## 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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assume atlas basis boundary bundle called chart circle closed cohomology collection compact complex compute condition connected constant containing continuous coordinate countable curve defined definition denote determinant diffeomorphism differential forms dimension disjoint element equal equation equivalence exact Example existence exterior derivative Figure finite follows formula function given gives Hausdorff Hence homeomorphism homotopy identity immersion induced integral interval inverse isomorphism Lemma level set Lie algebra Lie group linear map locally manifold map F matrix maximal means multiplication neighborhood notation open set open subset orientation partition plane Problem projection Proof properties Proposition Prove pullback quotient rank regular submanifold relation relative represented respectively restriction Riemann integral sequence Show sides smooth standard Suppose tangent space tangent vector theorem topological space topology vector field vector space write zero