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Tables of Contents for Foundations of Differential Geometry
Chapter/Section Title
Page #
Page Count
Interdependence of the Chapters and the Sections
xi
Differentiable Manifolds
Differentiable manifolds
1
16
Tensor algebras
17
9
Tensor fields
26
12
Lie groups
38
12
Fibre bundles
50
13
Theory of Connections
Connections in a principal fibre bundle
63
4
Existence and extension of connections
67
1
Parallelism
68
3
Holonomy groups
71
4
Curvature form and structure equation
75
4
Mappings of connections
79
4
Reduction theorem
83
6
Holonomy theorem
89
3
Flat connections
92
2
Local and infinitesimal holonomy groups
94
9
Invariant connections
103
10
Linear and Affine Connections
Connections in a vector bundle
113
5
Linear connections
118
7
Affine connections
125
5
Developments
130
2
Curvature and torsion tensors
132
6
Geodesics
138
2
Expressions in local coordinate systems
140
6
Normal coordinates
146
5
Linear infinitesimal holonomy groups
151
3
Riemannian Connections
Riemannian metrics
154
4
Riemannian connections
158
4
Normal coordinates and convex neighborhoods
162
10
Completeness
172
7
Holonomy groups
179
8
The decomposition theorem of de Rham
187
6
Affine holonomy groups
193
5
Curvature and Space Forms
Algebraic preliminaries
198
3
Sectional curvature
201
3
Spaces of constant curvature
204
5
Flat affine and Riemannian connections
209
16
Transformations
Affine mappings and affine transformations
225
4
Infinitesimal affine transformations
229
7
Isometries and infinitesimal isometries
236
8
Holonomy and infinitesimal isometries
244
4
Ricci tensor and infinitesimal isometries
248
4
Extension of local isomorphisms
252
4
Equivalence problem
256
11
Appendices
1. Ordinary linear differential equations
267
2
2. A connected, locally compact metric space is separable
269
3
3. Partition of unity
272
3
4. On an arcwise connected subgroup of a Lie group
275
2
5. Irreducible subgroups of O(n)
277
4
6. Green's theorem
281
3
7. Factorization lemma
284
3
Notes
1. Connections and holonomy groups
287
4
2. Complete affine and Riemannian connections
291
1
3. Ricci tensor and scalar curvature
292
2
4. Spaces of constant positive curvature
294
3
5. Flat Riemannian manifolds
297
3
6. Parallel displacement of curvature
300
1
7. Symmetric spaces
300
4
8. Linear connections with recurrent curvature
304
2
9. The automorphism group of a geometric structure
306
2
10. Groups of isometries and affine transformations with maximum dimensions
308
1
11. Conformal transformations of a Riemannian manifold
309
4
Summary of Basic Notations
313
2
Bibliography
315
10
Index
325
5
Errata for Foundations of Differential Geometry, Volume I
330
1
Errata for Foundations of Differential Geometry, Volume II
331