Spaces of Constant Curvature

Preface......Page 5
Preface to the Sixth Edition......Page 8
Notes to the Reader......Page 9
Contents......Page 11
Chapter 1. Affine Differential Geometry......Page 17
1.1 Differentiable manifolds......Page 18
1.2 Vector fields......Page 19
1.3 Differential forms......Page 21
1.4 Maps......Page 23
1.5 Lie groups......Page 25
1.6 The frame bundle: parallelism and geodesics......Page 30
1.7 Curvature, torsion and the structure equations......Page 39
1.8 Covering spaces......Page 47
1.9 The Cartan-Ambrose-Hicks theorem......Page 58
Chapter 2. Riemannian Curvature......Page 61
2.1 The Levi-Cività connection......Page 62
2.2 Sectional curvature......Page 68
2.3 Isometries and curvature......Page 73
2.4 Models for spaces of constant curvature......Page 78
2.5 The 2-dimensional space forms......Page 90
2.6 Finite rotation groups......Page 99
2.7 Homogeneous space forms......Page 104
2.8 Appendix: The metric space structure of a riemannian manifold......Page 107
Part II. The Euclidean Space Form Problem......Page 113
3.1 Discontinuous groups on euclidean space......Page 114
3.2 The Bieberbach Theorems on crystallographic groups......Page 116
3.3 Application to euclidean space forms......Page 121
3.4 Questions of holonomy......Page 123
3.5 Thee dimensional euclidean space forms......Page 127
3.6 Thee attacks on the classification problem for flat compact manifolds......Page 140
3.7 Flat homogeneous pseudo-riemannian manifolds......Page 147
Part III. The Spherical Space Form Problem......Page 153
4.1 Basic definitions......Page 154
4.2 The Frobenius-Schur relations......Page 155
4.3 Frobenius reciprocity and the group algebra......Page 157
4.4 Divisibility......Page 161
4.5 Tensor products and dual representations......Page 163
4.6 Two lemmas on representations over algebraically non-closed fields......Page 166
4.7 Unitary and orthogonal representations......Page 167
5.1 Vincent's program......Page 170
5.2 Preliminaries on p-groups......Page 172
5.3 Necessary conditions on fixed point free groups......Page 175
5.4 Classification of the simplest type of fixed point free groups......Page 178
5.5 Representations of finite groups in which every Sylow subgroup is cyclic......Page 181
5.6 A partial solution to the spherical space form problem......Page 187
Chapter 6. The Classification of Fixed Point Free Groups......Page 188
6.1 Zassenhaus' work on solvable groups with cyclic odd Sylow subgroups......Page 189
6.2 The binary icosahedral group......Page 197
6.3 Non-solvable fixed point free groups......Page 211
7.1 Representations of binary polyhedral groups......Page 214
7.2 Fixed point free complex representations......Page 219
7.3 The action of automorphisms on representations......Page 227
7.4 The classification of spherical space forms......Page 234
7.5 Spherical space forms of low dimension......Page 240
7.6 Clifford translations......Page 243
Chapter 8. Riemannian Symmetric Spaces......Page 247
8.1 Lie formulation of locally symmetric spaces......Page 248
8.2 Structure of orthogonal involutive Lie algebras......Page 250
8.3 Globally symmetric spaces and orthogonal involutive Lie algebras......Page 256
8.4 Curvature......Page 261
8.5 Cohomology......Page 263
8.6 Cartan subalgebras, rank and maximal tori......Page 268
8.7 Hermitian symmetric spaces......Page 273
8.8 The full group of isometries......Page 279
8.9 Extended Schläfli-Dynkin diagrams......Page 280
8.10 Subgroups of maximal rank......Page 291
8.11 The classification of symmetric spaces......Page 302
8.12 Two point homogeneous spaces......Page 309
8.13 Appendix: Manifolds with irreducible linear isotropy group......Page 316
Chapter 9. Space Forms of Irreducible Symmetric Spaces......Page 319
9.1 Feasibility of space form problems......Page 320
9.2 Grassmann manifolds as symmetric spaces......Page 322
9.3 Grassmann manifolds of even dimension......Page 323
9.4 Grassmann manifolds of odd dimension......Page 330
9.5 Symmetric spaces of positive characteristic......Page 335
9.6 An isolated manifold......Page 341
Chapter 10. Locally Symmetric Spaces of Non-negative Curvature......Page 344
10.1 The structure theorems......Page 345
10.2 Application of the structure theorems......Page 349
Chapter 11. Spaces of Constant Curvature......Page 353
11.1 The classification of finite space forms......Page 354
11.2 The geometry of pseudo-spherical space forms......Page 357
11.3 Homogeneous finite space forms......Page 363
11.4 The lattice space forms......Page 370
11.5 A wild Lorentz signature......Page 382
11.6 The classification for homogeneous manifolds of constant curvature......Page 386
12.1 Reductive Lie groups......Page 390
12.2 Examples of locally isotropic manifolds......Page 396
12.3 Structure of locally isotropic manifolds......Page 401
12.4 A partial classification of complete locally isotropic manifolds......Page 405
Appendix to Chapter 12......Page 412
References......Page 418
Additional References......Page 424
Index......Page 429