¢(E. D.) is injective. Fact 1 shows it is also surjective so that k€IIAz'

-N

H l E . , D.

Hodge theorem is proved. Let us apply Hodge theorem to the DeRham complex on a compact oriented Riemann manifold

0 ->Q0(M)

d

d

> Qllml

>--- d>Qn(M)->0, n

dimer.

SY* ( M ) ->§2/c>o( k

7

(p)

(up), W

m.

G

The above definition can be strengthened by the following nontrivial result. For a proof we refer to [50, Sec. I.8] or [127, Sec. 33]. Theorem 11.1.5 (Uniform boundedness principle). Let 6° be an open subset ofR". Suppose that UP 6 C"°°(@'l, j We 6 @(@l the sequence

>

0, i5 a sequence

of

generalized functions such that,

(U (p) has a finite limit Llcp) E C Then the following hold.

(1) The correspondence @(@) 9 go »-> L((,ol 6 (C i5 continuous in the sense ofDefinition I1.1.1 and thus defines a generalized function. (2) If K C @ i5 a compact set and (got) 6 Q K W ) is a sequence of smooth functions converging uniformly togethere with all their derivatives to a smooth function go 6 %oo

525

Spectral Geometry

Definition 11.1.6. Any p.d.o. L 9(6) -> 2(6) extends to a linear operator

L C1-°°(@) -> C1-°°(@), according to the rule

..II

( L u , 95)

(u,L*(p), Vu 6 C-°°(@'), go 6 @(@),

where go denotes the complex conjugate of up. In particular, for any multi-index a 6 Z">0 and any u 6 C-00(6) the distributional derivative d a y of u is determined by the equality

(0,

0, :z: ] R

f(wl Then QL do

ac,

a2>0,

0,

a2

up

( u ) , Vu 6

Q

C-°°(@'0)7

Qlgo) C @(@1l.

c-

(/

Z -|-

R"

do

u(2)Ty (WI dz do,

y. If we now think of u, 'U as

(u*v,go) = ('v,(u,Ty(,0)).

Observe that for any u p

C-OO

luv) and any up 6 9(]R") the function

R" 9 y »-> W(yl is smooth. Since

'U

== (u)T:u(/)) 6 R

has compact support we can pair it with

w as in (11.1.2).

Definition 11.1.13. For u 6 c~°°(R") and 'U 6 c°°°(R")p we define u

*U

6 C'°°(]R"')

by the equality ( u * v , c , 0 ) := (v,(u,Ty(p)

Exercise 11.1.14. Let u 6

c-oo

)

= (7u,,(v,Tycp)).

(11.13)

(RI. Show that for any ICGZzQ we have

5§,'") * u =

u(k)

in the sense of distributions.

Remark 11.1.15. More generally, suppose that C C R " is a closed convex cone satisfying the properness condition

C N ( K - C ) is compact for any compact subset K C C . The condition (11.1.4) is satisfied if for example if the map C

X

C9(x,y)>a:+yeRTL

(11.1.4)

528

Lectures on the Geometry

of Manzfolds

is proper. Note that if u, 'u 6 C°°(]R'") are such that supp u, s u p p l C C, then, Vi 6 R" the function R" 9 y »->'u,(ac-y)'u(y) 6 (C

has compact support so the convolution of these functions. Suppose now that SO 6 WRn), K := supp up, and

My)

u(2)(/)(2 -|- y)dz, We 6 R"

Rn

Note that u(2)¢(2+y)¢0=>

60,3/+26

Z

K

-->»y6

K

C

so that suppzp 6 K - C.

In particular, supp v(y)¢(y) C K O (K - C) is compact so the integral Rn

'u(y)¢(y)dy

is well defined. More generally if u, 'U 6 C"°°(lR") are supported in a closed convex cone satisfying the properness condition (1 l.1.4), then the convolution 'U * 'U is well defined.

11.1.2 Currents For any open subset 6" C R" and any k < n we denote by Qk..F(6°) the space of kdimensional currents on 6 i.e., the topological dual of the space Qkpt W) of complex valued, degree k-forms on 6" with compact support. More precisely, a linear functional u:Qkpt(@°)->lC,

(p*->((/w)

belongs to QM) if it satisfies the continuity condition (11.1.1).

Example 11.1.16. (a) The bijection 9(6) 9 (or-> (pdf Ad;u1 /\

determines a natural bijection

C`°°(@') ((pA diff 1 /\

---

9 u »->

[u] 6 Qnlgl /\ d i m , M) = (we), WE

In particular, we have a bijection c-°(1R> ->

6

QW)-

QUO (R).

(b) If S C 6 is a properly embedded' k-dimensional oriented submanifold, possibly with

boundary, then integration along S defines a k-dimensional current [S], o n

(Q),

S

907 W )

E Qkpt(l°

We say that [S] is the integration current or current of integration defined by S. (c) A form 17 6 Q"-'*@) defines a Is-dimensional current [17], by setting (90»

n

SO

777

W E Q kc p t ( 6 l '

l A subset S C @ is properly embedded if the inclusion i : S W is a proper map.

529

Spectral Geometry

Definition 11.1.17 (Boundary operator). The boundary operator

3 : QI¢W')

-)

Qk_1(6')

is the dual of the exterior derivative d : Qkpt1 (@°) -> Q1°pt(6'), i.e., for any u E Qk(@) we have

(Q), [0

oo],

p)

Pal

:

sup X€R

n(£

U)k

lol$

k

l@§'S0(a2ll.

Then

in = 5'(R") := {(p 6 C'°°(R"l; pa,((,o) < oo, 'v'k E

ZZ()

The functions in Yn are called Schwartz functions. Let us observe that 2

QW) C in, 6-1x1 E in. The space in is equipped with a natural topology defined by the metric l . - m l n (1»PI o o i f f lim Pk( S0u -S0 l=0, 'v'k 2 0 . We have the following result, [66, Lemma 7.1.8]. v->oo

Lemma 11.1.19. The subspace Q (RTL) $5 dense in 57r1-

I.

530

of Manzfolds

Lectures on the Geometry

Definition 11.1.20. A temperate distribution on Rn is a continuous linear functional 'u,:5'n->C, i.e., there exists k 6 ZZ() and C > 0 such that

I

I

(Fu, (p) S Cpk((/0), W ) G YnWe denote by BYI the space of temperate distributions. We have strict inclusions

Ccp£°(R") C Y' C C'°°(]R"). We denote by (-, -) the canonical inner product on R" .

Definition 11.1.21. The Fourier transform on R" is the linear map

8? : Yn -> C'°°(]R"), Yn 9 up »-> ¢ = 5"[(p], e 'i(€'X)90(a:)d:z:,

6(6)

Rn

we 6 R"

We list below a few fundamental results concerning the Fourier transform. For a proof we refer to [39, Thm. l4.8].

Theorem 11.1.22. (a) For any go G Yn, we have QUO 6 in and the resulting linear map 5? : Y" -> Yn i5 continuous. (b) For any ( / ) , W € § ' n (3(=v)1D(1v)d1v

(0(w)(w)d2v-

R"

(11.l.5)

R"

(c) (Parsevalformula) we have ((p »1 P> L2 (R

")

1 (2¢f)" (¢,3)L2,,,

I

i.e.,

1 Rn

¢(w)W(w)dI =

(g71fln

Rn

@(go,z(gldg.

(d) (Fourier inversion formula) For any go 6 in we have 1 @(§»@)¢(§)dg. (,f)(;v) = (gffln Rn

(d) For any ¢,¢€%n, their convolution up * up*

is)

I

w is also a Schwartz function and A

¢( (u,

(no,

0.

Now choose ¢ 0 € 0 3 ° ( I ) such that (M I, Vo) Set CO

Lu,

Then u = colll- Indeed, if .

satisfies the linear dnterential equation

du dt

(1l.l.6)

+Awu=f(m

then u 6 C1 (I, RTL), i.e., 'LL is a classical solution of(II.1.6).

Proof. Choose

E 6 c°°(1,End(R">

)

such that E = EA, and E(tl is invertible for any t 6 I. We deduce that

d (E(t)u(t)l

I v E c1(I,R ") satisfies

U/

I

E(t)f(t)-

E(t)f(t l» then we deduce d

(E(ilu»(i) - v(t)) = 0.

Theorem 11.1.24 implies that there exists C 6 R" such that E(t)u(t) = v(t)

Hence 'u,(t) son.

+ C.

Eltl-1fvltl + E(t)'1c is a classical solution of the above differential equa-

Corollary 11.1.26. Suppose that I C R is an open interval, f ( t ) 6 00(1) [to 6 C"00(Il satisfies the deferential equation d1"u dim then 'U 6

com (In, i.e.,

'U

_I.

m-1 j

0

007

7

defy

as do

f(t)»

is a classical solution of the above equation.

Proof. Observe that the (vector valued) generalized function u(tl =

(Vu),1/(tl, . . . ,W-1))

6

c-oolnm

satisfies a linear first order ordinary differential equation with smooth coefficients and continuous nonhomogeneous term. Corollary 11.1.25 implies that u is a classical solution.

Corollary 11.1.27. Suppose that Q C R" is an open set and I C R is an open interval. We denote by x1, . . . , ac" the Cartesian coordinates on Q and by t the Cartesian coordinate on I. If u 6 C-©°(I X Q) is a generalized function satisfying the dnterential equation 3t'LL = 0, then there exists 'Ag 6 C-00(9) such that

(U, (ton al)

I

(u0(11)» (1502 ac))dt, WE E

on

X

Q).

(11.1.7)

533

Spectral Geometry

Proof. For any

w 6 9(9) define UP/» 6 C-00(1) by setting (UWP) =

W ) 6 Qu).

( l b / 2 ) ,

d Then EW = 0. Using Theorem 11.1.24 we deduce that Up is a constant function. The map

@(SO)

@C"°°(I)

9¢1->'Ll,¢

is obviously linear and continuous from the usual topology of @(SO) to the weak topology of C-00(I). The image of this map is the l-dimensional subspace consisting of constant functions so that the resulting map

@(Q)

We E R

9 ¢ l->

C-OOIQI satisfying (11.1.7)

is continuous. This defines a generalized function U0 6 M Q ( I X Q) of the form (W, I) (/0(t)¢(11)» i.e.,

(U, W )

Z

/

I

for

Z

u¢c,o(t)dt =

(/)(t)¢(I))dt~

(U0(2Ul»

The desired conclusion follows by observing that the linear span of functions of the form go(tl1D(:u), go E QU), AD 6 9(9) is dense in QU X Q).

For any open interval I C R, any open set 6" C R" we denote by CO,-OOII space of (weakly) continuous maps I 9 t -> it 6 C'°°(6'), i.e., t I->

X

@) the

SO)

(Ut e

is continuous for any up 6 2(6) Note that we have inclusions inclusion c0(I

X

al

C

CO, °°( I

X

@)) C 0-°o( I

X

WI.

The first inclusion associates to each continuou s function Fu, 6 CO(I of continuous functions on H, ut(a2) = u(t,;z:), Vat 6 6. Moreover

Vu

Q

c0»-°°

age

Q

X

c0»-°° 0,

0

plays an important role. In particular, we will use less familiar properties of this function. They can be found in [84, 134]. An important concept we will use in the sequel is that of holomorphicfamily ofgeneralizedfunctions. A family of generalized functions l u a l a e A 6 C'°°(lR) parametrized by an open subset A C (C is called holomorphic if, for any go 6 9 (R) the function

A 9a»->(ua,(p)e(C is holomorphic. The concept of meromorphic family is defined in a similar fashion.

11.2.1 Some classical generalized funetions on the real axis If a G C , Re a > -1, then the function ;z3 _I_:]R->lC defined by

a2>0,

0,

;USO,

o f

ata,

is locally integrable and thus defines a generalized function 1/3+ E C"°°(]R). A simple computation shows that x - x + - x + +1 , ' v ' R e a > - 1 ,

(11.2.1a)

d -1 x+=ax+ ,'v'Rea>0. do:

lb)

We want to extend the definition of 513+ to all a € ( C . For QSQQUR) the function OO

{Rea

> -1}

9 a»->Ia(gz5)

(iv"+» QS)

:z:"cp(a3ld:u 0

is holomorphic. Using (11.2.1b) iteratively we deduce

la

_

la((l))= ( a + 1

1

(a + k) '"*'" ( Q5(/ - l ,

k E Z20.

(11.2.2)

If a 6 C \ Z < 0 then we can define

I@((/2)

5 .. -1

(a +

l(a + k ) I " + / " ( ¢ ( k ) )»

Vi > max(0, -1

Rea

.

The equality (11.2.2) shows that the right hand side of the above equality is independent of the choice of k > 1nax(0, -1 - Re a).

537

Spectral Geometry

This gives a meromorphic extension of the function a »-> la((p) with simple poles located at Z!

¢('"-"(0), (1 l.2.3)

so that (-l)/

The correspondence

{R e a > - l } 9 a » - > X + E C'°°(]R) and in this half-plane it satisfies the equality d X++1

I

X+.

We can use this equality to extend X+ analytically to all a E (C,

doat

X+-I-n

(11.2.5)

538

of Manzfolds

Lectures on the Geometry

where n is a positive integer such that n -|- Re a > - l . Since X0+ is the Heaviside function we deduce do k l 5§,' ° ) (1 l.2.6) 60, We E Z20. X+ dank

Similarly, for Re a > - l we can define zuCL

0,

an 2 0,

txt",

cz:

< 0,

and for any a l

$U>

x-a (f l = Xa+l- so) = I`(a+ II 0,lxl07

;z:


Equivalently

(xa p) = (x+,¢),

Va) 6 QQR>,

where g(:z:) = go(-ac), Van E R.

Note that

in

a

da2'*X'+

n

=(-l)

n

a

x--

Observe that for Re a > - l we have :H+-I-:1:a = laval"'.

Let co 6

m). The function a *->(£6"'+» (ma ,g25) is meromorphic on (C with simple poles at

(k

CL

and the residue at a

1,

is 1 ( k _ l )¢E(k-1)(0)

la

1 _

1>!

(IS'"*), (15)-

Thus the function a »->

Gwlal :z

(l22l"',(l5)

is meromorphic on (I with simple poles at a

-1, - 3 , . . . , - ( 2 j + 1 ) ,

7

j e Z20.

539

Spectral Geometry

(Quo + l) is

The residue at a

2

(2)! The function a

»_>

1

(6é23), 0 and an integer N E Z>0 such that lf(;u-{-iy)l

< Clyl - N

a

Va: -i-i?/GZ,

then f(o + i y ) has limit f0 6 C'°°(]R) when y -> 0. Moreovelg this limit i5 a generalized function of order at most N -|- l . We denote it by f (co + i0).

540

Lectures on the Geometry of Manzfolds

Proof. Fix Z0

E

Z and consider

F(2)

f(c)dc250

Then

N >1

C01/ 11'N ,

IF(2)l
(I whose restriction to Z is holomorphic and such that dN-l-1

F(2l

dzN+1 G(z).

This implies that for fixed y we have dN-I-1

f(w

.

-|-

d:z:N+1 G(:z: -|- iy).

iy)

Since G( -|- iy) converges in the sense of generalized functions to a continuous function :u i0), we deduce that f(:z: + in) converges as y \, 0 to the generalized function G; d +1 N -|- l. d:v"+1 G(:c -|- 110) which has order I

_

Theorem 11.2.1 implies that (11.2.9) holds for all CL E C \ % (at dz i0)"' is a holomorphic extension to C of the, a priori meromorphic, family a *-> :c+

+€i0nia

.

