# Introduction to Smooth Manifolds

Springer Science & Business Media, 2003 - Mathematics - 628 pages
This 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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User Review  - Leonhard Euler - Goodreads

Should be pithier. Read full review

#### Review: Introduction to Smooth Manifolds

Should be pithier. Read full review

### 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 Index 601 Copyright