Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
,Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
Foundations of Hyperbolic Manifolds
Third Edition
Solution Manual
John G. Ratcliffe
Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
,Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
Contents
1 Euclidean Geometry 1
1.1 Euclid’s Parallel Postulate ......................................................................... 1
1.2 Independence of the Parallel Postulate ..................................................... 1
1.3 Euclidean n-Space ....................................................................................... 3
1.4 Geodesics ...................................................................................................... 9
1.5 Arc Length.................................................................................................. 15
2 Spherical Geometry 22
2.1 Spherical n-Space....................................................................................... 22
2.2 Elliptic n-Space ......................................................................................... 26
2.3 Spherical Arc Length ................................................................................ 31
2.4 Spherical Volume ....................................................................................... 31
2.5 Spherical Trigonometry .............................................................................. 36
3 Hyperbolic Geometry 42
3.1 Lorentzian n-Space .................................................................................... 42
3.2 Hyperbolic n-Space.................................................................................... 48
3.3 Hyperbolic Arc Length ............................................................................. 54
3.4 Hyperbolic Volume .................................................................................... 56
3.5 Hyperbolic Trigonometry ........................................................................... 62
4 Inversive Geometry 71
4.1 Reflections ................................................................................................... 71
4.2 Stereographic Projection............................................................................ 74
4.3 Möbius Transformations............................................................................. 76
4.4 Poincar´e Extension ..................................................................................... 80
4.5 The Conformal Ball Model ....................................................................... 85
4.6 The Upper Half-Space Model .................................................................. 89
4.7 Classification of Transformations .............................................................. 94
5 Isometries of Hyperbolic Space 107
5.1 Topological Groups .................................................................................. 107
5.2 Groups of Isometries................................................................................ 113
5.3 Discrete Groups ........................................................................................ 123
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, Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
5.4 Discrete Euclidean Groups...................................................................... 126
5.5 Elementary Groups .................................................................................. 134
6 Geometry of Discrete Groups 137
6.1 The Projective Disk Model.................................................................... 137
6.2 Convex Sets .............................................................................................. 140
6.3 Convex Polyhedra..................................................................................... 145
6.4 Geometry of Convex Polyhedra .............................................................. 147
6.5 Polytopes ................................................................................................... 149
6.6 Fundamental Domains ............................................................................. 155
6.7 Convex Fundamental Polyhedra ............................................................. 157
6.8 Tessellations .............................................................................................. 162
7 Classical Discrete Groups 166
7.1 Reflection Groups..................................................................................... 166
7.2 Simplex Reflection Groups...................................................................... 169
7.3 Generalized Simplex Reflection Groups ................................................ 175
7.4 The Volume of a Simplex ........................................................................ 178
7.5 Crystallographic Groups ......................................................................... 180
7.6 Torsion-Free Linear Groups .................................................................... 182
8 Geometric Manifolds 185
8.1 Geometric Spaces ..................................................................................... 185
8.2 Clifford-Klein Space-Forms.................................................................... 188
8.3 (X, G)-Manifolds ..................................................................................... 193
8.4 Developing ................................................................................................. 194
8.5 Completeness ............................................................................................ 198
9 Geometric Surfaces 202
9.1 Compact Surfaces .................................................................................... 202
9.2 Gluing Surfaces ........................................................................................ 202
9.3 The Gauss-Bonnet Theorem................................................................... 204
9.4 Moduli Spaces .......................................................................................... 206
9.5 Closed Euclidean Surfaces ...................................................................... 212
9.6 Closed Geodesics ...................................................................................... 215
9.7 Closed Hyperbolic Surfaces .................................................................... 220
9.8 Hyperbolic Surfaces of Finite Area ........................................................ 224
10 Hyperbolic 3-Manifolds 231
10.1 Gluing 3-Manifolds .................................................................................. 231
10.2 Complete Gluing of 3-Manifolds ............................................................ 236
10.3 Finite Volume Hyperbolic 3-Manifolds .................................................. 237
10.4 Hyperbolic Volume .................................................................................. 242
10.5 Hyperbolic Dehn Surgery ........................................................................ 248
11 Hyperbolic n-Manifolds 252
11.1 Gluing n-Manifolds .................................................................................. 252
ii
Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
,Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
Foundations of Hyperbolic Manifolds
Third Edition
Solution Manual
John G. Ratcliffe
Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
,Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
Contents
1 Euclidean Geometry 1
1.1 Euclid’s Parallel Postulate ......................................................................... 1
1.2 Independence of the Parallel Postulate ..................................................... 1
1.3 Euclidean n-Space ....................................................................................... 3
1.4 Geodesics ...................................................................................................... 9
1.5 Arc Length.................................................................................................. 15
2 Spherical Geometry 22
2.1 Spherical n-Space....................................................................................... 22
2.2 Elliptic n-Space ......................................................................................... 26
2.3 Spherical Arc Length ................................................................................ 31
2.4 Spherical Volume ....................................................................................... 31
2.5 Spherical Trigonometry .............................................................................. 36
3 Hyperbolic Geometry 42
3.1 Lorentzian n-Space .................................................................................... 42
3.2 Hyperbolic n-Space.................................................................................... 48
3.3 Hyperbolic Arc Length ............................................................................. 54
3.4 Hyperbolic Volume .................................................................................... 56
3.5 Hyperbolic Trigonometry ........................................................................... 62
4 Inversive Geometry 71
4.1 Reflections ................................................................................................... 71
4.2 Stereographic Projection............................................................................ 74
4.3 Möbius Transformations............................................................................. 76
4.4 Poincar´e Extension ..................................................................................... 80
4.5 The Conformal Ball Model ....................................................................... 85
4.6 The Upper Half-Space Model .................................................................. 89
4.7 Classification of Transformations .............................................................. 94
5 Isometries of Hyperbolic Space 107
5.1 Topological Groups .................................................................................. 107
5.2 Groups of Isometries................................................................................ 113
5.3 Discrete Groups ........................................................................................ 123
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, Solutions Manual for Foundations of Hyperbolic Manifolds 3rd Edition by John G. Ratcliffe
5.4 Discrete Euclidean Groups...................................................................... 126
5.5 Elementary Groups .................................................................................. 134
6 Geometry of Discrete Groups 137
6.1 The Projective Disk Model.................................................................... 137
6.2 Convex Sets .............................................................................................. 140
6.3 Convex Polyhedra..................................................................................... 145
6.4 Geometry of Convex Polyhedra .............................................................. 147
6.5 Polytopes ................................................................................................... 149
6.6 Fundamental Domains ............................................................................. 155
6.7 Convex Fundamental Polyhedra ............................................................. 157
6.8 Tessellations .............................................................................................. 162
7 Classical Discrete Groups 166
7.1 Reflection Groups..................................................................................... 166
7.2 Simplex Reflection Groups...................................................................... 169
7.3 Generalized Simplex Reflection Groups ................................................ 175
7.4 The Volume of a Simplex ........................................................................ 178
7.5 Crystallographic Groups ......................................................................... 180
7.6 Torsion-Free Linear Groups .................................................................... 182
8 Geometric Manifolds 185
8.1 Geometric Spaces ..................................................................................... 185
8.2 Clifford-Klein Space-Forms.................................................................... 188
8.3 (X, G)-Manifolds ..................................................................................... 193
8.4 Developing ................................................................................................. 194
8.5 Completeness ............................................................................................ 198
9 Geometric Surfaces 202
9.1 Compact Surfaces .................................................................................... 202
9.2 Gluing Surfaces ........................................................................................ 202
9.3 The Gauss-Bonnet Theorem................................................................... 204
9.4 Moduli Spaces .......................................................................................... 206
9.5 Closed Euclidean Surfaces ...................................................................... 212
9.6 Closed Geodesics ...................................................................................... 215
9.7 Closed Hyperbolic Surfaces .................................................................... 220
9.8 Hyperbolic Surfaces of Finite Area ........................................................ 224
10 Hyperbolic 3-Manifolds 231
10.1 Gluing 3-Manifolds .................................................................................. 231
10.2 Complete Gluing of 3-Manifolds ............................................................ 236
10.3 Finite Volume Hyperbolic 3-Manifolds .................................................. 237
10.4 Hyperbolic Volume .................................................................................. 242
10.5 Hyperbolic Dehn Surgery ........................................................................ 248
11 Hyperbolic n-Manifolds 252
11.1 Gluing n-Manifolds .................................................................................. 252
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