In particular, we deduce that 1 Zi sin tra

€avri(x _

( 1 _ .

imp

e -a7r1l(x

:IJ-I-'0

Qzslntrall

Z

Ia

_|_

imp l

7

la? - in)a

(11.2.10a)

(11.2.10b)

Let us observe that d (33 -|- i0)0' = a(;z: -|- i0)a-1 do

Proposition 11.2.2. )E(§)

IX"I (5) =

e:F i " ( 2 + 1 )

1/72a +1IXI -

a (

:F in) '(0-I-1)

(ll.2.lla)

+1)(§), VGQQ.

(ll.2.llb)

541

Spectral Geometry

Proof. Let Rea > - l and 5 > 0. Set f5(;c) := e"'"';c+ so that fE -> to space Y' of Schwartz generalized functions. The Fourier transform of fS is

a + in the

e-as(6+i€);l:"'da3.

f2(§) 0

Set C:

5

+ig so that OO

f2@)=c-"

e-"o~C 'i1r(a-I-1) 2

(5 - i n ) - < >

Hence

fins)

'i1r(a+1)

I`(a

Z

1

-|-

2

(6

in)

-(a+1)

Arguing in a similar fashion we deduce that for Re a > -l the Fourier transform of ma is I`(a + l )e

'i1r(a+1)

Hence the Fourier transform of 1/a(:z:) =

me) = I`(a + 1)(e-

'i'r(a-I-l) 2

2

(5 + i0)'("+1).

Ill" is

(s - io)- + 6

€(a+1l1r'i€l0+1l

2 c o s l or(a+ q

+e

(a+1)

1)\ lél

2

(5 + i0)'("+1)

(€+(a+1) -I- 6 - ( a + 1 ) i § ( a + 1 )

'i'r(a+l) 2

Zsin

'i1r(a-I-l)

)

l§l-(01+1

l

At this point we invoke the reflection formula for the Gamma function sin to conclude that 1 I`(a_l_

MS

F(- %>F(1 + 7T

F(l +

l

go F(-3)

)

l§l'(0+1

go 'ii

F(1 +

3)

Thus

Ixl" (5)

27rII(a -|- l)

F ("§1>1"(1 + go |x|'("'+1>(€)-

IXI'("'+1)l§).

542

of Manzfolds

Lectures on the Geometry

Using the duplication formula we deduce

\/-'rrI`(a -|-

2"'I`

l)

I

a -I- l

2

a )1-(1+-) 2

Hence lx="(§)=¢%2"+1uxl-("+"(®,

vRe @ > 1 ,

aw.

The equality (11.2.1 lb) now follows by unique continuation.

Corollary 11.2.3. X+*

xi

X++b-I-l , Va, b.

Proof. Pass to Fourier transforms and then use (11.2.1 la). If we define

I + :z X+(1+0) 7

QECC,

then we deduce

and

I [3+

I

I+ 0-I-B7

'v'a,B€@.

Noting that

I+ k *f

Z

65" * f

f, of

6

m),

we could conclude that the operator

.@(R)9ul-> I + a * u€C°°(R) can be viewed as a fractional derivative of order a. The distributions II and satisfy similar properties.

a

, are defined as

L

•

11.2.2

Homogeneous generalizedfunetions

Suppose Y C R", X C RN are open sets and \I/ : X -> Y is a submersion. Denote by y = (y1, . . . ,if/"I the Euclidean coordinates on Y and by :u = (5131, . . EN the Euclidean ordinates on X . We have a natural pushforward map 7

q;* :

x) -> QW)

motivated follows. The co-area formula (9.l.l2) implies that for any 9(Y) we have

X

(@*u)()¢(x)ldgvl

/

Y

u(y)

U

l

dV

-1

(p Q

Q(X),u Q

ldyl,

543

Spectral Geometry

where IV\Ifl(a:) denotes the Jacobian of \If at the point QUO E X, and n a v y ) ( x ) n denotes the Euclidean volume density on the fiber \If' 1 (ac). More precisely, IV\I/l(;v)

\ det(D\I1)(;u)*(D\I/)(x)

I

where D\I1(13) : RN -> R" denotes the differential of IV\I/l(x) is indeed the length of the gradient of \11 at x. For any go E 9(X) we set

*(/2(9)

1

..II

\If

\11

at ac. Note that if n

dV

.

-1

l, then

(ll.2.l2)

More conceptually, the density \If*(p(ylldyl on Y is the pushforward of the density (p(x) Idol on X. Note that \I/*9, E 9(Y). We define the pullback *

: 0-°° -> 0-°°(x>

where for u 6 c-°(y) we define \If*u 6 c-° 0.

Equivalently, this means that for any go E Q(X) we have t"(u,(p)

I

(U, l.

(11.2. 14)

If we differentiate the last equality with respect to t at t = 1 we deduce the Euler identity (U»+")(U»S0)+(U»

Ecp)=0,

where E is the Euler vector field TL

a

E j 1

V(,o€9(X),

(11.2. 15)

545

Spectral Geometry

Conversely, the equality (1 l.2.l5) implies (1 l.2.l4). Indeed, for t 6 R define the continuous linear operator

St I C'0°(X) ->C'°°(X), Stu: e`t"%e*tu. Then SO

I

]17

St0+t17 \V/t0,t1 E R.

o Sto =

St()

The equality (11.2. 15) is equivalent to

d dt

Stu t

0.

0

so that l lim #So - SoIl

I

h->0

in the sense of generalized functions. Hence 1 0 sO lim #So - S0)'LL)

I

lim

0,

1

h->0

h->0

l5t0+h -

$t0) U-

Hence d dt Hence the function t »-> Stu E

Stu = 0, We. t=t0

C"00

(X) is constant, i.e., u is homogeneous of degree a.

Example 11.2.5. (a) The generalized functions Mai, (at :t in)", and XI are homogeneous of degree a on R \ 0. (b) The Dirac generalized function We have a pushforward and by q+ the restriction of q to @+. The set 6+ is Lo m -invariant. map q+ : 91¢(@+) * 9A:(R)> k < 1 . On the other hand, the map q+ : 5+ -> R is a submersion. Thus there is a push-forward

map

q+ : Qgptlg)-I-) -> Qlpt (R) given by a formula very similar, in fact equivalent, to (11.2. 12). More precisely .|.

_

_

_

ii (u) - v q d q , '1

q+ q d q where is the so-called Gelfand-Leray residue of 17 along the level set {q+ : q}. This differential form is represented by the restriction to this level set of any (n - 1)-form w defined in a neighborhood U of this fiber of q and satisfying the equality

go

17

dqAw.

The above equation has many solutions w, but all their restrictions to the fiber are the same. In particular, we have a dual pullback map

(QW IQKRI -> 9n(@'+). The image of (q+l! consists of Low-invariant currents. We want to prove that, in fact, any invariant current on @1+ lies in the image of (q-*l!. Let 'l+ = {t > 0} C R " On this open set the functions (Q, 5131, . . . 7 23rd define a global coordinate system. More precisely, 'l+ can be identified with the open set in the (q, a l l , . . . , as)-space defined by the inequality laval2 -{-q

Fix a function a E

COOlIR1,TTll,

>

0.

such that supp a C 7+ and for any A E R we have

supp a re {q = ,\} is compact, a(A, £ U 1 , . . . , ac7,,)da:1 /\ Run

- - - /\ dace

1.

552

Lectures on the Geometry

of Manzfolds

Now set 'Y

Oddélil

A

•

.

/\

dayTTI

E

Q"-1 b(A, $ U 1 7 . . . , 23rd

VAQR.

553

Spectral Geometry

has compact support. Using the Poincaré lemma for compact supports Theorem 7.2.1, or rather its proof, we can find a family of forms Bq Q Qcpt 1 (no), depending smoothly on q -E R such that, Vq 6 R we have

{ la:l2 -|- q > 0}

supp Bq C b(q,a3ld;z:1 /\

- - /\ dat77

7

- d,,Bq,

where das denotes the exterior derivative on the space RM with coordinates at = (al, . . . , mm). We can regard the family lgql as defining an (m - 1)-form B on the space 'T+ with coordinates (q, ac) and as such we have

3

d ( B /\ dql

and 0)l11

(7],T

.3.8)

-

l. Let Re a > m - 1. Then R; is locally integrable and defines a Low-invariant generalized function on R1,m_ We denote by Q55 this generalized function. It is homogeneous of degree a - n > -2 and its support is contained in C+. Corollary 11.3.6 shows that R+ is the unique extension to R1,m of the homogeneous generalized function d777»l0lX-l-2

where n = m

W)

(to

=

c,,,(¢3

-

l¢l2

II?"

+ l, do(al

+l

F

6 c-°°(02e.1»m

\0>,

r(@§'1> cm(a)

m

2 7r"II(a1

In particular l

dmlZl

27r

m

l

2

Observe that for any a, b > m - l the convolution R+ * Rb of the locally integrable functions R+, Rl is well defined. It is a locally integrable, Low-invariant function, homogeneous of degree (a -|- b) - n with support in C+ -|- C+ c+. We deduce from Corollary l 1.3.6 that there exists a constant c(a, 6) such that

R+ * RE; = c(a, b)R++L. We can be more precise.

Proposition 11.3.8. R+*R€;=R++", ` v ' a , b > m - 1.

556

Lectures on the Geometry

of Manzfolds

Proof. We follow the approach in [39, p. 160]. Observe that

* RW, ac)dtd;z:

e-to -*_

b(t7 ;z:)e-tdtda:. R1,m

R1,m

If we set

then the identity et

et

868

e-tR+(t, x)dtda:,

II

T(al

R1,m

implies that e-tR+

* R§(t, ;u)dtd;13 =

T(@)T(bl

R1,m

so that T(o

(t2 - l:z:l2)

a

m 2

/

do

dt

Iwlét t

Um 1

1

6-t

R ,

(f

(¢2 -

r2)

a

m 2

1

r'"lldr

if,

0

where of denotes the area of the unit sphere in Sm Qtr Um

9

7rn-I-1

2

m

l

F( 3+ )

Now observe that t

(t2

a

m

1

1

2

7,2

1

ta-1

m - d T'

Z

7'

2

m

-|-

1.

1 satisfy the

557

Spectral Geometry

Invoking (11.2. 16) we deduce that the above equality holds on R1,77l We have thus proved the following result.

Proposition 11.3.9. 53+ *Qi = 92++b, r/

pa-2

+

'+

on

v Rea,Reb

>m

1,'rn

>

'v'Rea

7

1,

( l 1.3.l2a)

m-I-1.

(ll.3.l2b)

We can now use the above equalities to define = + 6 C'°°(R1»*") for any a 6 setting

+

Q;

k@++2k7

(C

by

(ll.3.l3)

where is is a positive integer such that Re a + 2k > m - 1. The family

C 9a1->%+ €C"°0(]R1'M) is holomorphic. Moreover, for any a € l C the generalized function %+ is homogeneous of degree a, it is LoT"-invariant and supp

+ C C+ .

Moreover, the equalities (11.3.l2a) and (11.3.12b) extend by unique continuation to all a, b G (C. The family lgilaec is called the wave family and the generalized functions Q; are the Riesz generalized functions, [114].

Proposition 11.3.10. (a - 2>a@+ = -x,92+-2, pa

Q

Q, 1 < i < m .

(ll.3.l4)

Proof. To prove (11.3. 14) we observe that both sides of this equality are generalized functions depending holomorphically on CL. Hence it suffices to prove it only for Re CL >> 0. In this case a

a@+

Z

_

_

dm(a)@(q+)*x:

a do(al _ 2) d,n(a _ 2l(q+l*X

-2.21% dm(a

251%

F(";1)F(a- 2)92+-2

r(a)r(@21)

2

251%

2 2

n

2

= dm(@)(3m»q)(q+)*x TL

21/1i

dmlal

Q

-2

dmn(a-2) + Q-2

a

1)2(a

Theorem 11.3.11. *be unique fundamental solution of in C+ is the Riesz generalized function Qi.

on

2) +

1,'rnJ

a

z

2

53+

.

m > 1, with support

558

Lectures on the Geometry

of Manzfolds

Proof. Observe first that the function a +-> drn1(a)

F("'$1) 'ii"II(aI 7 Rea> m m

1

2

extends to a holomorphic map on (C. Indeed, the function a +-> defines a holomoiphic function on (C with zero set Z$0. at a = -k E ZS() Each of these zeroes is a simple zero. Moreover, the derivative of is (-Ilk kl. The function a »-> F l a - l - l l admits a meromorphic extension to (C. All its poles are simple and are located at -1 -|- 2Z$0. Thus the function dm71(0) extends to a holomorphic function on (C. It has only simple zeros situated at ZZ$0. We obtain in this fashion a holomorphic family of generalized functions

ga

ga

a

@9a+->%+

dm(@)(q+)*x+

n

2

6 C1-°°(R""

\ 0).

Note that

Re a

1.

By unique continuation we deduce that the above equality holds for all a E (C. Using (11.3. l 2b) we deduce that r/

Since dm(a)

+

1,m

on

\0,

`v'a€

Ofor a E ZZ$0, we deduce

f; = 0

on

R1,m

\ 0.

Hence supp $3

= {0}.

Since $3 is homogeneous of degree -n we deduce that %3 Then

C60,

for some constant C.

@+=@+*@9=c@+*60=c@+.

This shows that C = l.

We set E C2 -path

Qi. Using Proposition 11.1.28 with f (0, o f ) 9 t »-> E ( t )

Q

= 0 we deduce that there exists a

c-°° .

Proposition 11.3.12. The family ( E ( t ) )t>@ i5 a Cauchy fundamental solution of the wave operator, i.e., it satisfies the conditions (II.3.Ia), (II.3.II9) and (II.3.Ic).

559

Spectral Geometry

Proof. We follow the approach in [66, Sec. 6.2]. The equality (1 l.3.la) follows from the fact that = 50. We have to show that D = 0 and 6 = 50. To achieve this we first need to have a clearer idea on the nature of the generalized functions E(t). Let us compute the restriction of l * 2271, l * 1 2m 2,,,(m-1)/29 X-|-

to the punctured space

Rl-|-7r

2,,T(m-1>/29

X-|-

\ 0. Note that Ivq(t, H1)l = 2 \ t2 + l6212~

For t > 0 we have q'1(sl re c+

\0 = { (t, Go);

t

\

S

+lwl2, s+la:l > 0} s > 0 . 7

Thus q-1 (3) is the graph of the function uS : RM -> [0,oo), 'u,S(a:)

\

s+

l/z:l2.

The volume element on this graph is

S

I vI + 116I Idol

8

+ 21wI + l:ul

Note that along q-1(s) we have IVql = 2 i/ ' s + 2l:z:l2 so ,

1

IVQI If go

Q

1

ldVq-1(5) |

2\

+ I 12 id:z:l.

S

@lR1+m \ 0), then l

q*(/)(5l

2

(/)(

(5 +

l2:l2)1/2 ,ac

)(5 + |w|2)'1/2|d:v|-

Using spherical coordinates (T, w) we have :c

= re t =

\,5

+ 'r2,

7°

=

V / 2 - 8, .dt

pdf

= tl/2

'rd. (8_l_7'2)1/2

We set ¢(t, T)

Tm

2

(p(t , 1 u)ldwl lwl=1

and we get

¢>(t,(t2 - 3)1/2)dt. t2>s

Hence

(E+ ,(,0)

Z

1

)/2

l'fn-1 47T

Sl/2)

)it,

WE Q Q. ( 11.3.15)

For any Q E QQ" we set

Z(t~)

,am 2 l(»JI=1

¢(f~~»)ld»»l.

560

Lectures on the Geometry

The equality (1 l.3.15) shows that for t 1

(E+(t),¢)

> 0 and W Q @(R") we have 1-m

47 r(M _1 )/2

Z

of Manzfolds

( X+

2

(s),2((t2

sr/2)

-

)

(11.3.l6)

We rewrite (11.3.l5) as

(E+»(P)

( E ) , (Mt, ;z:))dt,

0

WE 6

Q

i

"

0).

We claim that (E+=(/')

( E ) , (/.(w:))dt, W) e

0

Q.

