Introduction to Smooth Manifolds

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 631 pages
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Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma trices, as easily as we think about the familiar 2-dimensional sphere in ]R3.
 

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Contents

Smooth Manifolds
1
Smooth Maps
30
Tangent Vectors
60
Vector Fields
80
The Lie Algebra of a Lie Group
93
Local and Global Sections of Vector Bundles
109
The Cotangent Bundle
124
Line Integrals
138
Tensors
260
Differential Forms
291
Orientations
324
Integration on Manifolds
349
De Rham Cohomology
388
The de Rham Theorem
410
Integral Curves and Flows
434
Lie Derivatives
464

Problems
151
ConstantRank Maps Between Manifolds
166
Level Sets
180
Lie Subgroups
194
Group Actions
207
Embedding and Approximation Theorems
241
Integral Manifolds and Foliations
494
Lie Groups and Their Lie Algebras
518
Review of Prerequisites
540
References
597
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