## Introduction to Smooth ManifoldsManifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma trices, as easily as we think about the familiar 2-dimensional sphere in ]R3. |

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### Contents

1 | |

Smooth Maps | 30 |

Tangent Vectors | 60 |

Vector Fields | 80 |

The Lie Algebra of a Lie Group | 93 |

Local and Global Sections of Vector Bundles | 109 |

The Cotangent Bundle | 124 |

Line Integrals | 138 |

Tensors | 260 |

Differential Forms | 291 |

Orientations | 324 |

Integration on Manifolds | 349 |

De Rham Cohomology | 388 |

The de Rham Theorem | 410 |

Integral Curves and Flows | 434 |

Lie Derivatives | 464 |

Problems | 151 |

ConstantRank Maps Between Manifolds | 166 |

Level Sets | 180 |

Lie Subgroups | 194 |

Group Actions | 207 |

Embedding and Approximation Theorems | 241 |

Integral Manifolds and Foliations | 494 |

Lie Groups and Their Lie Algebras | 518 |

Review of Prerequisites | 540 |

597 | |

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arbitrary basis called chapter closed cohomology compact compactly supported compute connected containing continuous map coordinate chart coordinate representation Corollary countable covector covering map defined definition denote derivative diffeomorphism dimension domain embedded submanifold equivalent Euclidean space Example exists Figure finite function f given GL(n global homotopy implies induced injective integral curve integral manifold inverse isomorphism left-invariant Lemma Let G Lie algebra Lie group Lie group homomorphism Lie subgroup linear map manifold with boundary map F matrix measure zero n-dimensional n-manifold neighborhood open set open subset oriented Problem Proof Proposition prove pushforward Q Exercise Rham Riemannian manifold satisfies smooth chart smooth coordinate smooth covering map smooth function smooth manifold smooth map smooth real-valued function smooth structure smooth vector field submersion subspace Suppose surjective symplectic tangent space tangent vector tensor field theorem topological manifold topological space U C M vector bundle