(ll.3.l7)

To see this observe that since limt\0 E(t) exists weakly we deduce from the uniform boundedness principle that the right-hand side of (11.3.17) exists and defines a homogeneous distribution of degree l - n on R" = l+'". This coincides with the homogeneous distribution E+ on the punctured space R" \ 0. We conclude from Proposition that they coincide on R" . To show that E0 - 0 and ET 60 we argue as in the proof of Proposition 11.3.1. Let (P = (Mt, or) G 9(R1-|-m)- We have

+»S0)=(

SOl0»0l=(50,0»(P)=( (11.3.17)

lim

e\,0

lim

e\,0

lim

e\,0

L /

5

l°

90>

+=

( +(t)»( (0)(t»-))dt OO

(am), ( A , -) )it ( A E t , (p(t, -I

/

)d¢ -

-|-

lim

s\,0

lim (E+( 0,

a@i =

@i(0+> = 0,

60.

(11.3. 18)

Thus Q; describes the propagation of a wave generated by a disturbance at t = 0 localized near (or at) the origin. If ac Q Run 0, then go;(t, 32) = 0 for > t. In other words, the disturbance at 0 takes t = seconds to reach the point x. I f m i s o d d , m = 2 k + 1 , k > 0,then

txt

txt

X+k

60(1¢-1)

so that

Qi

Z

W)

dm(2)xJ(ti -

has support on 3C+. In this case disturbance leaves the point ac immediately after it reaches this point. This is sometimes referred to as Huygens' Principle. If m is even, the disturbance will linger at Hz: forever, with diminishing strength.

Here is a simple but important application of the fundamental solution

Theorem 11.3.14. For arbitrary gag, (al E RM and f G C°°([0,ool problem u=f,

on (0,oo) X

X

Run) the Cauchy

m 7

u(0,a2) = go0(a:), 3tu(0,:z:l = go1(a:), Va: € RM 7 has a unique solution u 6 COO([0, ool

u(t, -) = @L(t)

X

Run) given by

d * 901 + * 901 + -§z§(¢l dt

(

(ll.3.l9)

)

t

0

§22~l¢ -

5)

* f(s)ds.

(1133.20)

Proof. Existence. From Proposition 11.3. 12 we deduce that Q; is a Cauchy fundamental solution for the wave operator. Using Proposition 11.1.28 we deduce that the correspondence

(0, OO) 9 t »-> %§(t) 6 C"°°(]Rm) is smooth. Moreover the support of Q; (t) is contained in the ball {l:13l < t}. Thus, the expression in the right-hand side of (11.3.20) is a well defined smooth function on [0, oo) X Run. The fact that u given by (1 l.3.20) is a solution of the initial value problem (1 l.3.19) follows by direct computation using the equalities d dt Uniqueness. There are many approaches. Ours is based on finite speed propagation of solutions. We follow closely the presentation in [32, VI.6.l]. For every T > 0 we consider the dependence cone (see Figure l l . l )

Qi

O i n (0, OO)

CT

m

X

7

92i(0+- = 0, -92i(0+) = 60.

•

{lt 'i

>§

o' tél

) T ] ' IQ;l < IT -¢

}

562

of Manzfolds

Lectures on the Geometry

A

r

T

CT

all

*

an

:I

:I

1

:I

up

_ _ ___ _- - ` I

In

¢

s

s

>

l

x2 .f

Fig. l 1.1

The dependence cone CT

We denote by 3+ CT conical part of the boundary of the cone and by

as

( 11.3.2l)

0.

'i=1

Along the base of the cone, described by {t = 0, laval g T}, the unit other normal is = -3¢. For any smooth function u = u(t,a2l we set

fig

m

Jo :

Utd t

771

ldul

uxdat,

-|-

2

2 Ut -|-

2 Um

»

i= 1

i=l

We have 2'Ll,t

u = %2"(t, 5y(rv)) as a smooth family of generalized functions defined on a neighborhood origin in R X ]R"". Equivalently, we can identify a neighborhood of the diagonal in M X M with a neighborhood of the zero section in TM. For fixed y 6 M the generalized function %?gl(t, sy(x)) is defined on T y M . Identifying UP0 with a neighborhood of 0 E Run, we can further view this family as a generalized function on a neighborhood of the origin in R X RM X R'rn,

%3'*(¢, sy(w)) 6

0-oo(

( e, c)

X

Upo

X

Upo

>-

Moreover, this generalized function depends holomorphically on a E C. Observe that if Re a >> 0, then we can identify ,%a2"(t, sy(:z:)) with the classical locally integrable function TL

dml2alx""' (to - dist9(:12, yl2 2

Thus %')§"(t, Sys-ll

Z

Qalt

5x(yll, Re a

>> 0

and by unique continuation we deduce that the above symmetry holds for any a,

@2a(¢, 8y(a2)) =

Qian, s,,,(y)),

Va E (C.

(1 l.3.45)

Using Proposition 11.3. 19 we deduce the following result.

Proposition 11.3.22. There exist

..

a positive number

C

= e(M)

>

0 and

a sequence ofsmoothfunetions U, 6

coo (M

X

M),u

l ,2,

with the following properly: (U1 U,§(a3)

U, (33, y) satisfy the equations (U1,h) and

c, where

h=9(V10>lgl ,

We deduce that for (t, functions

v

y)

Q

( oo, c)

X

M

X

y

).

( I I .3.46)

M we have the equality of generalized

N

(63 -|- A )

Ukl£E,

y>§z?g]R, é"(7',:z:,y)

We($)Wk(/)~ As UV, i.e., (P¢u)(;u)

M

0 such that for any A > l

,\' 0 such that

llullLm>

§

0kll L UIIL§(m)»

Vu

Q

(11.4.5)

c°°(m).

Let u E C°°(M), and set U r := PTu. Then

IILkUTIIL2(M) < (T + 1)kll"~'llL2(m)I so that

IIUIIL2

2k

(m)

k

< C'(T -|- l) lIUIIL2(m)

where C' is the product of the constants 00, 02 , m (11.4.3) we deduce that if 2k > 1/ + 2 we have

02k 2

7

defined by ( l 1.4.5). Using

fluff?m> -< C'(T + 1)"+ lnuni2(m, S C'('r-|- 1)"+nnunui2(m).

(11.4.6)

Fix a point Po E M and geodesic coordinates sz: at Po~ Observe that for any multi-index a such that lal = U we have

3§u(p0) = (u, V o , p 0 ) L 2 ( m ) » where

VaaP0)

: M -> R is given by Va7pQlq

I

z

lc'9§'\Pj(Po>l\PJ(q

agéaflp0, q )

>.

E(T) =

Ep,ql7"

(sign 7'l 0 , 2% c:os{l, 5), /Z < 0. sin(l, §>>

(27f)%

\I//;(f7)

1;

1,

l

We have l

6§32

I

2 sin(k, 9) sin(k, ¢) -I- (2/T)m

2

_|_

IZ,/TlM

-D

(mm

-D

-D

cosllg,

-»

l1?l (5)e-("»5°) ld60l.

QV

[0,27r]'"

The above series converges in coe. The sum in the left-hand side of (11.4.11) is q1>. We have 1 (0)=~ no c . (Q)

Now observe that Ca =

_Z

khZ'"

/

_»

¢(§)e->

ld@l Run

[0,2'/r]M-k

¢(§')€'""°'7>ld9l = (/7)~

If in Poisson's formula we choose a = 2'/TR, then we deduce l

$ oo, L2(M), defined as follows. Pick u E I;2(M). Then u has a Fourier decomposition

u = Z u k W k , up

= (u,\Pk),

k>0

where

n

llulli2

2

k>0

We set cos

lttlu

( I I .4.l3)

COS

k>0

Note that 2

II cos(t\ A)uu=i2(m)

(to

2 A/nuuu2 < llullL2(m>

k>0

so that the operator cos(t \ A) is indeed well defined and bounded L2 (M) -> L2(M). Note that if u E COO(M), then for any N E Z20 we have ANd,

Z k>o

Akn'Ll,k\Ifk

581

Spectral Geometry

and we deduce cos(tv-A)An'u, = AN cos(tv-A)u

A

cos(tvk)uk\Pk(

p ).

k>0

This shows that lIAncoslt\

A)ulnL2(m)

< HAN u m m ) ,

VNQZzo.

If we formally differentiate (1 l.4.l3) we respect to t, we deduce

(-WA\ cos(tvk)uk\I1k(

3t23(cos(tvX)u)( p ) =

p ).

1 _0

The above discussion shows that if u E

coo

( M ) , then

cos(t\/-A)u E C'°°(]R x M). For any u,

'u

E Cfoo(M) we define u IZ U E C°°(M

( u ® v ) ( p , q)

Z

X

M) by the equality

u( P )v(q ), iv'p , q € M -

The Schwarz kernel Kt(P7 q) of cos(t1/-A) is the generalized function

K l p,q) =

cos(tvk)(u, \Ifk)L2(v, \I/k».I)L2 . k>0

The family of generalized function Kt(p, q) E C-OO(M X M), t E R is called the wave kernel of the Riemann manifold (M, g). For fixed p, q E M we obtain a temperate generalized function on R t »-> K t ( p , q defined by 9

.

(ll.4.l4)

k

Thus, an approximation for Kt will yield an approximation for the Fourier transform of the derivative of the modified spectral function l

T

+-> -(sign 7'l Kt (p, q )€ C'°°(M).

The a priori estimates described in our next result will play a key role in our search for an approximation for Kt. Proposition 11.4.5. Let 'U 6 COO([0, 1] equation

X

of

M ) be a solution

the non-homogeneous wave

( 3 - i - A ) u = h on [ 0 , 1 ] x M

'v(0»p)

I

Utl0)P)

I

(ll.4.l6)

0, Vp G M.

Assume that 8g'h(0,pl = 0, V p 6 M, Then there exists Co

>

j < k.

(ll.4.l7)

0 independent of h such rat for any t 6 (0, 1 ) we nave

k-I-1

k

||@ + j

1-

'

311(t, ' l H L 2 ( M )

0

S Co

(f

0

t

k-1

(ll.4.l8)

.

lla'°h(s, ' ) ' I L 2 ( m ) d 8 + Z!l@£"--f

'l'IL2(m)

j=0

Proof. For any t 6 ([0, 1] we have 1 d 2 if

(ll@uui2(m) + ( A s ,

I7/¢¢,U¢lL2(M)

UlL2(M)

+ lA'U,'utlL2lA/fl

(in, u¢)L2(m) . Observing that lAU»UlL2(m)

-

2 lldv1IL2(m)9

we deduce by integrating that t

2

(h,as@)L2(mlds 0

1

t

< 2K

ll1(S,-)llL2(M)dS, 0

lIUtllL2(Ml ds.

K 0

583

Spectral Geometry

Since 1


> 0 we set 1

(M) SO

that M)

¢?(at) has support in

§;(t)Kt

i v

2

i

£

a 9

0 we have

Uk(y, y ) d m ( 2 / < ) ( t ) l x l 2 ' + " ( t ) .

k=1

5 The Fourier-Laplace transform of a smooth compactly supported function is an entire function.

(11.4.36)

590

Lectures on the Geometry

of Manzfolds

>> 0 the remainder

More precisely for N

N

QM)

(t)K5-

@(t)0. In other words we need

Next, we need to estimate terms of the form to estimate functions of the form

fpl7rl

11 a

2'rrlTI

Z

a


0.

50

lflé

The function /,L is nondecreasing and defines a Lebesgue-Stiltjes measure observe that d

du on R

We

l

((lea * m)(T' du * ( M )

R (MT

Z

s)du(sl

'

Hence &`0CL

7r-I-%

-1

du * kw)

d/,L 0, we divide (0, s) h1Mu 0

a

k=l

On the other hand, Theorem 11.4.6 implies that -

3;*65K(w,y)l=y=(»

A»m,a,@,klxl

k

m

2k

(1 l.4.46)

la+Bl+ (t),

0

where Am,a,,B,0 is a universal constant depending only on

10 -|- BI is odd.

MY

a709 which is equal to 0 if

Lemma 11.4.10. (a) Foranyr G Z a n d a n y N > 0 we have l

T`

T`

T'

ase->0.

-(IxI ,pa)=@ ((=al , w ) + 0 ( N ) ) 6 /\

( b ) For every positive integer

T'

A

we have

v',/T21-r

(l)cl"l»17)

fig)

l7'IT-1'wl7'ld7'_ R

Proof. (a) For transparency we will use the integral notation for the pairing between a generalized function and a test function. We have

(lxl"»171//Q

51

L

: g

/

lxl""(t)p(tl1F(t/@)dt lxl°°lt)p(€t)u7(t)dt

67(IXI',

i

R

Now observe that pE'z?

_w

lxl"°(@tlp((°2tl17(t)dt

R

Pg.]>g

Mt)

i

p(etl.

Mk = '®(p6 - 1) -> 0 in 5(]R). More precisely for k > 0 we have

-(p - 1) = O((=:NtN) as 5 -> 0.

Mk

This implies that

(uxl'x@(p

1) ) = O ( a N )

Asa->

0,

so that

(lxl'Hpv7)

1

(lxl'°»17)+(lxl'U'u7(»06 1))

1

(IxI

IN) + O(aN) as 5 -> 0.

(b) We have -7°

_1°

0lM,

(

11.4.51

otherwise.

0,

We can now rewrite (11.4.50) as

Dv7z,a,B

5-"*-|o+

@|»;|(q

KA(P, q)

j

Ag) consisting of real valued

where l'I'jl1$jSn(A) is any orthonormal basis of her(/\ eigenfunctions. This means that

( P All(Pl

7

1

K A ( P , q Lu(q )ldVg(q )|-

M

As in the previous subsections we describe the spectrum of Ag as a nondecreasing sequence 0

A0 < AS < AS
[0,

w(t)

A

599

Spectral Geometry

A

>

Fig. l 1.3

A smooth approximation o f the indicator function of [ - l , l].

We can think of w(t) as a smooth approximation of the indicator function of the interval [-1, 1], see Figure 11.3. As in the previous subsection we set

few p, q)

we) K A(P41 ).

w(@JE)@k(p)\pk(q) = k>0

A6spec(A)

The above sum consists of finitely may terms since it has only contributions from eigenvalues < 6 - 2 so it has O(e'"') terms. Fix a point p E M and normal coordinates ( i l near p. From (1 l.4.41) with lal = = l we deduce

um

8g;i 8y.7 (gas

(at, Yllrv=y=0

5

m

2

6

m

2

l

(mm

(f

Run

60 (2w)""

Run

w(lxl)xWdx

+ O(82)

w(lwl>(w1)2da: -I-O(82)

I

;Brow)

The positive constant Be(wl can be determined using (1 l.4.52). More generally, for two vector fields Xi, X2 E Vect (M) we have X1(3U)X2(3/)é"€(£€»3/)lw=y=0

=

X13mX (9)3g3 g5t (Lu, 311 l:v=y=0 "KJ

5

'rn

2

1

l m

6 ix;(0>xg

(Bow) + 0 )

id 6

'rn

2

1

(2,,mgp(x1( p),xi( p ) ) ( 8 m ( w ) + 0 ( @ 2 ) ) .

FoIIX1,X2 E Vect(M) fixed, the above estimate is uniform in p E M . We set dim ke1°(A - All.

0(6) A 0 sujiciently small the map F E : M -> RW), p

I

€""-*2(27r)M

F €

Be (w)

wls\/kl

Wk(p )

E RDl5l 7

(1 l.4.54)

k

0.

•

Proof. Denote by the canonical inner product in RD(€). For p E M a n d X E TpMwe denote by DX F e(p ) the value of the differential of F 5 at p on the tangent vector X (p) . Explicitly,

g

d F 5 (v(t)) E ]RD(@) 7 dt t 0 where by : (-5, 5) -> M is any smooth path such that 7(0) = P and w) = X(p)The estimates (11.4.53) show that, for any X E Vect (M) and any p 6 M, we have

Dx F e(P )

Do F a( p)

.

