# Fundamentals of Differential Geometry

Springer Science & Business Media, Sep 21, 2001 - Mathematics - 540 pages
The 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.

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### Contents

 CHAPTER I 3 CHAPTER II 22 4 Manifolds with Boundary 39 3 Exact Sequences of Bundles 52 4 Operations on Vector Bundles 58 2 Vector Fields Curves and Flows 88 4 The Flow of a Spray and the Exponential Map 105 2 Lie Derivative 122
 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 395 CHAPTER XV 397 2 Covariant Derivatives 407 3 The Jacobian Determinant of the Exponential Map 412

 4 The Poincare Lemma 137 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 7expY 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
 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 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 Index 531

### References to this book

 An Introduction to Symplectic GeometryRolf BerndtLimited preview - 2001
 Means of Hilbert Space OperatorsLimited preview - 2002
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