Geometry: from isometries to special relativity

✍ Scribed by Lee N.-H

• Publisher: Springer
• Year: 2020
• Tongue: English
• Leaves: 264
• Series: Undergraduate Texts in Mathematics
• Category: Library

✦ Table of Contents

Preface......Page 7
Contents......Page 10
Dependence Chart......Page 12
1.1 Isometries......Page 13
Exercises......Page 17
1.2 Three Reflections Theorem......Page 18
Exercises......Page 22
1.3 Rotations and Translations......Page 23
Exercises......Page 28
1.4 Glide Reflections and Orientation......Page 29
Exercises......Page 34
2.1 The Sphere S2 in R3......Page 35
2.2 Isometries of the Sphere S2......Page 41
Exercises......Page 47
2.3 Area of a Spherical Triangle......Page 48
2.4 Orthogonal Transformations of Euclidean Spaces......Page 55
3.1 Stereographic Projection......Page 58
3.2 Inversions on the Extended Plane......Page 67
Exercises......Page 78
3.3 Inversions on the Sphere S2......Page 79
Exercises......Page 85
3.4 Representation of the Sphere in the Extended Plane......Page 88
Exercises......Page 96
4.1 Poincaré Upper Half-Plane H2......Page 97
4.2 H2-Shortest Paths and H2-Lines......Page 103
4.3 Isometries of the Hyperbolic Plane......Page 110
Exercises......Page 115
4.4 Hyperbolic Triangle and Hyperbolic Area......Page 116
Exercises......Page 120
4.5 Poincaré Disk......Page 122
4.6 Klein Disk......Page 129
4.7 Euclid's Fifth Postulate: The Parallel Postulate......Page 132
Exercises......Page 137
5 Lorentz–Minkowski Plane......Page 138
5.1 Lorentz–Minkowski Distance......Page 139
Exercises......Page 143
5.2 Relativistic Reflections......Page 144
Exercises......Page 150
5.3 Hyperbolic Angle......Page 151
Exercises......Page 157
5.4 Relativistic Rotations......Page 158
5.5 Matrix and Isometry......Page 163
5.6 Relativistic Lengths of Curves......Page 170
Exercises......Page 174
5.7 Hyperboloid in R2,1......Page 175
5.8 Isometries of R2,1......Page 181
Exercises......Page 191
6.1 R3,1 and the Special Relativity of Einstein......Page 192
Exercises......Page 195
6.2 Causality......Page 196
6.3 Causal Isometry......Page 203
Exercises......Page 211
6.4 Worldline......Page 212
6.5 Kinetics in R3,1......Page 221
Exercises......Page 232
Chapter 1......Page 233
Chapter 2......Page 238
Chapter 3......Page 239
Chapter 4......Page 244
Chapter 5......Page 247
Chapter 6......Page 253
Bibliography......Page 258
Index......Page 260
Symbol Index......Page 263