I

Do F 6( p) = g p ( x ( p) , x ( p) ) ( 1 + 0 ( @ 2 ) ) a s e \ , 0 ,

uniformly in p 6 M. This proves that for 5 > 0 sufficiently small, the map FS is an immersion and the metric gE on M induced by FS is C0-close to g. To prove that FE is injective for 6 > 0 sufficiently small we argue by contradiction. Assume that this is not true. This means that for any n E N there exist points pn, qn E M such that \I/I e _ / 52 / > ( 1 - 152)/.

This shows that the scalar curvature function 36 does not converge uniformly to 0, the scalar curvature of the flat Euclidean metric. As a curiosity, note that

3E(7°) = 0, VT
0.

( 11.4.6l)

Proof. Using (11.4.43) we deduce 1

5

'of;jj

Q(K nil

described by an explicit formula6 in [28, Eq. (1.12)], such that, for any Riemann metric g on M, the quantity Q(Kg,g,,)eA2TpM

(X)

A2TpM

coincides with the Riemann curvature tensor of g at p. The sectional curvatures Kp of the metric g5 at p converges to the sectional curvature g and go -> up. Denote by Rp the Riemann curvature of g5 at p. We deduce that p E

Re

%(K9»95) -> 9?(Kp,9")

Rp-

Corollary 11.4.18. The spectral data determine the Riemann curvature of the original metric g.

Remark 11.4.19. (a) Using the very deep results in [5, 109] one can show that the C0convergence of the metrics g5 coupled with the convergence of sectional curvatures imply that g5 converges to g in Coo. We can rephrase this by saying that, for 5 > 0 sufficiently small, the map F 5 : M -> ]RD(6) defined in (1 l.4.54) is a nearly isometric embedding. (b) Let us observe that for any even, nonnegative temperate function w 6 5'(]R) we can define a map

m9p+F

6777,-1-2 (2w)'"'

> k>0

Be (w)

w/TQ)

@k(z»)@k Q L 2 ( m , 9 ) .

When w has compact support, the range of this map is contained in a finite dimensional subspace. _ 2 . . . . The case w(v°) e or was first investigated in [17] where, among many other things, the authors show that for small 6 > 0 the map F w5 is a nearly isometric embedding of M in 6 The

explicit formula of ,Q is rather complicated, but it is polynomial in K and h so it is continuous.

608

Lectures on the Geometry

of Manzfolds

the Hilbert space L2 (M, g). The image of this embedding is not be contained in any finite dimensional subspace. 52 Observe that in this case, if we set t we have HtlXv

y)

e-* >I fk (p )\I 'k( q

II

5"/¥(p, q )

k>0

>.

(11.4.75>

This is so called heat kernel of the Riemann manifold. (c) Let us point out that the above results solve more than the original spectral perestroika problem.

The geometry of the compact Riemann manifold ( M , g) is completely determined by the associated neat kernel. More precisely, the metric and its curvature are determined by the behavior as t \ 0 of the jets of order 4 along the diagonal of the heat kernel Ht (ac, y). This is a rather unexpected conclusion: we can learn the "shape" of the manifold by investigating how the heat propagates or, equivalently, how the diffusion takes place. Remark 11.4.20. (a) As explained in Remark 4.2.30, the correlators associated to embeddings have probabilistic interpretations. For example, when w is as in Figure 11.3, the correlate 55 (p, q ) is associated to random functions on M of the form K

f

Xgq/k,

/nu

where (X/§)k2 are independent random variables with mean zero and variances 6m-I-2

variance IX; )

l2,/Tlm

Be (w)

w(€

l.

As 5 \, 0 this random function converges in a precise sense to the "perfect chaos", the so called called Gaussian white-noise. A plastic way of phrasing the conclusions of the previous results is that there is a pattern in devolution to chaos which is very sensitive to the background geometry. (b) For every p 6 M we have a random symmetric bilinear form

GS := dU5(pl

(X)

dUe(pl : TpM

X

TpM -> R.

For 5 > 0 small, its expectation is the metric be induced by the embedding F 5 defined in (1 l.4.54). We are inclined to believe that the random variable G5 converges almost surely and L1 to the (deterministic) g. In other words GE is highly concentrated around its mean a s \ , 0. (c) Here is another little gem hidden under the probability cloak. Assume M oriented and even dimensional m = d i M = 211. The differential of the random function US is a random section dU5 of the cotangent bundle. One can show that, with probability 1, dUE intersects the zero section of T*M transversally, i.e., dU5 has nondegenerate zeros. To such

609

Spectral Geometry

a nondegenerate zero p we can associate a local index €(U6, p) = i t (see Exercise 7.3.47) and from the Poincare-Hopf formula in Corollary 7.3.48 we deduce that (11.4.76)

€(U5, p) = x ( M ) dUelPl

0

In fact, above we have produced quite a bit more, namely a 0-dimensional current €(U6, P)6p E

Zdzfs dUelPl=0

The "integral" of a given 0-form

f

over this random current is the random variable

6((&, p)f(p)-

(f, ZdUE ) dUfslP)

0

The expectation of this random variable has a rather striking geometric description [106]. More precisely we have

IE{(f, ZdU,5 )]

M

(1 l.4.77)

f€(Vge),

where e(V9@) is the Euler form associated to the Levi-Civita connection of the metric 969

e(V9)

l pf (mm/2

(

RE)

7

where Rs is the Riemann curvature tensor of the metric ge- Thus l Pf ( R) 1E[(f» ZdU»5 (2wl""/2 M f Note that when f

7

(1 l.4.78)

1, then

€lU6,

(1»Zdu)

(11.4.76)

p)

XlM)-

dUelPl=0

This is no longer a random quantity. Using this in (11.4.78) we deduce 1

x(M)

/

R). 5

(2w)"*/2 M pf ( This is the Gauss-Bonnet theorem for the metric ge, On the other hand, we know that R E` converges uniformly to the Riemann curvature R of the original metric g. If we let 5 \ 0 we obtain the Gauss-Bonnet theorem for the original metric g ! This "accident" is a manifestation of a more general phenomenon. In particular, the general Gauss-Bonnet theorem has a probabilistic interpretation. For details we refer to [105, 106].

This page intentionally left blank

Chapter 12

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. We will first describe their general features, and then we will spend the remaining part discussing some frequently encountered examples.

12.1 The structure of Dirac operators 12.1.1

Basie definitions and examples

Consider a Riemann manifold (Mg), and a smooth vector bundle E -> M .

Definition 12.1.1. A Dirac operator is a first order p.d.o.

D C°°(E) -> COO(E), such that D2 is a generalized Laplacian, i.e., (t(D2)(w,€) =

-151311E am) 6 T*m.

The Dirac operator is said to be graded if E splits as E = E0 ® El, and D(C°°(E¢ll C COO (E(1+1> mod Zl' In other words, D has a block decomposition D

0 A B 0

[ ]

Note that the Dirac operators are first order elliptic p.d.o.-s.

_

Example 12.1.2 (Hamilton-Floer). Denote by E the trivial vector bundle RQ" over the circle S1. Thus cOO (E) can be identified with the space of smooth functions u:S1->]R2" Let J : cOO (E) -> sition

cOO

(E) denote the endomorphism of E which has the block decompo-

II

r-w

611

612

Lectures on the Geometry

with respect to the natural splitting operator

RQ"

R" ® R".

=

dz

do2

We define the Hamilton-Floer

du

.]_

3': C°°(E) -> COO(E), 9"u, Clearly, 9:2

of Manzfolds

Vu 6 C°°(E).

do

is a generalized aplacian.

Example 12.1.3 (Cauchy-Riemann). Consider the trivial bundle Q2 over the complex plane (C equipped with the standard Euclidean metric. The Cauchy-Riemann operator is the first order p.d.o. D : Q2 -> Q2 defined by

H

0 > 2 82 0

Q'

(M)~

613

Dirac Operators

Let D : coo (E) -> cOO (E) be a Dirac operator over the oriented Riemann manifold (M, g). Its symbol is an endomorphism 07(D) : 'rr*E -> 'rr*E, where 'ii : T*M -> M denotes the natural projection. Thus, for any Hz: 6 M, and any E Tas M, the operator

0(€)

0(D)(€,1v)

is an endomorphism of Ex depending linearly upon §. Since D2 is a generalized Laplacian we deduce that @(§)2

I

0(D2)(1a€) =

-l€l§11E,,

To summarize, we see that each Dirac operator induces a bundle morphism e : T*M (X) E -> E

such that @(€)2

Z

(§,e) »-> c(§)e,

-l€l2, From the equality

¢@+n)2 = -la+vl

2

v m @ T , m , :UE M

we conclude that {0(€)»0(v)} = -29(€,'7)]1E,

where for any linear operators A, B we denoted by {A, B} their anticommutator

{A, B} :z AB + BA. Definition 12.1.6. (a) A Clifford structure on a vector bundle E over a Riemann manifold (M, g) is a smooth bundle morphism e : T*M (X) E -> E, such that

{0(€l» c(t7)}

= -2g(€»vl]1E,

We

6

Mm),

where for every 1-form a we denoted by c(al the bundle morphism e(a) : E -> E given

by e(a)u

e(a,ul, Vu E C°°(E).

The morphism e is usually called the ClqiItord multiplication of the (Clifford) structure. A pair (vector bundle, Clifford structure) is called a ClyjL'ord bundle. (b) A Z2-grading of a Clifford bundle E -> M is a splitting E = E+ ® E' such that, Va E Q1 ( M ) , the Clifford multiplication by a is an odd endomorphism of the superspace c°°(E+) e c°°(E_), i.e., c(0¢)C°°(E*l C c°°(E*).

Proposition 12.1.7. Let E -> M be a smooth vector bundle over the Riemann manifold ( M , g). Then the following conditions are equivalent. (a) There exists a Dirac operator D : cOO (E) -> cOO (El. (b) The bundle E admits a Clutord structure.

614

Lectures on the Geometry

of Manzfolds

Proof. We have just seen that ( a ) ( b ) . To prove the reverse implication let c:T*M

(X)

E->E

be a Clifford multiplication. Then, for every connection

V : C°°(E)->C°°(T*M

(X)

E),

the composition

D

CO

V

e

V : c°(E) -> c°°(T*m (X) E) -> c°°(E)

is a first order p.d.o. with symbol C. Clearly D is a Dirac operator.

Example 12.1.8. Let (M, 9) be a Riemann manifold. For each QCEM, and § € T x M define

0(5) A°T;M -> A°T;M by

= (€ - i€)W» where 6€ denotes the (left) exterior multiplication by g, while it denotes the interior differentiation along 5* e T,,,]l/I - the metric dual of €. The Exercise 2.2.55 of Section 2.2.4 shows that c defines a Clifford multiplication on A°T*M. If V denotes the Levi-Civita connection on A'T*M , then the Dirac operator COV is none other than the Hodge-DeRham operator. 0(€)(»J

Exercise 12.1.9. Prove the last asse ion in the above example. 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. In the following subsections we will address precisely this issue.

12.1.2 Clifford algebras The first thing we want to understand is the object called Clifford multiplication. Consider a real finite dimensional, Euclidean space (V, g). A Cluj'ord multiplication associated with (V, g) is then a pair (E, pa , where E is a K-vector space, and p : V -> End (E) is an R-linear map such that

{0('u»l,0(v)} = -2g(u,v)]lE, We Q VL If (Q ) is an orthonormal basis of V, then p is completely determined by the linear operators ,0i = P(@¢) which satisfy the anti-commutation rules {Pi>Pj}

I

-26ij]1E.

The collection ( p ) generates an associative subalgebra in End (E), and it is natural to try to understand its structure. We will look at the following universal situation.

Definition 12.1.10. Let V be a real, finite dimensional vector space, and

q:V

X

V-HR

615

Dirac Operators

a symmetric bilinear form. The Clifford algebra Cl(V, q) is the associative R-algebra with unit, generated by V, and subject to the relations

{'u,,'u}

up

-

-2q(u,ul ]1 'v'u,u 6 V

Vu

.

Proposition 12.1 11. The Clijtord algebra Cl(V, q) exists, and is uniquely defined by its universality properly: for every linear map j : V -> A such that A is an associative Ralgebra with unit, and {](u), ](v)} = -2q('u,, Fu) - ]l, there exists an unique morphism of algebras i15 : C1(V, q) -> A such that the diagram below is commutative.

V

'L

>

Cl(V, q)

J

A 'L

denotes the natural inclusion V --> C1(V, q).

Sketch of proof. Let A = ®k20V®k (V®0 with unit generated by V. Set

R) denote the free associative R-algebra

cllv, Q) = A/I, where I is the ideal generated by {u®v+'u®u+2q('u,,'u)®1;

'u,,v E

V},

The map Z is the composition V --> A -> Cl(V, q) where the second arrow is the natural projection. We let the reader check the universality prope y.

Exercise 12.1.12. Prove the universality prope y. Remark 12.1.13. (a) When q E 0 then C1(V, 0) is the exterior algebra A°V. (b) In the sequel the inclusion V Cl(V, q) will be thought of as being part of the definition of a Clifford algebra. This makes a Clifford algebra a structure richer than merely an abstract R-algebra: it is an algebra with a distinguished real subspace. Thus when thinking of automorphisms of this structure one should really concentrate only on those automorphisms of R-algebras preserving the distinguished subspace.

Corollary 12.1.14. Let ( V , i t ) ( i = 1, 2) be two real, finite dimensional vector spaces endowed with quadratieforms ii : V -> R. Then, any linear map T : VI -> V2 such that q2(T"u) = QUO (u), Vu E VI induces a unique morphism of algebras T# : Cl(v1,qll -> Cl(V2, (12) such that T#(V1l C Vg, where we view V; as a linear subspace in C1(V/, Cal. The correspondence T »-> T# constructed above is funetorial, i.e., ( l m )# = ]1C1(\&qi), and ( S o TO# = S# o T#, for all admissible S, and T. The above corollary shows that the algebra Cl(V, q) depends only on the isomorphism class of the pair (V, q)= vector space -|- quadratic form. It is known from linear algebra that the isomorphism classes of such pairs are classified by some simple invariants:

(dim V, rank q, sign q) .

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We will be interested in the special case when dim V = rankq = signq = n, i.e., when q is an Euclidean metric on the n-dimensional space V. In this case, the Clifford algebra Cl(V, q) is usually denoted by Cl(V), or Cin. If (611 is an orthonormal basis of V, then we can alternatively describe Cln as the associative R-algebra with 1 generated by (€¢l» and subject to the relations € ¢ € j -I- € j € ¢ -2(Sj. Using the universality property of Cin, we deduce that the map V -> c1(v>, U »-> -u E Cl(V) extends to an automorphism of algebras a : Cl(V) -> Cl(V). Note that a is involutive, i.e., 0 2 = ]l. Set CMV) := ker(a - ill, COW) = ker(a + II). Note that Cl(V) = Cl0(V) ® cll, and moreover I

c1 - c1"(v) C

c1(€+*1) mod

2(v>,

i.e., the automorphism a naturally defines a Z2-grading of Cl(V). In other words, the Clifford algebra Cl(V) is naturally a super-algebra. Let (Cl(V) , +, *I denote the opposite algebra of Cl(V). Then Cl(V) coincides with Cl(V) as a vector space, but its multiplication * is defined by cc

*y

y :u Va2,y E

CW),

where "-" denotes the usual multiplication in Cl(V). Note that for any u, 'U 6 V u-v+v-Iu,=u*'v+'v*7u,,

so that, using the universality property of Clifford algebras, we conclude that the natural injection V C--) Cl(V) extends to a morphism of algebras Cl(V) -> Cl(V). This may as well be regarded as an antimorphism Cl(V) -> C1(V) which we call the transposition map, 11: »-> mb. Note that $1

U2

"uL/1,)7

'01,

'Ufr

\V/Ui

E

V.

