Introduction to Smooth ManifoldsThis book is an introductory graduatelevel 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  
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 finitedimensional ﬁrst function f given GL(n global homotopy implies induced injective integral curve integral manifold inverse isomorphism leftinvariant Lemma Lie algebra Lie group Lie group homomorphism Lie subgroup linear map manifold with boundary map F matrix measure zero nmanifold 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