Topology, Geometry, and Gauge Fields: Foundations

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Springer Science & Business Media, Apr 24, 1997 - Mathematics - 396 pages
Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development ofnewcourses is a natural consequence of a high levelof excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface In Egypt, geometry was created to measure the land. Similar motivations, on a somewhat larger scale, led Gauss to the intrinsic differential geometry of surfaces in space. Newton created the calculus to study the motion of physical objects (apples, planets, etc.) and Poincare was similarly impelled toward his deep and far-reaching topological view of dynamical systems.
 

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Contents

Physical and Geometrical Motivation
1
03 The Hopf Bundle
11
04 Connections on Principal Bundles
20
05 NonAbelian Gauge Fields and Topology
23
Topological Spaces
27
12 Quotient Topologies and Projective Spaces
48
13 Products and Local Products
57
14 Compactness Conditions
70
43 Smooth Maps on Manifolds
193
44 Tangent Vectors and Derivatives
198
45 Submanifolds
207
46 Vector Fields and 1Forms
215
47 Matrix Lie Groups
229
48 VectorValued 1Forms
245
49 Orientability
263
410 2Forms and Riemannian Metrics
267

15 Connectivity and Covering Spaces
76
16 Topological Groups and Group Actions
87
Homotopy Groups
101
22 Path Homotopy and the Fundamental Group
102
23 Contractible and Simply Connected Spaces
112
24 The Covering Homotopy Theorem
127
25 Higher Homotopy Groups
142
Principal Bundles
165
32 Transition Functions
167
33 Bundle Maps and Equivalence
169
34 Principal GBundles Over Spheres
177
Differentiable Manifolds and Matrix Lie Groups
185
42 Differentiable Manifolds
189
Gauge Fields and Instantons
289
52 Curvature and Gauge Fields
307
53 The YangMills Functional
314
54 The Hodge Dual for 2Forms in Dimension Four
324
55 The Moduli Space
333
Motivation
345
57 Associated Fiber Bundles
348
58 Matter Fields and Their Covariant Derivatives
353
SU2 and SO3
365
References
377
Symbols
381
Index
387
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