Fundamentals of Differential Geometry

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1 - Serge Lang Fundamentals of Differential Geometry With 22 luustrations

2 Contents Foreword Acknowledgments v 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 The Inverse Mapping Theorem 15 CHAPTER II Manifolds Atlases, Charts, Morphisms Submanifolds, Immersions, Submersions Partitions of Unity Manifolds with Boundary 39 CHAPTER III Vector Bundles Definition, Pull Backs The Tangent Bündle Exact Sequences of Bundles 52 xiii

3 XIV CONTENTS 4. Operations on Vector Bundles Splitting of Vector Bundles 63 CHAPTER IV Vector Fields and Differential Equations Existence Theorem for Differential Equations Vector Fields, Curves, and Flows Sprays The Flow of a Spray and the Exponential Map Existence of Tubulär Neighborhoods Uniqueness of Tubulär Neighborhoods 112 CHAPTER V Operations on Vector Fields and Differential Forms Vector Fields, Differential Operators, Brackets Lie Derivative Exterior Derivative The Poincare Lemma Contractions and Lie Derivative Vector Fields and 1-Forms Under Seif Duality The Canonical 2-Form Darboux's Theorem 151 CHAPTER VI The Theorem of Frobenlus Statement of the Theorem Differential Equations Depending on a Parameter Proof of the Theorem The Global Formulation Lie Groups and Subgroups 165 PART II Metrics, Covarlant Derivatives, and Riemannlan Geometry 171 CHAPTER VII Metrics Definition and Functoriality The Hubert Group Reduction to the Hubert Group Hilbertian Tubulär Neighborhoods The Morse-Palais Lemma The Riemannian Distance The Canonical Spray 192 CHAPTER VIII Covarlant Derivatives and Geodeslcs Basic Properties 196

4 CONTENTS XV 2. Sprays and Covariant Derivatives Derivative Along a Curve and Parallelism The Metrie Derivative More Local Results on the Exponential Map Riemannian Geodesic Length and Completeness 221 CHAPTER IX Curvature The Riemann Tensor Jacobi Lifts Application of Jacobi Lifts to Texp* Convexity Theorems Taylor Expansions 263 CHAPTER X Jacobi Lifts and Tensorial Splitting of the Double Tangent Bündle Convexity of Jacobi Lifts Global Tubulär Neighborhood of a Totally Geodesic Submanifold More Convexity and Comparison Results Splitting of the Double Tangent Bündle Tensorial Derivative of a Curve in TX and of the Exponential Map The Flow and the Tensorial Derivative 291 CHAPTER XI Curvature and the Variation Formula The Index Form, Variations, and the Second Variation Formula Growth of a Jacobi Lift The Semi Parallelogram Law and Negative Curvature Totally Geodesic Submanifolds Rauch Comparison Theorem 318 CHAPTER XII An Example of Seminegative Curvature Pos (R) as a Riemannian Manifold The Metrie Increasing Property of the Exponential Map Totally Geodesic and Symmetrie Submanifolds 332 CHAPTER XIII Automorphisms and Symmetries The Tensorial Second Derivative Alternative Definitions of Killing Fields Metrie Killing Fields Lie Algebra Properties of Killing Fields Symmetrie Spaces Parallelism and the Riemann Tensor 365

5 XVI CONTENTS CHAPTER XIV Immersions and Submerslons The Covariant Derivative on a Submanifold 369 2, The Hessian and Laplacian on a Submanifold The Covariant Derivative on a Riemannian Submersion The Hessian and Laplacian on a Riemannian Submersion The Riemann Tensor on Submanifolds The Riemann Tensor on a Riemannian Submersion 393 PART III Volume Forms and Integration 395 CHAPTER XV Volume Forms Volume Forms and the Divergence Covariant Derivatives The Jacobian Determinant of the Exponential Map The Hodge Star on Forms Hodge Decomposition of Differential Forms Volume Forms in a Submersion Volume Forms on Lie Groups and Homogeneous Spaces Homogeneously Fibered Submersions 440 CHAPTER XVI Integration of Differential Forms Sets of Measure Change of Variables Formula Orientation The Measure Associated with a Differential Form Homogeneous Spaces 471 CHAPTER XVII Stokes' Theorem Stokes' Theorem for a Rectangular Simplex Stokes' Theorem on a Manifold Stokes' Theorem with Singularities 482 CHAPTER XVIII Applications of Stokes' Theorem The Maximal de Rham Cohomology Moser's Theorem The Divergence Theorem The Adjoint of d for Higher Degree Forms Cauchy's Theorem The Residue Theorem 507

6 CONTENTS XV11 APPENDIX The Spectral Theorem Hubert Space Functionals and Operators Hermitian Operators 515 Blbliography 523 Index 531

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