โ Scribed by Kobayashi S., Nomizu K.
No coin nor oath required. For personal study only.
One of two volumes which lay the foundations for understanding differential geometry. This work familiarizes readers with various techniques of computation.
Contents ......Page 6
Interdependence of the Chapters and the Sections ......Page 10
Differentiable manifolds ......Page 12
Tensor algebras ......Page 28
Tensor fields ......Page 37
Lie groups ......Page 49
Fibre bundles ......Page 61
Connections in a principal fibre bundle ......Page 74
Existence and extension of connections ......Page 78
Parallelism ......Page 79
Holonomy groups ......Page 82
Curvature form and structure equation ......Page 86
Mappings of connections ......Page 90
Reduction theorem ......Page 94
Holonomy theorem ......Page 100
Flat connections ......Page 103
Local and infinitesimal holonomy groups ......Page 105
Invariant connections ......Page 114
Connections in a vector bundle ......Page 124
Linear connections ......Page 129
Affine connections ......Page 136
Developments ......Page 141
Curvature and torsion tensors ......Page 143
Geodesics ......Page 149
Expressions in local coordinate systems ......Page 151
Normal coordinates ......Page 157
Linear infinitesimal holonomy groups ......Page 162
Riemannian metrics ......Page 165
Riemannian connections ......Page 169
Normal coordinates and convex neighborhoods ......Page 173
Completeness ......Page 183
Holonomy groups ......Page 190
The decomposition theorem of de Rham ......Page 198
Affine holonomy groups ......Page 204
Algebraic preliminaries ......Page 209
Sectional curvature ......Page 212
Spaces of constant curvature ......Page 215
Flat affine and Riemannian connections ......Page 220
Affine mappings and affine transformations ......Page 236
Infinitesimal affine transformations ......Page 240
Isometries and infinitesmial isometries ......Page 247
Holonomy and infinitesimal isometries ......Page 255
Ricci tensor and infinitesimal isometries ......Page 259
Extension of local isomorphisms ......Page 263
Equivalence problem ......Page 267
Ordinary linear differential equations ......Page 278
A connected, local compact metric space is separable ......Page 280
Partition of unity ......Page 283
On arcwise connected subgroups of a Lie group ......Page 286
Irreducible subgroups of O(n) ......Page 288
Green's theorem ......Page 292
Factorization lemma ......Page 295
Connections and holonomy groups ......Page 298
Complete affine and Riemannian connections ......Page 302
Ricci tensor and scalar curvature ......Page 303
Spaces of constant positive curvature ......Page 305
Flat Riemannian manifolds ......Page 308
Symmetric spaces ......Page 311
Linear connections with recurrent curvature ......Page 315
The automorphism group of a geometric structure ......Page 317
Groups of isometries and affine transformations with maximum dimensions ......Page 319
Conformal transformations of a Riemannian manifold ......Page 320
Summary of Basic Notations ......Page 324
Biblography ......Page 326
Index ......Page 336
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One of two volumes which lay the foundations for understanding differential geometry. This work familiarizes readers with various techniques of computation.