Introduction to Smooth Manifolds

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Springer Science & Business Media, 2003 - Mathematics - 628 pages
7 Reviews
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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Review: Introduction to Smooth Manifolds

User Review  - Leonhard Euler - Goodreads

Should be pithier. Read full review

Review: Introduction to Smooth Manifolds

User Review  - Joecolelife - Goodreads

This book is an antidote to the more common style of math text. So many math books feel like they were written by mathematicians, which is to say their authors prize being terse over being ... 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

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Riemannian Geometry
Peter Petersen
Limited preview - 2006
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