Differential Geometry: Bundles, Connections, Metrics and Curvature

Cover......Page 1
Contents......Page 8
1.1 Smooth manifolds......Page 16
1.2 The inverse function theorem and implicit function theorem......Page 18
1.3 Submanifolds of R[sup(m)]......Page 19
1.4 Submanifolds of manifolds......Page 22
1.5 More constructions of manifolds......Page 23
1.6 More smooth manifolds: The Grassmannians......Page 24
Appendix 1.1 How to prove the inverse function and implicit function theorems......Page 26
Additional reading......Page 28
2.1 The general linear group......Page 29
2.2 Lie groups......Page 30
2.3 Examples of Lie groups......Page 31
2.4 Some complex Lie groups......Page 32
2.5 The groups Sl(n; C), U(n) and SU(n)......Page 34
2.6 Notation with regards to matrices and differentials......Page 36
Appendix 2.1 The transition functions for the Grassmannians......Page 37
Additional reading......Page 39
3.1 The definition......Page 40
3.2 The standard definition......Page 42
3.3 The first examples of vector bundles......Page 43
3.4 The tangent bundle......Page 44
3.5 Tangent bundle examples......Page 46
3.6 The cotangent bundle......Page 48
3.7 Bundle homomorphisms......Page 49
3.8 Sections of vector bundles......Page 50
3.9 Sections of TM and T*M......Page 51
Additional reading......Page 53
4.1 Subbundles......Page 54
4.2 Quotient bundles......Page 55
4.3 The dual bundle......Page 56
4.4 Bundles of homomorphisms......Page 57
4.6 The direct sum......Page 58
4.7 Tensor powers......Page 59
Additional reading......Page 61
5.1 The pull-back construction......Page 63
5.2 Pull-backs and Grassmannians......Page 64
5.3 Pull-back of differential forms and push-forward of vector fields......Page 65
5.4 Invariant forms and vector fields on Lie groups......Page 67
5.5 The exponential map on a matrix group......Page 68
5.6 The exponential map and right/left invariance on Gl(n; C) and its subgroups......Page 70
5.7 Immersion, submersion and transversality......Page 72
Additional reading......Page 73
6.1 Definitions......Page 74
6.2 Comparing definitions......Page 75
6.3 Examples: The complexification......Page 77
6.4 Complex bundles over surfaces in R[sup(3)]......Page 78
6.6 Bundles over 4-dimensional submanifolds in R[sup(5)]......Page 79
6.8 Complex Grassmannians......Page 80
6.9 The exterior product construction......Page 83
6.10 Algebraic operations......Page 84
6.11 Pull-back......Page 85
Additional reading......Page 86
7 Metrics on vector bundles......Page 87
7.1 Metrics and transition functions for real vector bundles......Page 88
7.3 Metrics, algebra and maps......Page 90
Additional reading......Page 92
8.1 Riemannian metrics and distance......Page 93
8.2 Length minimizing curves......Page 94
8.3 The existence of geodesics......Page 96
8.4 First examples......Page 97
8.5 Geodesics on SO(n)......Page 100
8.6 Geodesics on U(n) and SU(n)......Page 104
8.7 Geodesics and matrix groups......Page 107
Appendix 8.1 The proof of the vector field theorem......Page 108
Additional reading......Page 109
9.2 The exponential map......Page 111
9.3 Gaussian coordinates......Page 113
9.4 The proof of the geodesic theorem......Page 115
Additional reading......Page 118
10.1 The definition......Page 119
10.2 A cocycle definition......Page 120
10.3 Principal bundles constructed from vector bundles......Page 121
10.4 Quotients of Lie groups by subgroups......Page 123
10.5 Examples of Lie group quotients......Page 125
10.6 Cocycle construction examples......Page 128
10.7 Pull-backs of principal bundles......Page 131
10.8 Reducible principal bundles......Page 133
10.9 Associated vector bundles......Page 134
Appendix 10.1 Proof of Proposition 10.1......Page 136
Additional reading......Page 139
11.1 Covariant derivatives......Page 140
11.2 The space of covariant derivatives......Page 141
11.3 Another construction of covariant derivatives......Page 142
11.4 Principal bundles and connections......Page 143
11.5 Connections and covariant derivatives......Page 149
11.6 Horizontal lifts......Page 150
11.7 An application to the classification of principal G-bundles up to isomorphism......Page 151
11.8 Connections, covariant derivatives and pull-back bundles......Page 152
