## An Introduction to ManifoldsManifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'. |

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### Contents

1 | |

3 | |

Chapter 2 Manifolds | 46 |

Chapter 3 The Tangent Space | 85 |

Chapter 4 Lie Groups and Lie Algebras | 163 |

Chapter 5 Differential Forms | 189 |

Chapter 6 Integration | 235 |

Chapter 7 De Rham Theory | 273 |

Appendices | 317 |

Solutions to Selected ExercisesWithin the Text | 361 |

Hints and Solutions to Selected EndofSection Problems | 366 |

List of Notations | 387 |

395 | |

397 | |

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### Common terms and phrases

basis bump function calculus called closed cochain coefficients compact compute constant rank coordinate chart Corollary covector defined definition denote diffeomorphism differential forms disjoint equivalence relation Euclidean space Example exterior derivative function f functor Hausdorff Hence homeomorphism homotopy immersion inclusion map integral curve inverse function theorem isomorphism k-form left-invariant vector fields Lemma Let F level set theorem Lie algebra Lie derivative Lie group linear map locally finite map F multiplication nonzero notation nowhere-vanishing open cover open interval open set open subset partition of unity permutation Problem Proof Proposition Prove pullback quaternions real number regular level set regular submanifold Rham cohomology Show smooth manifold smooth map standard coordinates subgroup submersion subspace topology supp Suppose surjective tangent bundle tangent space tangent vector topological space U,xl vector bundle vector field vector space wedge product zero set