For :u E C1(V) we set QQ* ¢ =

Ends(El. (b) A K-superspace E is said to be a K-Clifford s-module if there exists a morphism of s-algebras

p : Cl(V) -> Ends(El. (c) Let (E, p) be a K-Clifford module, p : Cl(V) -> EndK(E). The module (E, p) is said to be self-adjoint if there exists a metric on E (Euclidean if K = R, Hermitian if K = (C) such that

PW) = 0(a:)*

Vcc 6

c1.

We now see that what we originally called a Clifford structure is precisely a Clifford module.

Example 12.1.18. A°V is a self-adjoint, real C1(V) super-module. In the following two subsections we intend to describe the complex Clifford modules. The real theory is far more elaborate. For more information we refer the reader to the excellent monograph [83].

12.1.3 Clifford modules: the even case In studying complex Clifford modules it is convenient to work with the complexified Clifford algebras

mn Gun ®RC~ The (complex) representation theory of (CiTL depends on the parity of n so that we will discuss each case separately. The reader may want to refresh his/her memory of the considerations in Subsection 2.2.5.

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Let n = Zn, and consider an n-dimensional Euclidean space (v, 9)- The decisive step in describing the complex Cl(V)-modules is the following.

Proposition 12.1.19. There exists a complex Cl(V)-module S = §(V) such that

~

(Cl(V) = End@(S) as (C-algebras. (The above isomorphism is not natural; it depends on several auxiliary choices.)

Proof. Consider a complex structure on V, i.e., a skew-symmetric operator J : V -> V such that J2 = -]IV. Such a J exists since V is even dimensional. Let

{€l7.tl7°°°7€/ Ending (E) be a complex Clyora' module. Then E admits at least one Hermitian metric with respect to which p is self-adjoint.

622

Lectures on the Geometry

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Proof. Decompose E as § (X) W, and p as A (X) ]lW, for some isomorphism of algebras A : (Cl(V) -> End(S). The sought for metric is a tensor product of the canonical metric on S, and some metric on W. 12.1.4 Clifford modules: the odd case The odd dimensional situation can be deduced using the facts we have just established concerning the algebras C1;1'+1,

\I/(LE0

+:z:1) =

£1:0 -|- 60

- acl,

where £170 E Clipen, and :cl E Clod. We leave the reader to check that this is indeed an isomorphism of algebras.

Proposition 12.1.30. Let (V, g) be a (Zh + 1)-dimensional Euclidean space. Then there exist two complex, irreducible Cl(V)-modules S+ ( V ) and § ( V ) such that

~

Cl(V) = End(l;(S-L) ® End@(§`) as ungraded algebras. The direct sum § ( V ) = § ( V ) QB S' ( V ) is called the (odd) spinoza module. Proof. Fix an orientation on V and a positively oriented orthonormal basis 6 1 , 6 2 » - - - » € 2 k + 1 - Denote by S2k-l-2 the spinoza module of (Cl(V EB R), where V QB R is given the direct sum Euclidean metric and the orientation or(V ® R) = or /\ or(lR). Choose an isomorphism

p : (C1(V ® R) Then

S2k-I-2

_>

End@(§2k+2).

becomes naturally a super Cl2k,+2-module + 1

§2k+2

-

§2I ® End l § 2 k + 2 ) As in the previous section, we can choose Hermitian metrics on s:l:lv) such that the morphism 2

®Rleven

P : (co) -> End C (so)

)

is self-adj pint, i.e.,

QW) =

0( u) *, Vu Q (co).

The above result can be used to describe the complex (super)modules of

a(:z;) . V

-B-1 C

V },

where a : Cl(V) -> C1(V) denotes the involutive automorphism of C1(V) defining its Z2-grading. In general, the map

Cl(V) 9 u *P'°> 0(:z;)7u,a:-1 E Cl(V) is not an automorphism of algebras, but as we will see by the end of this subsection, if so: E I`(V), then px = :l:(p3,, and hence a posteriori this alteration has no impact. Its impact is mainly on the aesthetics of the presentation which we borrowed from the elegant paper [9]. By construction, the Clifford group rlvl comes equipped with a tautological representation p : FW) -> GL(V)

p(a2l : U »-> a(;r;)

'U

as

1

Proposition 12.1.34. kelp = (R*, -) C C1(V)*. Proof. Clearly R* C her p. To establish the opposite inclusion choose an orthonormal basis (€¢l of V, and let .CU 6 her p. The element .CU decomposes into even/odd components x

and the condition

&(.§U)€i.2U

61

Ito -

$0+$1,

translates into 61 (5130 -|- 5121),

5U1l€i

W.

This is equivalent with the following two conditions [110,

€]

=

200€i - €¢$0 =

£U1€¢

+ 61331

i

{I t , 61 }

i

0, W •

In terms of the s-commutator, the above two equalities can be written as one [@¢7$]s

= 0

W.

626

Since

of Manzfolds

Lectures on the Geometry

°7

ac] is a superderivation of Cl(V) we conclude that [y,;u]S = 0, 'v'y E Cl(V).

In particular, the even part following elementary fact.

lies in the center of Cl(V). We let the reader check the

5130

Lemma 12.1.35. The center of the ClQ§"ord algebra i5 the field of scalars R

C

Cl(V).

Note that since {act, et} = 0, then all should be a linear combination of elementary monomials en ets none of which containing et as a factor. Since this should happen for every i this means a l = 0, and this concludes the proof of the proposition.

---

Definition 12.1.36. The spinorial norm is the map N : C1(V) -> Cl M

N(;z:)=x>x°.

Proposition 12.1.37. The following hold. (a) nlrlvll C R*. (b) The map N : rlvl -> R* is a group morphism.

Proof. Let co: E F ( V ) . We first prove that ate E I`(V). Since 0(:z;lva2-1 E V, \7"v E V, we deduce that

aw)

'U

=v'1}b) = -a(:c) - u . x-1 6

v.

Using the fact that co: »-> :cb is an anti-automorphism, we deduce

Q( ( b ) - 1

-'u

- aw) 6

v,

so that, 'U

(1((1v')'1)

b

as E V ,

that is, (;u7)'1 G FM. Hence :u 6 F (V),Va:€ F (V). In particular, since 01(F(Vll C I`(Vl, we deduce N(II(Vll C FIG). For any U E V we have o¢(N(fv))

'U

(N(;v))l1

i

On the other hand, y = a(ac°) Hence 0((N(;v))

'U

a(:13>xl 'U

(N(11))'1

at

1

(wbffrl

Z

M) -

a

'u

•

a3'1} - (w

is an element in V, which implies YT = a(yb) = -y.

-aW) ~y* (wb)-1

-

-a(;cb) a(;z:b)-1 _UT -Mb

This means N(a:) E her p (b) If co, y E I`(V), then

f)

- (;z:b)-1

UT

'U.

R*

N(;u - y) = (22y)b(wy) = ybwbwy = yan(©v)y = N(x1)yby = N(;v)N(y)-

627

Dirac Operators

Theorem 12.1.38. (a) For every cc 6 I`(V), the transformation p(;c) of V is orthogonal. (IQ) There exists a short exact sequence of groups 1 -> R* F(V) £> O(V) -> 1.

--

(c) Every x 6 I`(V) can be written (in a non-unique way) as a product ac = 01 -up, Up 6 V . In particular; every element of r ( v l is Z2-homogeneous, i.e., it is either purely even, or purely odd.

Proof. (a) Note that, 'v'tJ 6 V, we have N(v) =

-IUI§. For every ac 6 I`(V) we get

-1

N(0((@3))N(x1)-1N(vl~ N(0>(11)('v)) N(a(x)va: ) N(0¢(11))N(v)N(33'1l On the other hand, $2 = 0(x2) = a(a1)2 so we conclude that I

N(a))2 = N(0(;z:))2.

He r e , N(p(;c)(v)) = :l:N(tv). Since both N(u) and N(px(u)) are negative numbers, we deduce that the only possible choice of signs in the above equality is -I-. This shows that 0(11) is an orthogonal transformation. (b) and (c) We only need to show p(II(V)) = O(V). For cz: 6 V with txtg = 1 we have 0(:z:)

51:

£12'1.

If we decompose 'U 6 V as Ask + u, where A 6 R and u

J_ co,

then we deduce

p(:I:)'v = -Ax + u. In other words, p(;z:) is the orthogonal reflection in the hyperplane through origin which is perpendicular to 111. Since any orthogonal transformation of V is a composition of such reflections (Exercise), we deduce that for each T € O(V) we can find 101, . . . , up 6 V such that

T

»0('v1)

a

°p('U/ Cl(V) is a smooth embedding. The proof of the following result is left to the reader.

Proposition 12.1.41. There exist short exact sequences 1 -> $2 -> Pin(V) -> O(V) -> 1 l -> Z2 -> Spin(V) -> SO(V) -> 1.

Proposition 12.1.42. The morphism p : Spin(V) -> SO(V) $5 a covering map. Moreover, the group Spin(V) $5 connected if dim V > 2, and simply connected if d i V > 3. In particular; Spin(V) is the universal cover of SO(V), when dim V > 3. Proof. The fact that p : Spin(V) -> SO(V) is a covering map is an elementary consequence of the following simple observations. (i) The map p is a group morphism, (ii) The map p is continuous, and proper, (iii) The subgroup her p is discrete.

Since SO(V) is connected if d i V > 2, the fact that Spin(V) is connected would follow if we showed that any points in the same fiber of p can be connected by arcs. It suffices to look at the fiber p-1ll) = { - l , 1}. If u, 'u 6 V are such that lul = lvl = l , u J. v, then the path 'y(t) = ( c o s t + ' s i f t ) ( u c o s t - sift), 0 < t g tr/2 lies inside Spin(Vl, because

lu cost :|:

tl =

'U sin

1, and moreover

'Mw/2)

'y(0l =: -1,

l.

To prove that Spin(Vl is simply connected, we argue by induction on dim V. If dim V = 3, then Spin(V) is isomorphic to the group of unit quaternions (see Example 12.1.55), and in particular, it is homeomorphic to the sphere S3 which is simply connected. Note that if V is an Euclidean space, and U is a subspace, the natural inclusion U V induces morphisms

Cll U) Cl(V), SO(U) + SO(V), Spin(U) --> Spin(V), such that the diagrams below are commutative

Spin(Vl

p

1

>

'Z

Spin(Ul

p

>

SO(V)

1

SO(U)

Spin(V)

" 1

Cl(V)

7

Spin(U)

>

Cl(U)

629

Dirac Operators

Hence, it suffices to show that if dim V > 3, then for every smooth, closed path

11: [0,1]-> Spin(V), 11(0) = 71(1) = 1, there exists a codimension one subspace U 6+ V such that 'EL is homotopic to a loop in Spin(U). Fix unit vector e 6 V, set u(t) := ,0fa(¢> 6 SO(V), and define v(t) := 'u,(t)e 6

V. The correspondence t »-> 'u(t) is a smooth, closed path on the unit sphere sdim v - 1 C Using Sard's theorem we deduce that there exists a vector so 6 sdim V-1, such that V.

v(t) go :l::c, Vt

6

[0, l].

In particular, the vectors u(t) and cc are linearly independent, for any t. Denote by f (t) the vector obtained by applying the Gramm-Schmidt orthonormalization process to the ordered linearly independent set {a:,'u(t)}. In other words,

f (t)

1

If0(t)l /0(t)' for)

where we denoted the inner product in V by

'v(t)-(v(t)

.

:r:):c,

We can find a smooth map

0 : [0, 1] -> [0,27r),

such that, 0(0) = 0(1) = 0, v(t) = accos 29(t)

+

f (t) sin 20(t).

Set

o(t) := cos 0 + asf sin 0 6 C1(V). Note that 07(t)

as COS

0 2

A

f

) I-$cos

0 2

s1n-

0 2

A

0

f s m 2-

7

so that c7(t) € Spin(V). On the other hand,

Ulf)-llf)

= casH - :of sin 0,

and (7(t)x Cl(V). Then spin(V) = q(A2V)~ The Lie bracket is given by the commutator in Cl(V).

Proof. The group I`(V) is a Lie group as a closed subgroup of the group of linear transformations of Cl(V). Since the elements of I`(V) are either purely even, or purely odd, we deduce that the tangent space at l 6 I`(V) is a subspace of E = { a ; 6 c1ev@"(v); :co

viz:

6 V , V u G V }.

Fix :z:€E, and let € 1 , . . . , € » n be an orthonormal basis of V. We can decompose 11: as co: =

£120 +€1£c1,

where kg 6 Cl(V)ev€", and C171 6 Cl(V)°dd are linear combinations of monomials involving only the vectors e t , . . . ,en. Since [:z:0, 81] = 0, and {:z:1, €1} = 0, we deduce 1 61561

2 [61»1U1]

1

2[e1,a:] EV.

In particular this means .2111 e R ® V e C 1 ( V ) . Repeating the same argument with every vector et we deduce that E C ] R € B s p a n { e - e j , 1 < i < j < dim V}

R€ 9 qlA2vl.

Thus,

T1F0(v) C R ® q(A2V). The tangent space to Spin(V) satisfies a further restriction obtained by differentiating the condition N(a3) = l. This gives

spin(V) C {$3 6 IR ® q (A2V) ,' we

-|-

x = 0} = q(A 2 V)-

Since dim spin(V) = dim QW) = dim AW we conclude that the above inclusion is in fact an equality of vector spaces. Now consider two smooth paths al)?! I (-(23€) -> Spin(V) such that 55(0) = y(0) = 1. The Lie bracket of x'(0) and 3)(0) is then found (using the Exercise 3.1.21) from the equality cl3(t)y(t)£u(tl'1y(tl

1 -|- [;ic(0), y°(0)]t2

-|-

o(t3) (as t -> 0),

631

Dirac Operators

where the above bracket is the commutator of 50(0) and 21(0) viewed as elements in the associative algebra Cl(V). To get a more explicit picture of the induced morphism of Lie algebras p*

: SPiH(V) -> QW),

we fix an orientation on V, and then choose a positively oriented orthonormal basis {et, . . . , en} of V, (n = dim V). For every an 6 spin(V), the element p*(ac°) 6 spin(V) acts on V according to

0*(x1)v

ac'

. 'U

'U

. co.

If .CU

=

.ClIi'€',1€j 7

iwA

g(Ae,;, €j)€i /\ € j i Cin extends by complex linearity to an automorphism of (Zin, while the anti-automorphism b : Cln -> Cln extends to (Cin according to the rule (u (X) z)b =

'u

(X)

Z.

As in the real case set QUT := a(a:)7, and N(ac) = b at. Let (V, g) be an Euclidean space. The complex Cluj'ord group I`°(V) is defined by a

F °(v) = {;u 6 (c1(v>*; 0(a:) p a : -1 E V Vu E v}. We denote by pC the tautological representation pC : I1e(V) -> GL(V, R). As in the real case one can check that pc(Iwc(V}} = O(V), kelpC = IC*

The spinorial norm N(:v) determines a group morphism N : I*C(V) -> ©*. Define Pin°(V) = {:z: 6 re(V> ; IN(:v)l We let the reader check the following result.

= 1}.

Proposition 12.1.49. There exists a short exact sequence 1 -> S1 -> Pin°(V) -> O(V) -> 1. Corollary 12.1.50. There exists a natural isomorphism

Pin°(v) where

"~ "

~= (Pin(Vl >< s1)/ ~,

is the equivalence relation (LU, z)

iv

we, z) G Pin(V)

(-at, -z)

X

51.

Proof. The inclusions Pin(V) C Cl(V), S1 C (I induce an inclusion (Pin(V)

X

SW ~-> co).

