## Differential and Riemannian ManifoldsThis is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. 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 differentiable maps in them (immersions, embeddings, isomorphisms, etc.). |

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### Contents

1 | |

Manifolds | 20 |

4 Manifolds with Boundary | 36 |

3 Exact Sequences of Bundles | 49 |

CHAPTER IV | 64 |

2 Vector Fields Curves and Flows | 86 |

3 Sprays | 94 |

4 The Flow of a Spray and the Exponential Map | 103 |

2 The Hilbert Group | 173 |

3 Reduction to the Hilbert Group | 176 |

4 Hilbertian Tubular Neighborhoods | 179 |

5 The MorsePalais Lemma | 182 |

6 The Riemannian Distance | 184 |

7 The Canonical Spray | 188 |

CHAPTER VIII | 190 |

Covariant Derivatives and Geodesics | 191 |

5 Existence of Tubular Neighborhoods | 108 |

6 Uniqueness of Tubular Neighborhoods | 110 |

CHAPTER V | 114 |

2 Lie Derivative | 120 |

3 Exterior Derivative | 122 |

4 The Poincaré Lemma | 135 |

5 Contractions and Lie Derivative | 137 |

6 Vector Fields and 1Forms Under Self Duality | 141 |

7 The Canonical 2Form | 146 |

8 Darbouxs Theorem | 148 |

CHAPTER VI | 153 |

2 Differential Equations Depending on a Parameter | 158 |

3 Proof of the Theorem | 159 |

4 The Global Formulation | 160 |

5 Lie Groups and Subgroups | 163 |

CHAPTER VII | 169 |

2 Sprays and Covariant Derivatives | 194 |

3 Derivative Along a Curve and Parallelism | 199 |

4 The Metric Derivative | 203 |

5 More Local Results on the Exponential Map | 209 |

6 Riemannian Geodesic Length and Completeness | 216 |

CHAPTER IX | 219 |

Curvature | 225 |

2 Jacobi Lifts | 233 |

3 Application of Jacobi Lifts to dexpx | 241 |

5 Taylor Expansions | 257 |

4 The Hodge Star on Forms | 273 |

2 Change of Variables Formula | 289 |

CHAPTER XII | 307 |

CHAPTER XIII | 321 |

5 Cauchys Theorem | 335 |

Bibliography | 355 |

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algebra apply assume automorphisms Banach space boundary C*-isomorphism called Chapter chart class CP commutative compact support concludes the proof contained continuous linear map coordinates Corollary covariant derivative curvature define denote differential equation differential form domain equal exists a unique exponential map expression fiber finite dimensional follows formula function functor Furthermore geodesic give given Hence Hilbert space induces initial condition integral curve inverse Jacobi lift kernel Lemma Let f Let x e locally map f morphism multilinear non-singular notation open ball open interval open neighborhood open set open subset operator oriented orthogonal orthonormal partitions of unity point x e positive definite Proposition 1.1 prove pseudo Riemannian representation Riemannian manifold Riemannian metric satisfies scalar product sequence spray Stokes theorem subbundle submanifold subspace tangent bundle tangent space Theorem 3.1 toplinear isomorphism topological trivial variable VB-morphism vector bundle vector field vector space