- Serge Lang

Fundamentals of Differential Geometry

With 22 lUustrations

Contents

Foreword v

Acknowledgments xi

PARTI General Differential Theory 1

CHAPTERI

Differential Calculus 3

§1. Categories 4 §2. Topological Vector Spaces 5 §3. Derivatives and Composition of Maps 8 §4. Integration and Taylor's Formula 12 §5. The Inverse Mapping Theorem 15

CHAPTER II

Manifolds 22

§1. Atlases, Charts, Morphisms 22 §2. Submanifolds, Immersions, Submersions 25 §3. Partitions of Unity 33 §4. Manifolds with Boundary 39

CHAPTER III

Vector Bundles 43

§1. Definition, Pull Backs 43 §2. The Tangent Bündle 51 §3. Exact Sequences of Bundles 52

xiii

XIV CONTENTS

§4. Operations on Vector Bundles 58 §5. Splitting of Vector Bundles 63

CHAPTER IV

Vector Fields and Differential Equations 66

§1. Existence Theorem for Differential Equations 67 §2. Vector Fields, Curves, and Flows 88 §3. Sprays 96 §4. The Flow of a Spray and the Exponential Map 105 §5. Existence of Tubulär Neighborhoods 110 §6. Uniqueness of Tubulär Neighborhoods 112

CHAPTER V

Operations on Vector Fields and Differential Forms 116

§1. Vector Fields, Differential Operators, Brackets 116 §2. Lie Derivative 122 §3. Exterior Derivative 124 §4. The Poincare Lemma 137 §5. Contractions and Lie Derivative 139 §6. Vector Fields and 1-Forms Under Seif Duality 143 §7. The Canonical 2-Form 149 §8. Darboux's Theorem 151

CHAPTER VI

The Theorem of Frobenlus 155

§1. Statement of the Theorem 155 §2. Differential Equations Depending on a Parameter 160 §3. Proof of the Theorem 161 §4. The Global Formulation 162 §5. Lie Groups and Subgroups 165

PART II Metrics, Covarlant Derivatives, and Riemannlan Geometry 171

CHAPTER VII

Metrics 173 §1. Definition and Functoriality 173 §2. The Hubert Group 177 §3. Reduction to the Hubert Group 180 §4. Hilbertian Tubulär Neighborhoods 184 §5. The Morse-Palais Lemma 186 §6. The Riemannian Distance 189 §7. The Canonical Spray 192

CHAPTER VIII

Covarlant Derivatives and Geodeslcs 196

§1. Basic Properties 196

CONTENTS XV

§2. Sprays and Covariant Derivatives 199 §3. Derivative Along a Curve and Parallelism 204 §4. The Metrie Derivative 209 §5. More Local Results on the Exponential Map 215 §6. Riemannian Geodesic Length and Completeness 221

CHAPTER IX

Curvature 231

§1. The Riemann Tensor 231 §2. Jacobi Lifts 239 §3. Application of Jacobi Lifts to Texp* 246 §4. Convexity Theorems 255 §5. Taylor Expansions 263

CHAPTER X

Jacobi Lifts and Tensorial Splitting of the Double Tangent Bündle 267

§1. Convexity of Jacobi Lifts 267 §2. Global Tubulär Neighborhood of a Totally Geodesic Submanifold 271 §3. More Convexity and Comparison Results 276 §4. Splitting of the Double Tangent Bündle 279 §5. Tensorial Derivative of a Curve in TX and of the Exponential Map 286 §6. The Flow and the Tensorial Derivative 291

CHAPTER XI

Curvature and the Variation Formula 294

§1. The Index Form, Variations, and the Second Variation Formula 294 §2. Growth of a Jacobi Lift 304 §3. The Semi Parallelogram Law and Negative Curvature 309 §4. Totally Geodesic Submanifolds 315 §5. Rauch Comparison Theorem 318

CHAPTER XII

An Example of Seminegative Curvature 322

§1. Pos„(R) as a Riemannian Manifold 322 §2. The Metrie Increasing Property of the Exponential Map 327 §3. Totally Geodesic and Symmetrie Submanifolds 332

CHAPTER XIII

Automorphisms and Symmetries 339

§1. The Tensorial Second Derivative 342 §2. Alternative Definitions of Killing Fields 347 §3. Metrie Killing Fields 351 §4. Lie Algebra Properties of Killing Fields 354 §5. Symmetrie Spaces 358 §6. Parallelism and the Riemann Tensor 365

XVI CONTENTS

CHAPTER XIV

Immersions and Submerslons 369

§1. The Covariant Derivative on a Submanifold 369 §2, The Hessian and Laplacian on a Submanifold 376 §3. The Covariant Derivative on a Riemannian Submersion 383 §4. The Hessian and Laplacian on a Riemannian Submersion 387 §5. The Riemann Tensor on Submanifolds 390 §6. The Riemann Tensor on a Riemannian Submersion 393

PART III Volume Forms and Integration 395

CHAPTER XV

Volume Forms 397

§1. Volume Forms and the Divergence 397 §2. Covariant Derivatives 407 §3. The Jacobian Determinant of the Exponential Map 412 §4. The Hodge Star on Forms 418 §5. Hodge Decomposition of Differential Forms 424 §6. Volume Forms in a Submersion 428 §7. Volume Forms on Lie Groups and Homogeneous Spaces 435 §8. Homogeneously Fibered Submersions 440

CHAPTER XVI

Integration of Differential Forms 448

§1. Sets of Measure 0 448 §2. Change of Variables Formula 453 §3. Orientation 461 §4. The Measure Associated with a Differential Form 463 §5. Homogeneous Spaces 471

CHAPTER XVII

Stokes' Theorem 475

§1. Stokes' Theorem for a Rectangular Simplex 475 §2. Stokes' Theorem on a Manifold 478 §3. Stokes' Theorem with Singularities 482

CHAPTER XVIII

Applications of Stokes' Theorem 489

§1. The Maximal de Rham Cohomology 489 §2. Moser's Theorem 495 §3. The Divergence Theorem 496 §4. The Adjoint of d for Higher Degree Forms 501 §5. Cauchy's Theorem 503 §6. The Residue Theorem 507

CONTENTS XV11

APPENDIX The Spectral Theorem 511

§1. Hubert Space 511 §2. Functionals and Operators 512 §3. Hermitian Operators 515

Blbliography 523

Index 531