## Introduction to Smooth ManifoldsThis book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997). |

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### Contents

Smooth Manifolds | xvii |

Topological Manifolds | 1 |

Topological Properties of Manifolds | 6 |

Smooth Structures | 9 |

Examples of Smooth Manifolds | 15 |

Manifolds with Boundary | 22 |

Problems | 26 |

Smooth Maps | 28 |

Differential Forms | 289 |

The Geometry of Volume Measurement | 290 |

The Algebra of Alternating Tensors | 292 |

The Wedge Product | 297 |

Differential Forms on Manifolds | 302 |

Exterior Derivatives | 305 |

Symplectic Forms | 314 |

Problems | 319 |

Smooth Functions and Smooth Maps | 29 |

Lie Groups | 35 |

Smooth Covering Maps | 38 |

Proper Maps | 43 |

Partitions of Unity | 47 |

Problems | 55 |

Tangent Vectors | 58 |

Tangent Vectors | 59 |

Pushforwards | 63 |

Computations in Coordinates | 67 |

Tangent Vectors to Curves | 73 |

Alternative Definitions of the Tangent Space | 75 |

Problems | 76 |

Vector Fields | 78 |

The Tangent Bundle | 79 |

Vector Fields on Manifolds | 80 |

Lie Brackets | 87 |

The Lie Algebra of a Lie Group | 91 |

Problems | 98 |

Vector Bundles | 101 |

Local and Global Sections of Vector Bundles | 107 |

Bundle Maps | 113 |

Categories and Functors | 116 |

Problems | 119 |

The Cotangent Bundle | 122 |

Covectors | 123 |

Tangent Covectors on Manifolds | 125 |

The Cotangent Bundle | 127 |

The Differential of a Function | 130 |

Pullbacks | 134 |

Line Integrals | 136 |

Conservative Covector Fields | 141 |

Problems | 149 |

Submersions Immersions and Embeddings | 153 |

Maps of Constant Rank | 154 |

The Inverse Function Theorem and Its Friends | 157 |

ConstantRank Maps Between Manifolds | 164 |

Submersions | 167 |

Problems | 169 |

Submanifolds | 171 |

Embedded Submanifolds | 172 |

Level Sets | 178 |

Immersed Submanifolds | 184 |

Restricting Maps to Submanifolds | 188 |

Vector Fields and Covector Fields on Submanifolds | 189 |

Lie Subgroups | 192 |

Vector Subbundles | 197 |

Problems | 199 |

Lie Group Actions | 204 |

Group Actions | 205 |

Equivariant Maps | 210 |

Proper Actions | 214 |

Quotients of Manifolds by Group Actions | 216 |

Covering Manifolds | 221 |

Homogeneous Spaces | 226 |

Applications | 229 |

Problems | 234 |

Embedding and Approximation Theorems | 239 |

Sets of Measure Zero in Manifolds | 240 |

The Whitney Embedding Theorem | 244 |

The Whitney Approximation Theorems | 250 |

Problems | 256 |

Tensors | 258 |

The Algebra of Tensors | 259 |

Tensors and Tensor Fields on Manifolds | 266 |

Symmetric Tensors | 269 |

Riemannian Metrics | 271 |

Problems | 283 |

Orientations | 324 |

Orientations of Vector Spaces | 325 |

Orientations of Manifolds | 327 |

The Orientation Covering | 329 |

Orientations of Hypersurfaces | 334 |

Boundary Orientations | 338 |

The Riemannian Volume Form | 342 |

Hypersurfaces in Riemannian Manifolds | 344 |

Problems | 346 |

Integration on Manifolds | 349 |

Integration of Differential Forms on Euclidean Space | 350 |

Integration on Manifolds | 353 |

Stokess Theorem | 359 |

Manifolds with Corners | 363 |

Integration on Riemannian Manifolds | 370 |

Integration on Lie Groups | 374 |

Densities | 375 |

Problems | 382 |

De Rham Cohomology | 388 |

The de Rham Cohomology Groups | 389 |

Homotopy Invariance | 390 |

The MayerVietoris Theorem | 394 |

Computations | 399 |

Problems | 407 |

The de Rham Theorem | 410 |

Singular Homology | 411 |

Singular Cohomology | 415 |

Smooth Singular Homology | 416 |

The de Rham Theorem | 424 |

Problems | 431 |

Integral Curves and Flows | 434 |

Integral Curves | 435 |

Global Flows | 438 |

The Fundamental Theorem on Flows | 440 |

Complete Vector Fields | 446 |

Regular Points and Singular Points | 447 |

TimeDependent Vector Fields | 451 |

Proof of the ODE Theorem | 452 |

Problems | 460 |

Lie Derivatives | 464 |

The Lie Derivative | 465 |

Commuting Vector Fields | 468 |

Lie Derivatives of Tensor Fields | 473 |

Applications to Geometry | 477 |

Applications to Symplectic Manifolds | 481 |

Problems | 491 |

Integral Manifolds and Foliations | 494 |

Tangent Distributions | 495 |

Involutivity and Differential Forms | 497 |

The Frobenius Theorem | 500 |

Applications to Partial Differential Equations | 505 |

Foliations | 510 |

Problems | 515 |

Lie Groups and Their Lie Algebras | 518 |

OneParameter Subgroups | 519 |

The Exponential Map | 522 |

The Closed Subgroup Theorem | 526 |

The Adjoint Representation | 529 |

Lie Subalgebras and Lie Subgroups | 530 |

Normal Subgroups | 535 |

The Fundamental Correspondence Between Lie Algebras and Lie Groups | 536 |

Problems | 537 |

Review of Prerequisites | 540 |

Linear Algebra | 558 |

Calculus | 581 |

References | 597 |

601 | |

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### Common terms and phrases

arbitrary basis bundle map 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 differential forms dimension domain embedded submanifold equivalent Euclidean space Example Exercise exists ﬁeld Figure ﬁnite finite-dimensional ﬁrst function f given GL(n global homotopy implies induced injective integral curve integral manifold inverse isomorphism left-invariant Lemma Lie algebra Lie group Lie group homomorphism Lie subgroup linear map manifold with boundary map F matrix measure zero n-manifold neighborhood open set open subset oriented Problem Proof Proposition prove pushforward Rham Riemannian manifold satisﬁes satisfying smooth chart smooth coordinate smooth covering map smooth function smooth manifold smooth map smooth structure smooth vector field submersion subspace Suppose surjective symplectic tangent space tangent vector tensor theorem topological manifold topological space trivial U C M vector bundle