Introduction to Smooth Manifolds

Sampul Depan
Springer Science & Business Media, 2003 - 628 halaman
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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LibraryThing Review

Ulasan Pengguna  - cpg - LibraryThing

The printing is not up to the standard of the writing By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written. But my copies arrived from Amazon this week, and ... Baca ulasan lengkap

Halaman terpilih

Isi

Smooth Manifolds
1
Topological Manifolds
3
Topological Properties of Manifolds
8
Smooth Structures
11
Examples of Smooth Manifolds
17
Manifolds with Boundary
24
Problems
28
Smooth Maps
30
Differential Forms
291
The Geometry of Volume Measurement
292
The Algebra of Alternating Tensors
294
The Wedge Product
299
Differential Forms on Manifolds
304
Exterior Derivatives
307
Symplectic Forms
316
Problems
321

Smooth Functions and Smooth Maps
31
Lie Groups
37
Smooth Covering Maps
40
Proper Maps
45
Partitions of Unity
49
Problems
57
Tangent Vectors
60
Tangent Vectors
61
Pushforwards
65
Computations in Coordinates
69
Tangent Vectors to Curves
75
Alternative Definitions of the Tangent Space
77
Problems
78
Vector Fields
80
The Tangent Bundle
81
Vector Fields on Manifolds
82
Lie Brackets
89
The Lie Algebra of a Lie Group
93
Problems
100
Vector Bundles
103
Local and Global Sections of Vector Bundles
109
Bundle Maps
115
Categories and Functors
118
Problems
121
The Cotangent Bundle
124
Covectors
125
Tangent Covectors on Manifolds
127
The Cotangent Bundle
129
The Differential of a Function
132
Pullbacks
136
Line Integrals
138
Conservative Covector Fields
143
Problems
151
Submersions Immersions and Embeddings
155
Maps of Constant Rank
156
The Inverse Function Theorem and Its Friends
159
ConstantRank Maps Between Manifolds
166
Submersions
169
Problems
171
Submanifolds
173
Embedded Submanifolds
174
Level Sets
180
Immersed Submanifolds
186
Restricting Maps to Submanifolds
190
Vector Fields and Covector Fields on Submanifolds
191
Lie Subgroups
194
Vector Subbundles
199
Problems
201
Lie Group Actions
206
Group Actions
207
Equivariant Maps
212
Proper Actions
216
Quotients of Manifolds by Group Actions
218
Covering Manifolds
223
Homogeneous Spaces
228
Applications
231
Problems
236
Embedding and Approximation Theorems
241
Sets of Measure Zero in Manifolds
242
The Whitney Embedding Theorem
246
The Whitney Approximation Theorems
252
Problems
258
Tensors
260
The Algebra of Tensors
261
Tensors and Tensor Fields on Manifolds
268
Symmetric Tensors
271
Riemannian Metrics
273
Problems
285
Orientations
326
Orientations of Vector Spaces
327
Orientations of Manifolds
329
The Orientation Covering
331
Orientations of Hypersurfaces
336
Boundary Orientations
340
The Riemannian Volume Form
344
Hypersurfaces in Riemannian Manifolds
346
Problems
348
Integration on Manifolds
351
Integration of Differential Forms on Euclidean Space
352
Integration on Manifolds
355
Stokess Theorem
361
Manifolds with Corners
365
Integration on Riemannian Manifolds
372
Integration on Lie Groups
376
Densities
377
Problems
384
De Rham Cohomology
390
The de Rham Cohomology Groups
391
Homotopy Invariance
392
The MayerVietoris Theorem
396
Computations
401
Problems
409
The de Rham Theorem
412
Singular Homology
413
Singular Cohomology
417
Smooth Singular Homology
418
The de Rham Theorem
426
Problems
433
Integral Curves and Flows
436
Integral Curves
437
Global Flows
440
The Fundamental Theorem on Flows
442
Complete Vector Fields
448
Regular Points and Singular Points
449
TimeDependent Vector Fields
453
Proof of the ODE Theorem
454
Problems
462
Lie Derivatives
466
The Lie Derivative
467
Commuting Vector Fields
470
Lie Derivatives of Tensor Fields
475
Applications to Geometry
479
Applications to Symplectic Manifolds
483
Problems
493
Integral Manifolds and Foliations
496
Tangent Distributions
497
Involutivity and Differential Forms
499
The Frobenius Theorem
502
Applications to Partial Differential Equations
507
Foliations
512
Problems
517
Lie Groups and Their Lie Algebras
520
OneParameter Subgroups
521
The Exponential Map
524
The Closed Subgroup Theorem
528
The Adjoint Representation
531
Lie Subalgebras and Lie Subgroups
532
Normal Subgroups
537
The Fundamental Correspondence Between Lie Algebras and Lie Groups
538
Problems
539
Review of Prerequisites
542
Linear Algebra
560
Calculus
583
References
599
Index
603
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