.

0 (hodnocen0 x )

BK

New York : Springer, 1999

xvii,535 s. : il

ISBN 0-387-98593-X (váz.)

Graduate texts in mathematics ; [vol.] 191

Bibliografie: s. 523-530

Geometrie diferenciální - učebnice vysokošk.

000032242

Contents // Foreword v // Acknowledgment* xi // PARTI // General Differential Theory 1 // CHAPTER I // Differential Calculus 3 // $1. Categories... 4 // §2 Topological Vector Space»... 5 // S3. Derivatives and Composition of Maps... 8 // §4 Integration and Taylor’s Formula... 12 // S5. The Inverse Mapping Theorem... 15 // CHAPTER II // Manifolds 22 // SI. Atlases. Chans. Morphism*... 22 // §2 Submanifolds. Immersions. Submersions... 25 // S3. Partitions of Unity... 33 // S». Manifolds with Boundary... 39 // CHAPTER HI // Vector Bundles 43 // SI. Definition, Pull Backs... 43 // ?2. TTse Tangent Bundle 51 // S3. Exact Sequences of Bundles... 52 // 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 Tubular Neighborhoods... 110 // §6. Uniqueness of Tubular 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 Self Duality... 143 // §7. The Canonical 2-Form... 149 // §8. Darboux’s

Theorem... 151 // CHAPTER VI // The Theorem of Frobenius... 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, Covariant Derivatives, and Riemannian Geometry... 171 // CHAPTER VII // Metrics... 173 // §1. Definition and Functoriality... 173 // §2. The Hilbert Group... 177 // §3. Reduction to the Hilbert Group... 180 // §4. Hilbcrtian Tubular Neighborhoods... 184 // §5. The Morse-Palais Lemma... 186 // §6. The Riemannian Distance... 189 // §7. The Canonical Spray... 192 // CHAPTER VIII // Covariant Derivatives and Geodesics... 196 // §1. Basic Properties... 196 // CONTENTS // XV // §2. Sprays and Covariant Derivatives... 199 // §3. Derivative Along a Curve and Parallelism... 204 // §4. The Metric 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 7exp.r... 246 // §4. Convexity Theorems... 255 // §5. Taylor Expansions... 263 // CHAPTER X // Jacobi Lifts and Tensorial Splitting ot the Double Tangent Bundle... 267 // §1. Convexity of Jacobi Lifts... 267 // §2. Global Tubular Neighborhood of a Totally Geodesic Submanifold 271 // §3. More Convexity and Comparison

Results... 276 // §4. Splitting of the Double Tangent Bundle... 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 Metric Increasing Property of the Exponential Map... 327 // §3. Totally Geodesic and Symmetric Submanifolds... 332 // CHAPTER XIII // Automorphisms and Symmetries... 339 // §1. The Tensorial Second Derivative... 342 // §2. Alternative Definitions of Killing Fields... 347 // §3. Metric Killing Fields... 351 // §4. Lie Algebra Properties of Killing Fields... 354 // §5. Symmetric Spaces... 358 // §6. Parallelism and the Riemann Tensor... 365 // XVI // CONTENTS // CHAPTER XIV // Immersions and Submersions... 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... 496 // §3. The Divergence Theorem... 497 // §4. The Adjoint of d for Higher Degree Forms... 501 // §5. Cauchy’s Theorem... 503 // §6. The Residue Theorem... 507 // CONTENTS ХѴІІ // APPENDIX // The Spectral Theorem... 511 // §1. Hilbert Space... 511 // §2. Functionals and Operators... 512 // §3. Hermitian Operators... 515 // Bibliography ... 523 // Index... 531