The image of this morphism lies obviously in F°(v)n{ INI = l} so that (Pin(V) X 51)/ can be viewed as a subgroup of PinC(V). The sought for isomorphism now follows from the exact sequence l -> S1 -> (Pin(V)

X

51)/ ~-> O(V) -> l.

We define Spin°(V) as the inverse image of SO(V) via the morphism pC : PinC(Vl -> O(V). Arguing as in the above corollary we deduce

Spin°(Vl

2

(Spin(V)

X

51)/

n

(Spin(V)

X

51)/%2.

Exercise 12.1.51. Prove Spine (V) satisfies all the conditions outlined in Subsection 10.1.5.

634

Lectures on the Geometry

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Assume now dim V is even. Then, any any isomorphism m@l(V) induces a complex unitary representation

r\/

End((;(S(V))

Spin°(V) -> Aut (S(V)) called the complex spinorial representation of Spine. It is not irreducible since, once we fix an orientation on V , the space End lslvll has a natural superstructure and, by definition, SpinC acts through even automorphism. As in the real case, §(V) splits into a direct sum of irreducible representations S:t (V). Fix a complex structure J on V. This complex structure determines two things.

(i) A canonical orientation on V. (ii) A natural subgroup

u(v,J) = {T 6 SO(V) ; [T, J] = 0} C SO(V). Denote by U

U(V, J) -> SO(V) the inclusion map.

kJ Proposition 12.1.52. There exists a natural group morphism U (V, J) -> Spin°(Vl such that the diagram below is commutative.

U(v,J) at > Spin°(V) 1.]

PC

SO(V) Proof. Let w 6 U(V), and consider a path by : [0, 1] -> U(V) connecting 11 to w. Via the inclusion U(V) SO(V) we may regard by as a path in SO(V). As such, it admits a unique lift ' : [0, 1] -> Spin(V) such that g) I 11. Using the double cover S1 -> S1, z »-> z2, we can find a path t »-> 6(t) 6 S1 such that

6(0)

=l

det *y(t).

and 65%)

Define §(w) to be the image of ('?(1), c5(1)) in Spin°(V). We have to check two things. (i) The map g is well defined. (ii) The map O' is a smooth group morphism.

To prove (i), we need to show that if 17 [0, 1] -> U(V) is a different path connecting 11 to w, and I \ : [ 0 , 1] - > S 1 is such that A(0) = l and A(t) = det 17(tl2 , then

(@(1),A(1))

I

('r(l)»6(l)) in Spin°(V).

The elements 5/(1) and 5(1) lie in the same fiber of the covering Spin(V) £> SO(V) so that they differ by an element in her p. Hence (1)==617(1%

6

:l:ll.

We can identify e as the holonomy of the covering Spin(V) -> SO(V) along the loop * 17' which goes from II to w along by, and then back to II along 17' (t) = n(1 - t).

*y

635

Dirac Operators

The map det : U(V) -> S1 induces an isomorphism between the fundamental groups (see Exercise 6.2.35 of Subsection 6.2.5). Hence, in describing the holonomy e it suffices to replace the loop *y * 17' C U(V) by any loop 1/(t) such that

det 1/(t) = det('y * it) = AA) 6 s1. Such a loop will be homotopic to by * 17 1 in U(V), and thus in S O ( V ) as well. Select 1/(t) of the form 1/(t)e1

= A(t)e1,

I/(t)€i

W > 2,

€

where (Q) is a complex, orthonormal basis of (V, J). Set f = Je With respect to the real basis (e1, f 1 , e 2 , f 2 , . . . ) the operator 1/(t) (viewed as an element of SO(V)) has the matrix description

cos 0(t) sin 0(t)

sin 0(t) cos 0(t)

7

]1

where 0 : [0, 1] -> R is a continuous map such that A(t) The lift of y(t) to Spin(V) has the form

€i0(t)

I

.

6(t) 9(t) 61f1 sin 2 2 We see that the holonomy defined by alt) is nontrivial if and only if the holonomy of the

DU)

COS

•

2

loop t »-> 6(t) in the double cover S1 i> S1 is nontrivial. This means that 6(1) and A(1) differ by the same element of Z2 as (1) and 17(1) This proves g is well defined. We leave the reader to check that g is indeed a smooth morphism of groups. 12.1.8 Low dimensional examples

In low dimensions the objects discussed in the previous subsections can be given more suggestive interpretations. In this subsection we will describe some of these interpretations. Example 12.1.53 (The case n = 1). The Clifford algebra C11 is isomorphic with the field of complex numbers C. The Z2-grading is Re (CaBIn C. The group Spin(l) is isomorphic with Z2 .

Example 12.1.54 (The case n = 2). The Clifford algebra C12 is isomorphic with the algebra of quaternions H. This can be seen by choosing an orthonormal basis {et, e2} in R2 . The isomorphism is given by 1 »->l,

€1

i->'i7

621-)

j7

61621->

k7

where i, j, and k are the imaginary units in H. Note that

Spin(2) = {a

-|-

bk

a,b€R,

0,2-1-62

The natural map Spin(1) -> SO(2) 2 S1 takes the form et

1} 2 So. »_>

et;

636

of Manzfolds

Lectures on the Geometry

Example 12.1.55 (The case n = 3). The Clifford algebra C13 is isomorphic, as an ungraded algebra, to the direct sum H ® H. More relevant is the isomorphism Cleven 3 C12 2 H given by r\/

1 P-> 17

6162

* ni,

6263

P->

j,

6361 |-)

k7

where {et, € 2 , e3} is an orthonormal basis in R3. Under this identification the operation as »-> mb coincides with the conjugation in H QUO

a-I-bi-l-cj-I-dkr->T

bi-cj-dk.

a

In particular, the spinorial norm coincides with the usual norm on H N ( a + b i + c j + d k ) =a2-l-52-I-C2-l-d2. Thus, any QUO 6 C13even \{0} is invertible, and

1 N(:z:l

1

23

b

Moreover, a simple computation shows that xR3x-1 C RB, Va: 6

cl3ven

\{0}, SO that

r0(n3) 5 H \ {0}. Hence Spin(3) E { a3€]H[; I:uI = l }

iv

SU(2).

The natural map Spin(3) -> SO(3) is precisely the map described in the Exercise 6.2.8 of Subsection 6.2.1. The isomorphism Spin(3) 2 SU(2) can be visualized by writing each q = a + bi + c j -|- dk as

q = u +j'u, u To a quaternion q

a+bi,v=

= 'LL + iv one associates the 2 X

C

di)€(C.

2 complex matrix

G SU(2).

Sq

Note that Sq' = Sq , 'v'q E H. For each quaternion q E H we denote by Lq (respectively Rq) the left (respectively right) multiplication. The right multiplication by i defines a complex structure on II-ll. Define T : II-ll -> (C2 by q='u,+ju»->Tq

H u

'U

A simple computation shows that T(Riq) = iTq, i.e., T is a complex linear map. Moreover, 'v'q 6 Spin(3) 2 $3, the matrix Sq is in SU(2), and the diagram below is commutative. H LQ

H

T ) C 2

Sq T)ac2

637

Dirac Operators

In other words, the representation

Spin(3) 9 q »-> LQ 6 GL€(H) of Spin(3) is isomorphic with the tautological representation of SU (2) on (C2 On the other hand, the correspondences

616263

l ® End (@al

6162

»-> s ® s E End

l(Q2

6263

»-> So ® 61 E End

So ® So E End ((C2) ® End ( R

® End

( End ((C2) ® End (@2). This proves that the tautological representation of SU (2) is precisely the complex spinorial representation §3. From the equalities [Re)Lq] Z { R j ) R i } = 0, we deduce that RE defines an isomoiphism of Spin(3)-modules Le

53 -> §3.

This implies there exists a Spin(3)-invariant bilinear map @¢s3

x§3->(c.

This plays an important part in the formulation of the recently introduced Seiberg-Witten equations (see [135]).

Exercise 12.1.S6. The left multiplication by ii introduces a different complex structure on H. Prove the representation Spin(3) 9 q »-> (Ra : H -> H) is (i) complex with respect to the above introduced complex structure on H, and (ii) it is isomorphic with the complex spinorial representation described by the left multiplication.

Example 12.1.57 (The case n = 4). The Clifford algebra C14 can be realized as the algebra of 2 X 2 matrices with entries in H. To describe this isomorphism we have to start from a natural embedding R4 M2(H-H) given by the correspondence I'

1

]HI""R49ac»->

A simple computation shows that the conditions in the universality property of a Clifford algebra are satisfied, and this correspondence extends to a bona-fide morphism of algebras C14 -> M2 (H). We let the reader check this morphism is also injective. A dimension count concludes it must also be surjective.

638

Lectures on the Geometry

Proposition 12.1.S8. Spin(4)%SU(2)

X

of Manzfolds

SU(2).

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) X SU(2) -> SO(4). Again we think of SU(2) as the group of unit quaternions. Thus each pair (quo, QQ) 6 SU(2) X SU(2) defines a real linear map TCII1,CI12

: H -> H, ac »->

T611,92513

= qlac'q2.

-

Clearly l;z:l = IC]1l lac°| . . = ITC11,112£8l Vi 6 H, so that each T611,92 is an orthogonal transformation of H. Since SU (2) X SU (2) is connected, all the operators TC11 ,QQ belong to the component of 0(4) containing II, i.e., T defines an (obviously smooth) group morphism T:S(/(2)

X

SU(2)->SO(4).

Note that kerT = {1, -l}, so that T is 2 : 1. In order to prove T is a double cover it suffices to show it is onto. This follows easily by noticing T is a submersion (veru'y this), so that its range must contain an entire neighborhood of II 6 SOM). Since the range of T is closed (veru'y t h i s ) 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 1.2.3). The above result shows that

~=

~=

U) ®

($l

{ d i g (p, Q); p, q G H, IpI

Ill

M)

spin(4)

Q

@(3) ® 2(3)-

~ Exercise 12.1.59. Using the identification C14 = M2 (H) show that Spin(4) corresponds to the subgroup 1 } C m2(H1.

Exercise 12.1.60. Let {et, et, et, e4} be an oriented orthonormal basis of R4. Let * denote the Hodge operator defined by the canonical metric and the above chosen orientation. Note that

-

* : A2R4 -> A2R4 is involutive, *Q

]1, so that we can split A2 into the :l:l eigenspaces of

4 2 AQR -A +R4 ® A-2RE

*

(a) Show that

As = spans{ Ni n 17? }

7

where l 171i= v§(€1 A€2=l: € 3 / \ € 4 l ,

to?

1

vi=

1 §l€1/\€3

dz

64 /\ 6 2 ) ,

/ \ € 4 :|: €2 I\€3l.

e (b) Show that the above splitting of A2R4 corresponds to the splitting under the natural identification A2n4 2 2(4

M)

@(3l®(3)

639

Dirac Operators

To obtain an explicit realization of the complex spinorial representations 34:|: we need to describe a concrete realization of the complexification (CL4. We start from the morphism of R-algebras

.

IHI9;z:=u-I-.ivr->Sx=

This extends by complexification to an isomorphism of (I-algebras

H

®1R

(C""]ll2((C).

~

(VerQ'y t h i s ) We now use this isomorphism to achieve the identification. ENdH(H ® H) ®R (C = MQIHI ®1R (C 2 End@(@2 ® (c2>. The embedding R4 -> (214 now takes the form r

H l *

n

9;z:»->Tx =

eEnd(@2 ®

.

(12.1.4)

Note that the chirality operator F = -61626364 is represented by the canonical involution

F »-> M32 e (-]1 p e GL(2; (c), S4 :SU(2) x SU(2) 9 (p,q) »-> q 6 GL(2;(C). Exactly as in the case of Spin(3), these representations can be given quaternionic descriptions. Exercise 12.1.61. The space $4 2 II-ll has a canonical complex structure defined by Re which defines (following the prescriptions in §10.1.3) an isomorphism e: -> End (A°(c2>. Identify C2 in the obvious way with A1lC2Aodd(C2, and with Aevenlc2 via the map 61 I-> 1 G A0(c2, 82 l-> 6 1 I \ € 2 . Show that under these identifications we have To = QUO 7 R4 o C(a2) id R Va: 6 (H in (12, 1.4). where Tx is the odd endomorphism of Q2 ® (Q2 defined

cu

-

N

~

Exercise 12.1.62. Let V be a 4-dimensional, oriented Euclidean space. Denote by q : A'V -> C1(V) the quantization map, and fix an isomorphism A : lc1(vl -> End (S(V)) of Z2-graded algebras. Show that for any 17 6 A2+(Vl, the image A o q (al 6 End (S(V)) is an endomorphism of the form T ® 0 6 End (S+(V)) ® End is-lvll, Exercise 12.1.63. Denote by V a 4-dimensional oriented Euclidean space. (a) Show that the representation SL (X) §4+of Spin(4) descends to a representation of SO(4) and moreover §+(V) ®(F: §+(V)§ (A°(v) ® A i m ) ®R (I as SO(4) representations. (b) Show that s+(V) 2 S+(Vl as Spin(4) modules. (c) The above isomorphism defines an element M be a Clifford bundle over the oriented Riemann manifold (M, g). We denote by e : Ql (M) -> End(E) the Clifford multiplication. (a) A Dirac structure on E is a pair (h, V) consisting of a Hermitian metric h on E , and a Clqord connection, i.e., a connection V on E compatible with h and satisfying the following conditions. (al) For any a 6 Q1 ( M ) , the Clifford multiplication by a is a skew-Hermitian endomor-

phism of E. (a2) For any a 6

Qllml, X

6 Vect (M), u 6 C°°(E)

Vi(c(a)u)

= c(Vxma)u

+ e(a)(VXu),

where VM denotes the Levi-Civita connection on T*M. (This condition means that the Clifford multiplication is covariant constant.) A pair (Clifford bundle, Dirac structure) will be called a Dirac bundle. (b) A Z2 -grading on a Dirac bundle (E, h, V) is a Z2 grading of the underlying Clifford structure E = E0 ® El, such that h = h0 ® h1, and V = V0 ® V1, where it (respectively Vfi) is a metric (respectively a metric connection) on E, .

The next result addresses the fundamental consistency question: do there exist Dirac bundles? Proposition 12.1.6S. Let E -> M be a ClQj'ord bundle over the oriented Riemann manifold ( M , go. Then there exist Dirac structures on E.

Proof. Denote by QD the (possible empty) family of Dirac structure on E. Note that if ( I W ) 6 D J = l , 2, and f 6 C°°(M), then

(fm + (1 - fm, fv1 + (1 - f)v2) 6 D. 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. Thus, it suffices to consider only the case when M is an open subset of R", and E is a trivial vector bundle. On the other hand, we cannot assume that the metric g is also trivial (Euclidean) since the local obstructions given by the Riemann curvature cannot be removed. We will distinguish two cases. A. n = dim M is even. The proof will be completed in three steps. Step 1. A special example. Consider the (complex) spinoza module (representation)

e : (Hn -> End (Sn).

Dirac Operators

641

We can assume that e is self-adjoint, i.e., c(uT) = c('u,)*, Vu 6 (CiTL

.

We will continue to denote by e the restriction of the Clifford multiplication to Spin(n) '--> (can.

Fix a global, oriented, orthonormal frame (60 of TM, and denote by ( e ) its dual coframe. Denote by w = (WW ) the connection 1-form of the Levi-Civita connection on T*M with respect to this moving frame, i.e., V67

we"

Win®€'L,

w e Qllml (X) @(11)~

ii

Using the canonical isomoiphism ,0* spin(n) -> (IJ

:

p*

1 2

1((»J)

M) we define

w e - e j €Q1(M)

(X)

spin(n).

i C1(T*M) we get a section

q(F) 6 C1(T*M) (X) End (E)On the other hand, the Clifford multiplication C : C1(T*M) -> End (E) defines a linear map c1

(X)

End (E) -> End(E), w

(X)

T »-> c(w) o T.

