## 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. |

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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 |

531 | |

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### Common terms and phrases

algebra apply assume Banach space bilinear called Chapter chart closed commutes compact complete concludes the proof condition connected constant contained continuous coordinates Corollary covariant derivative curvature curve define definition denote depends determined differential element equal equation equivalent example exists exponential map expression fact fiber finite dimensional fixed follows formula function geodesic given gives Hence Hilbert immediate integral interval invertible isomorphism Jacobi lift Killing Lemma length linear map locally means measure metric morphism namely neighborhood normal Note obtain open set operator oriented orthogonal positive Proposition prove Remark representation respect Riemannian manifold satisfies shows side spray submanifold Suppose symmetric tangent tensor Theorem theory translation unique variables variation vector bundle vector fields volume form write