Geometries and Groups

Title page......Page 1
Preface......Page 3
1. Formulating the problem......Page 7
2. Spherical geometry......Page 10
3.1. First acquaintance......Page 17
3.2. How to measure distances......Page 21
3.3. The study of geometry on a cylinder......Page 26
4. A world in which right and left are indistinguishable......Page 30
5.1. Description of the geometry......Page 36
5.2. Lines on the torus......Page 42
5.3. Some applications......Page 47
6.1. The definition of a geometry......Page 51
6.2. Superposing geometries......Page 56
7.1. Definition of equivalence by means of motions......Page 59
7.2. The geometry corresponding to a uniformly discontinuous group......Page 67
8.1. Motions of the plane......Page 72
8.2. Classification: generalities and groups of Type I and II......Page 78
8.3. Classification: groups of Type III......Page 82
9. A new geometry......Page 94
10. Classification of all 2-dimensional locally Euclidean geometries......Page 103
10.1. Constructions in an arbitrary geometry......Page 104
10.2. Coverings......Page 108
10.3. Construction of the covering......Page 113
10.4. Construction of the group......Page 119
10.5. Conclusion of the proof of Theorem 1......Page 123
11.1. Motions of 3-space......Page 127
11.2. Uniformly discontinuous groups in 3-space: generalities......Page 131
11.3. Uniformly discontinuous groups in 3-space: classification......Page 136
11.4. Orientability of the geometries......Page 145
12.1. Symmetry groups......Page 155
12.2. Crystals and crystallographic groups......Page 159
12.3. Crystallographic groups and geometries: discrete groups......Page 166
12.4. A typical example: the geometry of the rectangle......Page 172
12.5. Classification of all locally C_n or D_n geometries......Page 176
12.6. On the proof of Theorems 1 and 2......Page 190
12.7. Crystals and their molecules......Page 191
13.1. When are two geometries defined by uniformly discontinuous groups the same?......Page 193
13.2. Similarity of geometries......Page 197
14.1. Geometries on the torus and the modular figure......Page 202
14.2. When do two pairs of vectors generate the same lattice?......Page 208
14.3. Application to number theory......Page 212
15.1. The geometrical definition of complex numbers......Page 216
15.2. Similarity of lattices and the modular group......Page 221
16.1. \'Motions\'......Page 226
16.2. \'Lines\'......Page 229
16.3. Distance......Page 231
16.4. Construction of the geometry concluded......Page 238
17.1. Discreteness of the modular group......Page 244
17.2. The set of an geometries on the torus......Page 246
Historical remarks......Page 251
List of notation......Page 253
Index......Page 255