This map associates to the element q (F) an endomorphism of E which we denote by e(F). If (et is a local, oriented, orthonormal moving frame for T * M , then we can write .

Z'

FM

€

/\

€3 (X)

Fig 7

z' ) c ( ) m v 1

Z

C(€i)C(€3>V2VV

'As

i

I

II

ii

i>»i>c°°(c1(T*m)®End(w)) Show that 02W

v

+ C°°(End(E

(X)

w)).

644

Lectures on the Geometry

of Manzfolds

12.2 Fundamental examples This section is entirely devoted to the presentation of some fundamental examples of Dirac operators. More specifically we will discuss the Hodge-DeRham operator, the Dolbeault operator the spin and spin" Dirac. We will provide more concrete descriptions of the WeitzenbOck remainder presented in Subsection 10.1.9 and show some of its uses in establishing vanishing theorems. 12.2.1 The Hodge-DeRham operator

Let ( M , g ) be an oriented Riemann manifold and set A. T * M A'T*M C

(X)

C

For simplicity we continue to denote by Q' (M) the space of smooth differential forms on M with complex coefficients. We have already seen that the Hodge-DeRham operator

d+d*

Q. ( M ) -> Q. (M)

is a Dirac operator. In fact, we will prove this operator is a geometric Dirac operator. Continue to denote by g the Hermitian metric induced by the metric g on the complexification AcT*M. We will denote by VI all the Levi-Civita connection on the tensor bundles of M. 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. E.g., VT*M is the Levi-Civita connection on T * M . Proposition 12.2.1. The pair (g, V 9 ) defines a Dirac structure on the Clylord bundle A@T*M and d + d* i5 the associated Dirac operator Proof. In Subsection 4.1.5 we have proved that d can be alternatively described as the composition V

5

c°°lA°T*m> -> c°°lT*m (X) A°T*m1 -> c°°c°°(A2T*m

(X)

TM).

(X)

To),

We have a dual morphism

R : c°° 0°°(A2T*m uniquely determined by the equality

a(R(Y, Z)X)

(R(}/, Z) Q1(M). Hodge theory asserts that bi (M) = dim her A1 SO that, to find the first Betti number, we need to estimate the "number" of solutions of the elliptic equation A1'I']=0,

D 691

Using the Boehner-Weitzenbéck theorem and the equality (l2.2.l) we deduce that if 17 E her A1, then

V*Vn + Ric 17 = 0, onM. Taking the L2-inner product by 17, and then integrating by parts, we get

lvnl2dvg + M

(Ric 17,v)dug = 0.

(12.2.2)

M

Since Ric is non-negative definite we deduce V77 = 0, so that any harmonic 1-form must be covariant constant. In particular, since M is connected, the number of linearly independent harmonic 1-forms is no greater than the rank of T*M which is dim M. (b) Using the equality (l2.2.2) we deduce that any harmonic 1-form 17 must satisfy (Ric(:z:)17,,,, Urvlrv

0 Vcc 6 M.

If the Ricci tensor is positive at some £170 6 M, then 77(£U0) constant, and M is connected, we conclude that 17 E 0.

0. Since 17 is also covariant

Remark 12.2.6. For a very nice survey of some beautiful applications of this technique we refer to [13].

12.2.2 The Hodge-Dolbeault operator This subsection introduces the reader to the Dolbeault operator which plays a central role in complex geometry. 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. To define this operator we need a little more differential geometric background.

Definition 12.2.7. (a) Let E -> M be a smooth real vector bundle over the smooth manifold M . An almost complex structure on E is an endomorphism J : E -> E such that

JO

i

-HE.

(b) An almost complex structure on a smooth manifold M is an almost complex structure J on the tangent bundle. An almost complex manifold is a pair (manifold, almost complex structure).

Note that any almost complex manifold is necessarily even dimensional and orientable, so from the start we know that not any manifold admits almost complex structures. In

649

Dirac Operators

fact, the existence of such a structure is determined by topological invariants finer than the dimension and orientability.

Example 12.2.8. (a) A complex manifold M is almost complex. Indeed, the manifold M is locally modelled by C", and the transition maps are holomorphic maps C" -> C" . The multiplication by i defines a real endomorphism on RQ" = KI", and the differential of a holomorphic map C" -> C" commutes with the endomorphism of TQ" given by multiplication by ii in the fibers. Hence, this endomorphism induces the almost complex structure on TM. (b) For any manifold M, the total space of its tangent bundle TM can be equipped in many different ways with an almost complex structure. To see this, denote by E the tangent bundle of T M , E = T ( T M ) . The bundle E has a natural subbundle V, the vertical subbundle, which is the kernel of the differential of the canonical projection 'IT : T M -> M . If 11: 6 M , 'U 6 Ta,M, then a tangent vector in Tl,v/3,)TM is vertical if and only if it is tangent to a smooth path which entirely the fiber Tx3 M C TM. Note that we have a canonical identification

~

*'(v,x)

~-

TM.

Fix a Riemann metric h on the manifold TM, and denote by Uf C E, the subbundle of E which is the orthogonal complement of V in E. For every (u, iv) 6 TM, the differential of 'IT' induces an isomorphism 7T*

i

TxM r

Vv,:1:.

induces a bundle isomorphism A : IH -> V. Using the direct sum decomposition Fu ® V we define the endomorphism J of E in the block form

Thus,

E

. UMm)

Tr*

r

A

f\

1 1

J Exercise 12.2.9. Prove that a smooth manifold M of dimension 2n admits an almost complex structure if an only if there exists a 2-form w 6 Q2lml such that the top exterior power as" is a volume form on M . Let (M, J) be an almost complex manifold. Using the results of Subsection 2.2.5 we deduce that the complexified tangent bundle TM (X) (C splits as

TM (X) (C = (Tm)1»0 (X) (Tm)0»1. The complex bundle T M 0 is isomorphic (over (C) with ( T M , J). By duality, the operator J induces an almost complex structure in the cotangent bundle T*m, and we get a similar decomposition T*M (X) C (T*M)1,0 ® lT*Ml0,1 Z

In turn, this defines a decomposition

A@T*M

Ap» -> Q§§+1(m> splits as a direct sum d = &Bp+q=kdp'q, where dp»'1

{d : Qp»'1(m) -> Qp+1,qlml e Qp»q+1(M)},

Z

The component QW; -> Qp+1,q is denoted by 8 = 31°'q, while the component Qzmq -> Qp,q+1 is denoted by 5 = Gina.

Example 12.2.14. In local holomorphic coordinates ( ) the action of the operator described by

5

n A B d z A /\

doB

l-1l|A|

A,B

j,A,B

3*7AB . d

622

A

z

dzj

5 is

deB

It is not difficult to see that 0 Vp, q.

§p,q+1 O §p,q

In other words, for any 0 S p S dim() M, the sequence 5

0 ->Q12,0(M) 5>Q19,1(M)

>

is a cochain complex known as the p-th Dolbeault complex of the complex manifold M. Its cosmology groups are denoted by H§,q1(M)_

Lemma 12.2.15. The Dolbeault complex i5 an elliptic complex. Proof. The symbol of (§p,q is very similar to the symbol of the exterior derivative. For any x € M and any § e T * X

0(5'»q)(§) :

Ap» Ap»§0'1 A

This complex is the (Z-graded) tensor product of the trivial complex 1

0 -> AND ->

Ap»0

-> 0,

with the Koszul complex 0 ->

A0,0

(-1)P§>'1/\ A0,1 (-1)P§>'1/\

652

Lectures on the Geometry

of Manzfolds

Since §0'1 as 0, for any 5 go 0, the Koszul complex is exact (see Subsection 7.1.3). This proves that the Dolbeault complex is elliptic. To study the Dirac nature of this complex we need to introduce a Hermitian metric h on TM. Its real part is a Riemann metric g on M, and the canonical almost complex structure on TM is a skew-symmetric endomorphism with respect to this real metric. The associated 2-form Qh = -lm h is nondegenerate in the sense that Q" (n = dime; M) is a volume form on M. According to the results of Subsection 2.2.5 the orientation of M defined by Q" coincides with the orientation induced by the complex structure. We form the Hodge-Dolbeault operator 5) + 5* : Qp»'(m) -> Q2°»'(M).

5*> is a Dirac operator.

Proposition 12.2.16. The Hodge-Dolbeault operator \/§(§ + Proof. We need to show that

(u(@')((,g)

0t(5)(§)* )

2

l 4 {€(§)

-|-

i€(t7)

g* J -I-'i77*J }2

l

1 {( 6(5)- fi* J ) + 'i(€(17)+t1* J ) } l

Z {€(§)

2

l

2

=

;{

6(v)2 + i( c(€)6(t1) + c(f7)c(€) l}

Note that

c(@2=-(e(§)§* J

+e* J 6(0) =- l€ l 2

and

W2=WW J

+17* J 6(0)

lnl

2

7

o -

c(§) + we)

-

}

2

653

Dirac Operators

On the other hand, since 5 J_ 17, we deduce as above that

Cl lélul'+'&lwlcll'='0~ (Verify this!) Hence 1 1 2 -7(lsl2 W) = - 5 M 2 {Ct(@)(e) - U(5)(€)* } A natural question arises as to when the above operator is a geometric Dirac operator. Note first that the Clifford multiplication is certainly skew-adjoint since it is the symbol of a formally self-adjoint operator. Thus all we need to inquire is when the Clifford multiplication is covariant constant. Since _ 1 10

0(5)

(

.

) (e(§)+@e(v)

g* J

in* J

7

where 17 : Jo, we deduce the Clifford multiplication is covariant constant if VI J

0.

Definition 12.2.17. Let M be a complex manifold, and h a Hermitian metric on TM (viewed as a complex bundle). Then h is said to be a Kohler metric if VJ = 0, where V denotes the Levi-Civita connection associated to the Riemann metric Re h, and J is the canonical almost complex structure on TM. A pair (complex manifold, Kahler metric) is called a Kohler manifold. Exercise 12.2.18. Let (M, J) be an almost complex manifold and h a Hermitian metric on TM. Let Q = -lm h. Using the Exercise 12.2.13 show that if dpI, = 0, then the almost complex is integrable, and the metric h is Kohler. Conversely, assuming that J is integrable, and h is Kohler show that do = 0. We see that on a Kohler manifold the above Clifford multiplication is covariant constant. In fact, a more precise statement is true. Proposition 12.2.19. Let M be a complex manifold and h a Kohler metric on T M . Then the Levi-Civita induced connection on A0"T*M is a Clifford connection with respect to the above Clifford multiplication, and moreover; the Hodge-Dolbeault operator I§(5+5* I i5 the geometric Dirae operator associated to this connection.

Exercise 12.2.20. Prove the above proposition. Example 12.2.21. Let ( M , g l be an oriented 2-dimensional Riemann manifold (surface). The Hodge *-operator defines an endomorphism

*:TM->TM, satisfying *2 : -ATM, i.e., the operator * is an almost complex structure on M. Using the Exercise 12.2.18 we deduce this almost complex structure is integrable since, by dimensionality ds = 0, where Q is the natural 2-form Q(X, Y) = g(*X, Y) X, Y 6 Vect ( M ) . This complex structure is said to be canonically associated to the metric.

654

Lectures on the Geometry of Manu'olds

Example 12.2.22. Perhaps the favorite example of Kohler manifold is the complex projective space C". To describe this structure consider the tautological line bundle L1 -> CIP". It can be naturally viewed as a subbundle of the trivial bundle Q""-1 -> CIP". Denote by 110 the canonical Hermitian metric on Q - 1 , and by V0 the trivial connection. If we denote by P : Q""-1 -> L1 the orthogonal projection, then V = P o V0 | L1 defines a connection on L1 compatible with hi = h | L1. Denote by Ld the first Chern form associated to this connection ii F ( V )» CU 271' and set

hFS(X, alx

== -Wx(X7 J Y ) + i w ( X , y)

Vi 6

m, x, Y 6

TM.

Then h p is a Hermitian metric on KHP" (verify) called the Fubini-Study metric. It is clearly a Kohler metric since (see the xercise 12.2.l8) do/'L = -do : -de1(V) = 0.

Exercise 12.2.23. Describe lIeS in projective coordinates, and then prove that h is indeed a Hermitian metric, i.e., it is positive definite. Remark 12.2.24. Any complex submanifold of a Kohler manifold is obviously Kohler. In particular, any complex submanifold of CCIP" is automatically Kohler. A celebrated result of Chow states that any complex submanifold of CIP" is automatically algebraic, i.e., it can be defined as the zero set of a family of homogeneous polynomials. Thus, all complex nonsingular algebraic varieties admit a natural Kohler structure. It is thus natural to ask whether there exist Kohler manifolds which are not algebraic. The answer is positive, and a very thorough resolution of this problem is contained in the famous Kodaira embedding theorem which provides a simple necessary and sufficient condition for a compact complex manifold to be algebraic. For this work, Kodaira was awarded the Fields medal in 1954. His proofs rely essentially on some vanishing results deduced from the WeitzenbOck formulae for the Dolbeault operator 65 -|- 5*, and its twisted versions. A very clear presentation of this subject can be found in the beautiful monograph [55].

12.2.3 The spin Dirac operator Like the Dolbeault operator, the spin Dirac operator exists only on manifolds with a bit of extra structure. We will first describe this new structure. Let IM", g) be an in,-dimensional, oriented Riemann manifold. In other words, the tangent bundle TM admits an SO(n) structure so that it can be defined by an open cover (Ua), and transition maps gal? :

U05 -> SO(n)

satisfying the cocycle condition. The manifold is said to spinnable if there exist smooth maps §o:[-3

: U05 -> Spin(n)

655

Dirac Operators

satisfying the cocycle condition, and such that 0(§0¢-2)

I

QQB Va,

5,

where p : Spin(n) -> SO(n) denotes the canonical double cover. The collection Qo i-2 as above is called a spin structure. A pair (manifold, spin structure) is called a spin manifold. Not all manifolds are spinnable. To understand what can go wrong, let us start with a trivializing cover 'Ll = (UOf) for TM, with transition maps 901% and such that all the multiple intersection USB~~~v are contractible. In other words, U is a good cover. Since each of the overlaps U . , is contractible, each map gal? : UQ;-3 -> SO(n) admits at least one lift

Qop : U06 -> Spin(n). From the equality Pl§aB§Bv§v

(6_l

Note that for any (1,/3,%6 such that Ua,B76 65

-

6076

€057.

# (Z), we have

+ 6aB6 -

6067

0€%2

In other words, e, defines a tech 2-cocycle, and thus defines an element in the tech cohomology group I-I2(M, Z2). It is not difficult to see that this element is independent of the various choices: the cover 'LL the gluing data .gaB» and the lifts Qog. This element is intrinsic to the tangent bundle TM. It is called the second Stiefel-Whitney class of M, and it is denoted by W2 (Ml. We see that if w2(M) quo 0 then M cannot admit a spin structure. In fact, the converse is also true.

Proposition 12.2.25. An oriented Riemann manifold M admits a spin structure if and only

Exercise 12.2.26. Prove the above result. Remark 12.2.27. The usefulness of the above proposition depends strongly on the ability of computing W2. This is a good news/bad news situation. The good news is that algebraic topology has produced very efficient tools for doing this. The bad news is that we will not mention them since it would lead us far astray. See [83] and [98] for more details.

Remark 12.2.28. The definition of isomorphism of spin structures is rather subtle (see [95]). More precisely, two spin structures defined by the cocycles in and h., are isomorphic if there exists a collection sa 6 Z2 C Spin(n) such that the diagram below is

656

Lectures on the Geometry

of Manzfolds

commutative for all ZUGUQQ So

Spin(n)

\ J

Spin(n)

\-

I

l

§5a(@¢) 60

Spin(n)

11561

>

Spin(n)

The group H1(M, Z2) acts on Spin(M) as follows. Take an element 5 E H1lM,Z2l represented by a tech cocycle, i.e., a collection of continuous maps sag : Uog Z2 C Spin(n) satisfying the cocycle condition 6025 ' 551 '

€°v0¢

1.

