Differential forms and connections

Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1.1 Exterior Powers of a Vector Space ......Page 12
1.2 Multilinear Alternating Maps and Exterior Products ......Page 16
1.1 Exercises ......Page 18
1.4 Exterior Powers of a Linear Transformation ......Page 19
1.5 Exercises ......Page 23
1.6 Inner Products ......Page 24
1.7 The Hodge Star Operator ......Page 28
1.8 Exercises ......Page 31
1.9 Some Formal Algebraic Constructions ......Page 32
1.10 History and Bibliography ......Page 34
2.1 Tangent Spaces - the Euclidean Case ......Page 35
2.2 Differential Forms on a Euclidean Space ......Page 39
2.3 Operations on Differential Forms ......Page 42
2.4 Exercises ......Page 44
2.5 Exterior Derivative ......Page 46
2.6 Exercises ......Page 43
2.7 The Differential of a Map ......Page 52
2.8 The Pullback of a Differential Form ......Page 54
2.9 Exercises ......Page 58
2.10 History and Bibliography ......Page 60
2.11 Appendix: Maxwell\'s Equations ......Page 61
3.1 Immersions and Submersions ......Page 64
3.2 Definition and Examples of Submanifolds ......Page 66
3.3 Exercises ......Page 71
3.4 Parametrizations ......Page 72
3.5 Using the Implicit Function Theorem to Parametrize a Submanifold ......Page 75
3.6 Matrix Groups as Submanifolds ......Page 80
3.7 Groups of Complex Matrices ......Page 82
3.8 Exercises ......Page 83
3.9 Bibliography ......Page 86
4.1 Moving Orthonormal Frames on Euclidean Space ......Page 87
4.2 The Structure Equations ......Page 89
4 3 Fxercispc ......Page 90
4.4 An Adapted Moving Orthonormal Frame on a Surface......Page 92
4.5 The Area Form ......Page 96
4.6 Exercises ......Page 98
4.7 Curvature of n Surface ......Page 99
4.8 Explicit Calculation of Curvatures ......Page 102
4.9 Exercises ......Page 105
4.10 The Fundamental Forms: Exercises ......Page 106
4.11 History and Bibliography ......Page 108
5.1 Definition of a Differential Manifold ......Page 109
5.2 Basic Topological Vocabulary ......Page 111
5.3 Differentiable Mappings between Manifolds ......Page 113
5.4 Exercises ......Page 115
5.5 Submanifolds ......Page 116
5.6 Embeddings . ......Page 118
5.7 Constructing Submanifolds without Using Charts ......Page 121
5.8 Submanifolds-with-Boundary ......Page 122
5.9 Exercises ......Page 125
5.10 Appendix: Open Sets of a Submanifold ......Page 127
5.11 Appendix: Partitions of Unity ......Page 128
5.12 History and Bibliography ......Page 130
6.1 Local Vector Bundles ......Page 131
6.2 Constructions with Local Vector Bundles ......Page 133
6.3 General Vector Bundles ......Page 136
6.4 Constructing a Vector Bundle from Transition Functions ......Page 141
6.5 Exercises ......Page 143
6.6 The Tangent Bundle of a Manifold ......Page 145
6.7 Exercises ......Page 150
6.9 Appendix: Constructing Vector Bundles ......Page 152
7.1 Frame Fields for Vector Bundles ......Page 155
7.2 Tangent Vectors as Equivalence Classes of Curves ......Page 158
7.3 Exterior Calculus on Manifolds ......Page 159
7.4 Exercises ......Page 162
7.5 Indefinite Riemannian Metrics ......Page 163
7.6 Examples of Riemannian Manifolds ......Page 164
7.7 Orthonormal Frame Fields ......Page 167
7.8 An Isomorphism between the Tangent and Cotangent Bundles ......Page 171
7.9 Exercises ......Page 172
7.10 History and Bibliography ......Page 174
8.1 Volume Forms and Orientation ......Page 175
8.2 Criterion for Orientability in Terms of an Atlas ......Page 178
8.3 Orientation of Boundaries ......Page 180
8.4 Exercises ......Page 183
8.5 Integration of an n-Form over a Single Chart ......Page 185
8.6 Global Integration of n-Forms ......Page 189
8.7 The Canonical Volume Form for a Metric ......Page 192
8.8 Stokes\'s Theorem ......Page 194
8.9 The Exterior Derivative Stands Revealed ......Page 195
8.10 Exercises ......Page 198
8.12 Appendix: Proof of Stokes\'s Theorem ......Page 200
9.1 Koszul Connections ......Page 205
9.2 Connections via Vector-Bundle-valued Forms ......Page 208
9.3 Curvature of a Connection ......Page 213
9.4 Exercises ......Page 217
9.5 Torsion-free Connections ......Page 223
9.6 Metric Connections ......Page 227
9.7 Exercises ......Page 230
9.8 History and Bibliography ......Page 233
10.1 The Role of Connections in Field Theory ......Page 234
10.2 Geometric Formulation of Gauge Field Theory ......Page 236
10.3 Special Unitary Groups and Quaternions ......Page 242
10.4 Quaternion Line Bundles ......Page 244
10.5 Exercises ......Page 249
10.6 The Yang-Mills Equations ......Page 253
10.7 Self-duality ......Page 255
10.8 Instantons ......Page 258
10.9 Exercises ......Page 260
10.10 History and Bibliography ......Page 261
Bibliography ......Page 262
Index ......Page 264
Back Cover......Page 268