Additional reading......Page 153
12.1 Exterior derivative......Page 154
12.2 Closed forms, exact forms, diffeomorphisms and De Rham cohomology......Page 156
12.3 Lie derivative......Page 158
12.4 Curvature and covariant derivatives......Page 159
12.5 An example......Page 161
12.7 Connections and curvature......Page 163
12.8 The horizontal subbundle revisited......Page 165
Additional reading......Page 166
13.1 Flat connections......Page 167
13.2 Flat connections on bundles over the circle......Page 168
13.3 Foliations......Page 170
13.4 Automorphisms of a principal bundle......Page 171
13.5 The fundamental group......Page 172
13.7 The universal covering space......Page 174
13.8 Holonomy and curvature......Page 175
13.9 Proof of the classification theorem for flat connections......Page 177
Appendix 13.1 Smoothing maps......Page 179
Appendix 13.2 The proof of the Frobenius theorem......Page 181
Additional reading......Page 184
14.1 The Bianchi Identity......Page 185
14.2 Characteristic forms......Page 186
14.3 Characteristic classes: Part 1......Page 189
14.4 Characteristic classes: Part 2......Page 190
14.5 Characteristic classes for complex vector bundles and the Chern classes......Page 192
14.6 Characteristic classes for real vector bundles and the Pontryagin classes......Page 194
14.7 Examples of bundles with nonzero Chern classes......Page 195
14.8 The degree of the map g → g[sup(m)] from SU(2) to itself......Page 204
Appendix 14.1 The ad-invariant functions on M(n; C)......Page 205
Appendix 14.2 Integration on manifolds......Page 207
Appendix 14.3 The degree of a map......Page 212
Additional reading......Page 219
15.1 Metric compatible covariant derivatives......Page 220
15.2 Torsion free covariant derivatives on T*M......Page 223
15.3 The Levi-Civita connection/covariant derivative......Page 225
15.4 A formula for the Levi-Civita connection......Page 226
15.5 Covariantly constant sections......Page 227
15.6 An example of the Levi-Civita connection......Page 229
15.7 The curvature of the Levi-Civita connection......Page 231
Additional reading......Page 233
16.1 Spherical metrics, flat metrics and hyperbolic metrics......Page 235
16.2 The Schwarzchild metric......Page 238
16.3 Curvature conditions......Page 239
16.4 Manifolds of dimension 2: The Gauss–Bonnet formula......Page 242
16.5 Metrics on manifolds of dimension 2......Page 244
16.6 Conformal changes......Page 245
16.7 Sectional curvatures and universal covering spaces......Page 247
16.8 The Jacobi field equation......Page 248
16.9 Constant sectional curvature and the Jacobi field equation......Page 251
16.10 Manifolds of dimension 3......Page 253
16.11 The Riemannian curvature of a compact matrix group......Page 254
Additional reading......Page 259
17 Complex manifolds......Page 260
17.1 Some basics concerning holomorphic functions on C[sup(n)]......Page 261
17.2 The definition of a complex manifold......Page 262
17.3 First examples of complex manifolds......Page 263
17.4 The Newlander–Nirenberg theorem......Page 266
17.6 The almost Kähler 2-form......Page 270
17.7 Symplectic forms......Page 271
17.8 Kähler manifolds......Page 272
17.9 Complex manifolds with closed almost Kähler form......Page 273
17.10 Examples of Kähler manifolds......Page 274
Appendix 17.1 Compatible almost complex structures......Page 276
Additional reading......Page 282
18.1 Holomorphic submanifolds of a complex manifold......Page 283
18.2 Holomorphic submanifolds of projective spaces......Page 284
18.3 Proof of Proposition 18.2, about holomorphic submanifolds in CP[sup(n)]......Page 286
18.4 The curvature of a Kähler metric......Page 287
18.5 Curvature with no (0, 2) part......Page 290
18.6 Holomorphic sections......Page 292
18.7 Example on CP[sup(n)]......Page 294
Additional reading......Page 296
19.1 Definition of the Hodge star......Page 297
19.2 Representatives of De Rham cohomology......Page 298
19.3 A fairy tale......Page 299
19.4 The Hodge theorem......Page 300
19.5 Self-duality......Page 301
Additional reading......Page 302
List of lemmas, propositions, corollaries and theorems......Page 304
List of symbols......Page 306
D......Page 310
J......Page 311
R......Page 312
Z......Page 313