# Introduction to Smooth Manifolds

Springer, 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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### 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 300 Exterior Derivatives 303 Symplectic Forms 312 Problems 317

 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 322 Orientations of Vector Spaces 323 Orientations of Manifolds 325 The Orientation Covering 327 Orientations of Hypersurfaces 332 Boundary Orientations 336 The Riemannian Volume Form 340 Hypersurfaces in Riemannian Manifolds 342 Problems 344 Integration on Manifolds 347 Integration of Differential Forms on Euclidean Space 348 Integration on Manifolds 351 Stokess Theorem 357 Manifolds with Corners 361 Integration on Riemannian Manifolds 368 Integration on Lie Groups 372 Densities 373 Problems 380 De Rham Cohomology 386 The de Rham Cohomology Groups 387 Homotopy Invariance 388 The MayerVietoris Theorem 392 Computations 397 Problems 405 The de Rham Theorem 408 Singular Homology 409 Singular Cohomology 413 Smooth Singular Homology 414 The de Rham Theorem 422 Problems 429 Integral Curves and Flows 432 Integral Curves 433 Global Flows 436 The Fundamental Theorem on Flows 438 Complete Vector Fields 444 Regular Points and Singular Points 445 TimeDependent Vector Fields 449 Proof of the ODE Theorem 450 Problems 458 Lie Derivatives 462 The Lie Derivative 463 Commuting Vector Fields 466 Lie Derivatives of Tensor Fields 471 Applications to Geometry 475 Applications to Symplectic Manifolds 479 Problems 489 Integral Manifolds and Foliations 492 Tangent Distributions 493 Involutivity and Differential Forms 495 The Frobenius Theorem 498 Applications to Partial Differential Equations 503 Foliations 508 Problems 513 Lie Groups and Their Lie Algebras 516 OneParameter Subgroups 517 The Exponential Map 520 The Closed Subgroup Theorem 524 The Adjoint Representation 527 Lie Subalgebras and Lie Subgroups 528 Normal Subgroups 533 The Fundamental Correspondence Between Lie Algebras and Lie Groups 534 Problems 535 Review of Prerequisites 538 Linear Algebra 556 Calculus 579 References 595 Index 599 Copyright

### References from web pages

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Introduction to Smooth Manifolds
Introduction to Smooth Manifolds. by John M. Lee. Description · Table of Contents and sample chapter; Corrections to the book ( Updated! ...
www.math.washington.edu/ ~lee/ Books/ smooth.html

Corrections to Introduction to Smooth Manifolds (version 3.0) (Lee ...
Electronic literature gets more accessible and comfortable than the printed one: Dleex, the first Web 2.0 library
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Introduction to Smooth Manifolds. Home Page. Fall, 2006. Mark Brittenham. link to Lee's textbook at amazon.com. Handouts:. Initial course announcement ...
www.math.unl.edu/ ~mbrittenham2/ classwk/ 856f06/

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Corrections to Introduction to Smooth Manifolds
Introduction to Smooth Manifolds. by John M. Lee. January 14, 2008. Changes or additions made in the past twelve months are dated. ...
www.math.washington.edu/ ~lee/ Books/ Smooth/ errata.pdf

Introduction To Smooth Manifolds

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Newtonian limit of GR Text - Physics Forums Library
See for example John M. Lee, Introduction to Smooth Manifolds, Springer, for some good discussion of levels of structure in the theory of smooth manifolds. ...
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