..

Then the collection 5 §»» is a Spin(n) gluing cocycle defining a spin structure that we denote by 5 U. It is easy to check that the isomorphism class of 5 O' is independent of the various choice, i.e., the tech representatives for 5 and O'. Clearly, the correspondence

-

H1(M,Z2)

X

Spin(M) 9 (e,0) »->

5

0

6 Spin(M)

defines a left action of H1 ( M , Z2) on Spin(M). his action is transitive and free. Exercise 12.2.29. Prove the statements in the above remark.

Exercise 12.2.30. Describe the only two spin structures on S1 Example 12.2.31. (a) A simply connected Riemann manifold M of dimension > 5 is spinnable if and only if every compact orientable surface embedded in M has trivial normal bundle. (b) A simply connected four-manifold M is spinnable if and only if the normal bundle No: of any embedded compact, orientable surface Z] has even Euler class, i.e., 2

€(N2l

is an even integer. (c) Any compact oriented surface is spinnable. Any sphere S" admits a unique spin structure. The product of two spinnable manifolds is canonically a spinnable manifold. (d) U12 (mn) = 0 if and only if n E 3 (mod 4), while CIP" admits spin structures if and only if n is odd. Let ( M n , g) be a spin manifold. Assume the tangent bundle TM is defined by the open cover (Ua), and transition maps QQB : U05 -> SO(n).

Moreover assume the spin structure is given by the lifts

505 : U06 -> Spin(n).

657

Dirac Operators

As usual, we regard the collection .90 B as defining the principal SO(n)-bundle of oriented frames of TM . We call this bundle Pso( M) . The collection toa defines a principal Spin(n)-bundle which we denote by PSpin(M)° We can regard Pso(m) as a bundle associated to P S p i n ( M ) via p : Spin(n) -> SO(n). Using the unitary spinorial representation

An : Spin(n) -> Aut (in) we get a Hermitian vector bundle

S(M)

I

PSpin(M) X A n

5,¢

called the complex spinoza bundle. Its sections are called (complex) spinors. Let us point out that when M is even dimensional, the spinoza bundle is equipped with a natural Z/2-grading

§(m)=§+(m) ®

s-.

~

From Lemma 12.1.31 we deduce that § ( M ) is naturally a bundle of Cl(TM) modules. Then, the natural isomorphism Cl(TM) = Cl(T*M) induces on S ( M ) a structure of Cl(T*M)-module, c

c1(Tm) -> End@(§(m)).

Moreover, when M is even dimensional, the above morphism is compatible with the Z/2gradings of Cl(TM) and Ends(S(M)). As it turns out, the bundle S(M) has a natural Dirac structure whose associated Dirac operator is the spin Dirac operator on M. We will denote it by D. Since An is a unitary representation, we can equip the spinoza bundle S ( M ) with a natural metric with respect to which the Clifford multiplication e

c1(Tm) -> End@

'

Sp

T 3 Z2 > ) 9 6 »- Us

H1l I,

(T3

1N

6

U0-

Compute the spectrum of the spin Dirac operator 536 determined by the spin structure

12.2.4

0@

.

The spin" Dime operator

Our last example of Dirac operator generalizes both the spin Dirac operator, and the HodgeDolbeault operator. The common ingredient behind both these examples is the notion of spin" structure. We begin by introducing it to the reader. Let ( M n , g) be an oriented, n-dimensional Riemann manifold. As in the previous section we can regard the tangent bundle as associated to the principal bundle Pso(m) of oriented orthonormal frames. Assume Pso(m) is defined by a good open cover U = (Ua ) and transition maps .QQB : Una -> SO(nl.

The manifold M is said to posses a spin" structure (or complex spin structure) if there exists a principal Spin°(n)-bundle PSpin° such that Psouw) is associated to PSpin° via the natural morphism pC : Spin°(n) -> SO(n):

Pso(m) =

PSpin° X p c

Equivalently, this means there exist smooth maps cocycle condition, such that

SO(n).

Eton

: U05 -> Spin°(n), satisfying the

»o°(§a@) = 90B-

As for spin structures, there are obstructions to the existence SpinC structures, but they are less restrictive. Let us try to understand what can go wrong. We stick to the assumption that all the overlaps U05-~.w are contractible. Since Spin°(n) = (Spin(n) X S1l/Z2, lifting the S O ( n ) structure (Gan) reduces to finding smooth maps

hog : U0g -> Spin(n) and zag : Uog -> S1

7

such that P(IIGB) = Qa

and (6o0v>

CQBwI

h Spin(n)

X

S1/ ~,

2$n this subsection, by complex Hermitian line bundle we understand a complex line bundle equipped with a U(1)-structure.

662

Lectures on the Geometry

of Manzfolds

and L is given by the S1 cocycle Cog : UQ5 -> S1 then [ha@»zo@Co@] . Note that 9

det(c7 (X) L) = deto

(X)

O'

(X)

L is given by the cocycle

L2 7

so that

cg019 l det(a

(X)

L) ) = et1°p(det 0)

-|-

2@§op(L).

Proposition 12.2.38. The above action of Pic" (M) on Spine ( M l is free and transitive. Proof. Consider two spin" structures 0.1 and

02

defined by the good cover (Ua), and the

gluing cocycles (1)

(i)

1,2.

[ha,3' 20,3]7 Since I0°(h(1§) a sign) that

e ( ) (2) P (how) = .90B> we can assume (eventually modifying the maps h015 by

_

ho( 1 ) _ h (2). This implies that the collection ( )

Qag

(2) Zan/zale

is an S1-cocycle defining a complex Hermitian line bundle L. Obviously 02 = 0.1 (X) L. This shows the action of Pie0°(M) is transitive. We leave the reader verify this action is indeed free. The proposition is proved. Given two spin" structures 071 and 02 we can define their "difference" U2/0'1 as the unique complex Hermitian line bundle L such that (72 = 071 (X) L. This shows that the collection of spinC structures is (non-canonically) isomorphic with H2(M, Z) 2- Pic°° ( M ) . It is a sort of affine space modeled on H 2 ( m , Z) in the sense that the "difference " between two spinC structures is an element in H2(M, Z), but there is no distinguished origin of this space. A structure as above is usually called a H2(M, Z)-torsor. The set Spin°(M) is equipped with a natural involution Spin°(M) 9

07

»-> 6 6 Spin°(M),

defined as follows. If O' is defined by the cocycle [pa@» zaBI1] . Observe that O' O'

[§o@»ZaBL

then 6 is defined by the cocycle

det Spin(n) of the SO-structure to a spin structure canonically defines a spin" structure via the trivial morphism

Spin(n) -> Spin°(n)

XZ2

S1 n g »-> (g, 1) mod the Z2 - action.

We see that in this case the associated complex line bundle is the trivial bundle. This is called the canonical spin" structure of a spin manifold. We thus have a map

Spin(M) -> Spin°(M). Suppose we have fixed a spin structure on M given by a Spin-lift QuOf of an SO-gluing cocycle 96a .

To any complex Hermitian line bundle L defined by the S1-cocycle

sate the spin" structure O' (X) L defined by the gluing data {(to@»

(201Q

) we can asso-

ZQBI

The complex Hermitian line bundle deta (X) L associated to this structure is det O' (X) L = L®2. Since the topological Picard group Pic0°(M) acts freely and transitively on Spin°(]l/I), we deduce that to any pair (6, 0) e Spin(M) X Spin°(M) we can canonically associate a complex line bundle L = L 6 , 0 such that L®2 - det ct, i.e., L 6 , 0 is a square root of det O'.

~

~

Exercise 12.2.40. Show that for any O' E Spin°(M) there exists a natural bijection between the set Spin(]l/I), and the set of isomorphisms of complex line bundles L such that L®2 = det

0.

Exercise 12.2.41. Prove that the image of the natural map Spin(M) -> Spin°(M) coincides with the fixed point set of the involution 0' »->o. Exercise 12.2.42. The torus T3 is the base of a principal bundle Z3->I[-2.3->T3

7

so that, to any group morphism p : Z3 -> U(l) we can associate a complex line bundle

Lp -> T3. (a) Prove that Lp1°P2 E LP1 (X) LP27 Vp1, 01 6 Hom(Z3, S1), and use this fact to describe the first Chern class of Lp by explicitly producing a Hermitian connection on Lp. (b) Prove that any complex line bundle on T3 is isomorphic to a line bundle of the form Lp, for some morphism p E Hom(Z3, S1). (c) Show that the image of the map Spin(T3) -> Spin°(T3) consists of a single point.

Exercise 12.2.43. Prove that Spin(RIP3) consists of precisely two isomorphisms classes of spin structures and moreover, the natural map sp1n(R1p3) -> spin°(R1p)3) is a bijection.

664

Lectures on the Geometry

of Manzfolds

To understand why an almost complex manifold admits a canonical spin" structure it suffices to recall the natural morphism U (lc) -> SO(2k) factors through a morphism g : U(k) -> Spin°(2kl. The U(k) -structure of TM, defined by the gluing data hog : Uog -> U ( k ) , induces a spine structure defined by the gluing data €(h00l- Its associated line bundle is given by the S1-cocycle dot(clhogl : Uog -> S1 7 and it is precisely the determinant line bundle A'*0Tm. d€t@T1'0M The dual of this line bundle, d€t@lT*Ml1'0 = A'"00T*M plays a special role in algebraic geometry. It usually denoted by Km, and it is called the canonical line bundle. Thus the line bundle associated to this spin" structure is Kml dof KM . From the considerations in Subsections 11.1.5 and 11.1.7 we see that many (complex) vector bundles associated to the principal SpinC bundle of a spin" manifold carry natural Clifford structures, and in particular, one can speak of Dirac operators. We want to discuss in some detail a very important special case. Assume that (M, g) is an oriented, n-dimensional Riemann manifold. Fix O' 6 Spin°(M) (assuming there exist spinC structures). Denote by (Hoe) a collection of gluing data defining the SO structure Pso(m) on M with respect to some good open cover (Ua). Moreover, we assume O' is defined by the data hog : Ugh -> Spin°(n). Denote by A; the fundamental complex spinorial representation defined in Subsection 11.1.6,

An Spin°(nl -> Aut (§,,,), We obtain a complex Hermitian vector bundle

SuIMI = PSpinC

XA"

in,

which has a natural Clifford structure. This is called the bundle of complex spinors associated to o. Example 12.2.44. (a) Assume M is a spin manifold. We denote by (70 the spinC structure corresponding to the fixed spin structure. The corresponding bundle of spinors §0(M) coincides with the bundle of pure spinors defined in the previous section. Moreover for any complex line bundle L we have

aL :=SU ; S 0 ® L , where O' =

00 (X) L. Note that in this case L2 = det O' so one can write 2- §0 (X) (det 0 ) 1 / 2 . (b) Assume M is an almost complex manifold. The bundle of complex spinors associated to the canonical spine structure O' (such that det O' = KM1) is denoted by S@(M). Note that §@lml A0"T*M.

s.

r\/

665

Dirac Operators

We will construct a natural family of Dirac structures on the bundle of complex spinors associated to a spine structure. Consider the warm-up case when TM is trivial. Then we can assume .90 B E II, and

= (1h to a) •• U05 -> Spin(n) X S1 -> Spin°(n). The S1-cocycle lz0g) defines the complex Hermitian line bundle det O`. In this case something more happens. The collection (20B) is also an S1-cocycle defining a complex Hermitian line bundle L, such that L2 2- det O`. Traditionally, L is denoted by (det 0 ) 1 / 2 , though the square root may not be uniquely defined. We can now regard §0 (M) as a bundle associated to the trivial Spin(n)-bundle PSpiD and as such, there exists an isomorphism of complex Spin(n) vector bundles h

2

9

§O'lM)§§lMl

(X)

ldeto)1/2

As in the Exercise 12.1.68 of Subsection 11.1.9, we deduce that twisting the canonical connection on the bundle of pure spinors §0 (M) with any U(1)-connection on det 01/2 we obtain a Clifford connection on SO' ( M ) . Notice that if the collection {(»Ja

defines a connection on det

ct,

e u(1)

(X)

9l(Uo) }

i.e., dz2

we

+ W e over Ua0,

z25

then the collection 1

A

defines a Hermitian connection on /2 = det 071/2. Moreover, if F denotes the curvature of (w.), then the curvature of ( o ) is given by A

F

1 F. 2

(l2.2.5)

Hence any connection on det O' defines in a unique way a Clifford connection on SU (Ml. Assume now that TM is not necessarily trivial. We can however cover M by open sets (Url such that each TUo is trivial. If we pick from the start a Hermitian connection on det 0, this induces a Clifford connection on each SO(Uol. These can be glued back to a Clifford connection on §U(M) using partitions of unity. We let the reader check that the connection obtained in this way is independent of the various choices. Here is an equivalent way of associating a Clifford connection on SO' to any Hermitian connection on det O`. Fix an open cover (url of M consisting of geodesically convex open sets. The restriction of TM to any Ua is trivializable. Fixing such (orthogonal) trivializations leading to the gluing cocycle .BBQ : Uog -> SOM) , n :

dim M.

We can describe the Levi-Civita connection on TM as a collection AO( E 91(Ua)

(X)

20"),

666

Lectures on the Geometry

of Manzfolds

satisfying the transition rules AB

-(d-g,30¢lgBa -|- .9B0-A0g5 1

I

The spinC structure O' is described by a gluing cocycle

age

zeal E Spin(n)

: Uog -> Spin°(n), h e = [Q

X

U(1)/{i1}7

such that, if p : Spin(n) -> SO(n) denotes the natural 2 : l morphism, then 9601

Define A E Qllzfa)

(X)

Z

0(§5av)-

spin(n) by setting

A )

)

P* 1lA0(;v)

7

where p* : spin(n) -> M) is the natural Lie algebra isomorphism described in (l2.1.3). The collection (All satisfies the transition rules A

AB

A

Z

-1

'ld.9B S1, we denote by Lp the associated complex line bundle. We set Op := 0 (X) Lp, and we fix a Hermitian metric on Lp° The spinc structure up, and a Hermitian connection V on Lp determines spin" Dirac operator 9p(V) . (a) Prove that if is a Hermitian a unitary automorphism of Lp then ©p(7Vv'1) = 7©,,(v)7

In particular, the operators Dp('yV'7-1) and D,,(V) have the same spectrum. (b)* We say that two Hermitian connections V, V' on Lp are gauge equivalent if and only if there exists a unitary automorphism *y of L such that V' = '7V*y-1. Prove that two connections V and V' are gauge equivalent if and only if 1 lllllllll.ll.l

2tr

T3

del'

/\

(F(v ')

F( Vl ) E 2triZ,

*v'_j

1,2,3.

(c) Fix a Hermitian connection V on Lp. Describe the spectrum of ®/,(V) in terms of the real numbers l . do] Fw), j = l , 2 , 3 . . So Zfrz T3

This page intentionally left blank

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Index

(x :|; i0)", 540

0'r1/0'rg, 60 G l"U1, . . . ,"U7~l, 58 I :t , 542 e ( E ) , 287 Xa-I-» 537 Xa , 538 53, 45 det V, 49 det ind(L), 513

A*, 43 C°°(E), 35 C°°(U,E), 35 0-°°(6>)), 523 C°°(M), 7

Q

c+, 549 c-, 549 E0 m E l , 182 I°(g),337 Lk,p, 465 Re, 554 Add, 127 CTF, 29 AF, 29 Am , 163 End_(V), 61 GL(1K"), 18 GL(n,H