Fundamentals of Differential GeometryThe present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differen tiable maps in them (immersions, embeddings, isomorphisms, etc. ). One may also use differentiable structures on topological manifolds to deter mine the topological structure of the manifold (for example, it la Smale [Sm 67]). In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. ) and studies properties connected especially with these objects. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. In differential equations, one studies vector fields and their in tegral curves, singular points, stable and unstable manifolds, etc. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings. |
Contents
| 3 | |
39 | 22 |
4 Manifolds with Boundary | 41 |
4 Operations on Vector Bundles 63383 | 58 |
2 Vector Fields Curves and Flows | 88 |
4 The Flow of a Spray and the Exponential Map | 105 |
2 Lie Derivative | 122 |
4 The Poincaré Lemma | 137 |
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 | 396 |
CHAPTER XV | 397 |
2 Covariant Derivatives | 407 |
3 The Jacobian Determinant of the Exponential Map | 412 |
4 The Hodge Star on Forms | 418 |
8 Darbouxs Theorem | 151 |
5 Lie Groups and Subgroups | 165 |
APPENDIX | 173 |
3 Reduction to the Hilbert Group | 180 |
CHAPTER VIII | 195 |
2 Sprays and Covariant Derivatives | 199 |
5 More Local Results on the Exponential Map | 215 |
Curvature | 231 |
3 Application of Jacobi Lifts to Texpx | 246 |
5 Taylor Expansions | 263 |
CHAPTER XIV | 270 |
3 More Convexity and Comparison Results | 276 |
6 The Flow and the Tensorial Derivative | 291 |
2 Growth of a Jacobi Lift | 304 |
5 Rauch Comparison Theorem | 318 |
3 Totally Geodesic and Symmetric Submanifolds | 332 |
2 Alternative Definitions of Killing Fields | 347 |
6 Parallelism and the Riemann Tensor | 365 |
Immersions and Submersions | 369 |
2 The Hessian and Laplacian on a Submanifold | 376 |
3 The Covariant Derivative on a Riemannian Submersion | 383 |
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 |
Integration of Differential Forms | 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 | 475 |
2 Stokes Theorem on a Manifold | 478 |
3 Stokes Theorem with Singularities | 482 |
CHAPTER XVIII | 489 |
2 Mosers Theorem | 496 |
3 The Divergence Theorem | 497 |
4 The Adjoint of d for Higher Degree Forms | 501 |
5 Cauchys Theorem | 503 |
6 The Residue Theorem | 507 |
The Spectral Theorem | 511 |
| 531 | |
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Common terms and phrases
algebra assume automorphism Banach space bilinear chart class CP commutes compact support concludes the proof coordinates Corollary covariant derivative define denote differential equation differential form exponential map fiber finite dimensional follows formula function given Hence Hilbert space initial conditions integral curve invertible isometry Jacobi lift Killing fields Laplacian Lemma Let f Lie derivative linear map metric derivative morphism notation open neighborhood open set open subset operator oriented orthogonal orthonormal parallel translation partitions of unity positive definite Proposition 1.1 prove pseudo Riemannian Riemann tensor Riemannian manifold Riemannian metric satisfies scalar product seminegative curvature spray submanifold submersion subspace Suppose symmetric spaces tangent bundle tangent space Texp Theorem 3.1 toplinear totally geodesic variables vector bundle vector fields vector space volume form