This book is based on real inner product spaces X of arbitrary (finite or infinite) dimension greater than or equal to 2. With natural properties of (general) translations and general distances of X, euclidean and hyperbolic geometries are characterized. For these spaces X also the sphere geometries of M?bius and Lie are studied (besides euclidean and hyperbolic geometry), as well as geometries where Lorentz transformations play the key role. The geometrical notions of this book are based on general spaces X as described. This implies that also mathematicians who have not so far been especially interested in geometry may study and understand great ideas of classical geometries in modern and general contexts. Proofs of newer theorems, characterizing isometries and Lorentz transformations under mild hypotheses are included, like for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. Only prerequisites are basic linear algebra and basic 2- and 3-dimensional real geometry.
Author(s): Walter Benz
Edition: 1
Publisher: Birkhäuser Basel
Year: 2005
Language: English
Pages: 251
Contents......Page 5
Preface......Page 8
1.1 Real inner product spaces......Page 12
1.2 Examples......Page 13
1.3 Isomorphic, non-isomorphic spaces......Page 14
1.4 Inequality of Cauchy–Schwarz......Page 15
1.5 Orthogonal mappings......Page 16
1.6 A characterization of orthogonal mappings......Page 18
1.7 Translation groups, axis, kernel......Page 21
1.8 Separable translation groups......Page 25
1.9 Geometry of a group of permutations......Page 27
1.10 Euclidean, hyperbolic geometry......Page 31
1.11 A common characterization......Page 32
1.12 Other directions, a counterexample......Page 45
2.1 Metric spaces......Page 48
2.2 The lines of L.M. Blumenthal......Page 49
2.3 The lines of Karl Menger......Page 54
2.4 Another definition of lines......Page 56
2.5 Balls, hyperplanes, subspaces......Page 57
2.6 A special quasi-hyperplane......Page 61
2.7 Orthogonality, equidistant surfaces......Page 62
2.8 A parametric representation......Page 65
2.9 Ends, parallelity, measures of angles......Page 67
2.10 Angles of parallelism, horocycles......Page 72
2.11 Geometrical subspaces......Page 74
2.12 The Cayley–Klein model......Page 77
2.13 Hyperplanes under translations......Page 81
2.14 Lines under translations......Page 83
2.15 Hyperbolic coordinates......Page 85
2.16 All isometries of (X, eucl), (X, hyp)......Page 86
2.17 Isometries preserving a direction......Page 88
2.18 A characterization of translations......Page 89
2.19 Different representations of isometries......Page 90
2.20 A characterization of isometries......Page 91
2.21 A counterexample......Page 96
2.22 An extension problem......Page 97
2.23 A mapping which cannot be extended......Page 102
3.1 Möbius balls, inversions......Page 104
3.2 An application to integral equations......Page 107
3.3 A fundamental theorem......Page 109
3.4 Involutions......Page 113
3.5 Orthogonality......Page 118
3.6 Möbius circles, M[sub(N)]- and M[sup(N)]-spheres......Page 122
3.7 Stereographic projection......Page 131
3.8 Poincaré’s model of hyperbolic geometry......Page 134
3.9 Spears, Laguerre cycles, contact......Page 144
3.10 Separation, cyclographic projection......Page 150
3.11 Pencils and bundles......Page 155
3.12 Lie cycles, groups Lie (X), Lag (X)......Page 161
3.13 Lie cycle coordinates, Lie quadric......Page 165
3.14 Lorentz boosts......Page 170
3.15 M(X) as part of Lie (X)......Page 178
3.16 A characterization of Lag (X)......Page 181
3.17 Characterization of the Lorentz group......Page 183
3.18 Another fundamental theorem......Page 184
4.1 Two characterization theorems......Page 186
4.2 Causal automorphisms......Page 188
4.3 Relativistic addition......Page 192
4.4 Lightlike, timelike, spacelike lines......Page 195
4.5 Light cones, lightlike hyperplanes......Page 197
4.6 Characterization of some hyperplanes......Page 202
4.7 L(Z) as subgroup of Lie (X)......Page 204
4.8 A characterization of LM-distances......Page 205
4.9 Einstein’s cylindrical world......Page 208
4.10 Lines, null-lines, subspaces......Page 211
4.11 2-point invariants of (C(Z), MC(Z))......Page 213
4.13 2-point invariants of (Σ(Z), MΣ(Z))......Page 216
4.14 Elliptic and spherical distances......Page 219
4.15 Points......Page 221
4.16 Isometries......Page 223
4.17 Distance functions of X[sub(0)]......Page 226
4.18 Subspaces, balls......Page 228
4.19 Periodic lines......Page 229
4.20 Hyperbolic geometry revisited......Page 233
A Notations and symbols......Page 241
B Bibliography......Page 243
E......Page 248
M......Page 249
V......Page 250
Z......Page 251
