Manifolds and Differential Geometry *EFFREY-,EE 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/107 Manifolds and Differential Geometry Manifolds and Differential Geometry Jeffrey M. Lee Graduate Studies in Mathematics Volume 107 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 58A05, 58A10, 53C05, 22E15, 53C20, 53B30, 55R10, 53Z05. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-107 Library of Congress Cataloging-in-Publication Data Lee, Jeffrey M., 1956– Manifolds and differential geometry / Jeffrey M. Lee. p. cm. — (Graduate studies in mathematics ; v. 107) Includes bibliographical references and index. ISBN 978-0-8218-4815-9 (alk. paper) 1. Geometry, Differential. 2. Topological manifolds. 3. Riemannian manifolds. I. Title. QA641.L38 2009 516.36—dc22 2009012421 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Preface xi Chapter 1. Differentiable Manifolds 1 §1.1. Preliminaries 2 §1.2. Topological Manifolds 6 §1.3. Charts, Atlases and Smooth Structures 11 §1.4. Smooth Maps and Diffeomorphisms 22 §1.5. Cut-off Functions and Partitions of Unity 28 §1.6. Coverings and Discrete Groups 31 §1.7. Regular Submanifolds 46 §1.8. Manifolds with Boundary 48 Problems 51 Chapter 2. The Tangent Structure 55 §2.1. The Tangent Space 55 §2.2. Interpretations 65 §2.3. The Tangent Map 66 §2.4. Tangents of Products 72 §2.5. Critical Points and Values 74 §2.6. Rank and Level Set 78 §2.7. The Tangent and Cotangent Bundles 81 §2.8. Vector Fields 87 §2.9. 1-Forms 110 §2.10. Line Integrals and Conservative Fields 116 v vi Contents §2.11. Moving Frames 120 Problems 122 Chapter 3. Immersion and Submersion 127 §3.1. Immersions 127 §3.2. Immersed and Weakly Embedded Submanifolds 130 §3.3. Submersions 138 Problems 140 Chapter 4. Curves and Hypersurfaces in Euclidean Space 143 §4.1. Curves 145 §4.2. Hypersurfaces 152 §4.3. The Levi-Civita Covariant Derivative 165 §4.4. Area and Mean Curvature 178 §4.5. More on Gauss Curvature 180 §4.6. Gauss Curvature Heuristics 184 Problems 187 Chapter 5. Lie Groups 189 §5.1. Definitions and Examples 189 §5.2. Linear Lie Groups 192 §5.3. Lie Group Homomorphisms 201 §5.4. Lie Algebras and Exponential Maps 204 §5.5. The Adjoint Representation of a Lie Group 220 §5.6. The Maurer-Cartan Form 224 §5.7. Lie Group Actions 228 §5.8. Homogeneous Spaces 240 §5.9. Combining Representations 249 Problems 253 Chapter 6. Fiber Bundles 257 §6.1. General Fiber Bundles 257 §6.2. Vector Bundles 270 §6.3. Tensor Products of Vector Bundles 282 §6.4. Smooth Functors 283 §6.5. Hom 285 §6.6. Algebra Bundles 287 §6.7. Sheaves 288 Contents vii §6.8. Principal and Associated Bundles 291 Problems 303 Chapter 7. Tensors 307 §7.1. Some Multilinear Algebra 308 §7.2. Bottom-Up Approach to Tensor Fields 318 §7.3. Top-Down Approach to Tensor Fields 323 §7.4. Matching the Two Approaches to Tensor Fields 324 §7.5. Tensor Derivations 327 §7.6. Metric Tensors 331 Problems 342 Chapter 8. Differential Forms 345 §8.1. More Multilinear Algebra 345 §8.2. Differential Forms 358 §8.3. Exterior Derivative 363 §8.4. Vector-Valued and Algebra-Valued Forms 367 §8.5. Bundle-Valued Forms 370 §8.6. Operator Interactions 373 §8.7. Orientation 375 §8.8. Invariant Forms 384 Problems 388 Chapter 9. Integration and Stokes’ Theorem 391 §9.1. Stokes’ Theorem 394 §9.2. Differentiating Integral Expressions; Divergence 397 §9.3. Stokes’ Theorem for Chains 400 §9.4. Differential Forms and Metrics 404 §9.5. Integral Formulas 414 §9.6. The Hodge Decomposition 418 §9.7. Vector Analysis on R3 425 §9.8. Electromagnetism 429 §9.9. Surface Theory Redux 434 Problems 437 Chapter 10. De Rham Cohomology 441 §10.1. The Mayer-Vietoris Sequence 447 §10.2. Homotopy Invariance 449 viii Contents §10.3. Compactly Supported Cohomology 456 §10.4. Poincar´e Duality 460 Problems 465 Chapter 11. Distributions and Frobenius’ Theorem 467 §11.1. Definitions 468 §11.2. The Local Frobenius Theorem 471 §11.3. Differential Forms and Integrability 473 §11.4. Global Frobenius Theorem 478 §11.5. Applications to Lie Groups 484 §11.6. Fundamental Theorem of Surface Theory 486 §11.7. Local Fundamental Theorem of Calculus 494 Problems 498 Chapter 12. Connections and Covariant Derivatives 501 §12.1. Definitions 501 §12.2. Connection Forms 506 §12.3. Differentiation Along a Map 507 §12.4. Ehresmann Connections 509 §12.5. Curvature 525 §12.6. Connections on Tangent Bundles 530 §12.7. Comparing the Differential Operators 532 §12.8. Higher Covariant Derivatives 534 §12.9. Exterior Covariant Derivative 536 §12.10. Curvature Again 540 §12.11. The Bianchi Identity 541 §12.12. G-Connections 542 Problems 544 Chapter 13. Riemannian and Semi-Riemannian Geometry 547 §13.1. Levi-Civita Connection 550 §13.2. Riemann Curvature Tensor 553 §13.3. Semi-Riemannian Submanifolds 560 §13.4. Geodesics 567 §13.5. Riemannian Manifolds and Distance 585 §13.6. Lorentz Geometry 588 §13.7. Jacobi Fields 594 Contents ix §13.8. First and Second Variation of Arc Length 599 §13.9. More Riemannian Geometry 612 §13.10. Cut Locus 617 §13.11. Rauch’s Comparison Theorem 619 §13.12. Weitzenb¨ock Formulas 623 §13.13. Structure of General Relativity 627 Problems 634 Appendix A. The Language of Category Theory 637 Appendix B. Topology 643 §B.1. The Shrinking Lemma 643 §B.2. Locally Euclidean Spaces 645 Appendix C. Some Calculus Theorems 647 Appendix D. Modules and Multilinearity 649 §D.1. R-Algebras 660 Bibliography 663 Index 667 Preface Classical differential geometry is the approach to geometry that takes full advantage of the introduction of numerical coordinates into a geometric space. This use of coordinates in geometry was the essential insight of Ren´e Descartes that allowed the invention of analytic geometry and paved the way for modern differential geometry. The basic object in differential geometry (and differential topology) is the smooth manifold. This is a topological space on which a sufficiently nice family of coordinate systems or “charts” is defined. The charts consist of locally defined n-tuples of functions. These functions should be sufficiently independent of each other so as to allow each point in their common domain to be specified by the values of these functions. One may start with a topological space and add charts which are compatible with the topology or the charts themselves can generate the topology. We take the latter approach. The charts must also be compatible with each other so that changes of coordinates are always smooth maps. Depending on what type of geometry is to be studied, extra structure is assumed such as a distinguished group of symmetries, a distinguished “ten- sor” such as a metric tensor or symplectic form or the very basic geometric object known as a connection. Often we find an interplay among many such elements of structure. Modern differential geometers have learned to present much of the sub- ject without constant direct reference to locally defined objects that depend on a choice of coordinates. This is called the “invariant” or “coordinate free” approach to differential geometry. The only way to really see exactly what this all means is by diving in and learning the subject. The relationship between geometry and the physical world is fundamen- tal on many levels. Geometry (especially differential geometry) clarifies, xi xii Preface codifies and then generalizes ideas arising from our intuitions about certain aspects of our world. Some of these aspects are those that we think of as forming the spatiotemporal background of our activities, while other aspects derive from our experience with objects that have “smooth” surfaces. The Earth is both a surface and a “lived-in space”, and so the prefix “geo” in the word geometry is doubly appropriate. Differential geometry is also an appro- priate mathematical setting for the study of what we classically conceive of as continuous physical phenomena such as fluids and electromagnetic fields. Manifolds have dimension. The surface of the Earth is two-dimensional, while the configuration space of a mechanical system is a manifold which may easily have a very high dimension. Stretching the imagination further we can conceive of each possible field configuration for some classical field as being an abstract point in an infinite-dimensional manifold. The physicists are interested in geometry because they want to under- stand the way the physical world is in “actuality”. But there is also a discovered “logical world” of pure geometry that is in some sense a part of reality too.