Classical Geometries in Modern Contexts : Geometry of Real Inner Product Spaces

by Benz, Walter

• ISBN13: 9783764385408
• ISBN10: 3764385405
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-03-05
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Preface	p. ix
Preface to the Second Edition	p. xiii
Translation Groups	p. 1
Real inner product spaces	p. 1
Examples	p. 2
Isomorphic, non-isomorphic spaces	p. 3
Inequality of Cauchy-Schwarz	p. 4
Orthogonal mappings	p. 5
A characterization of orthogonal mappings	p. 7
Translation groups, axis, kernel	p. 10
Separable translation groups	p. 14
Geometry of a group of permutations	p. 16
Euclidean, hyperbolic geometry	p. 20
A common characterization	p. 21
Other directions, a counterexample	p. 34
Euclidean and Hyperbolic Geometry	p. 37
Metric spaces	p. 37
The lines of L.M. Blumenthal	p. 38
The lines of Karl Menger	p. 43
Another definition of lines	p. 45
Balls, hyperplanes, subspaces	p. 46
A special quasi-hyperplane	p. 50
Orthogonality, equidistant surfaces	p. 51
A parametric representation	p. 54
Ends, parallelity, measures of angles	p. 56
Angles of parallelism, horocycles	p. 61
Geometrical subspaces	p. 63
The Cayley-Klein model	p. 66
Hyperplanes under translations	p. 70
Lines under translations	p. 72
Hyperbolic coordinates	p. 74
All isometries of (X, eucl), (X, hyp)	p. 75
Isometries preserving a direction	p. 77
A characterization of translations	p. 78
Different representations of isometries	p. 79
A characterization of isometries	p. 80
A counterexample	p. 85
An extension problem	p. 86
A mapping which cannot be extended	p. 91
Sphere Geometries of Möbius and Lie	p. 93
Möbius balls, inversions	p. 93
An application to integral equations	p. 96
A fundamental theorem	p. 98
Involutions	p. 102
Orthogonality	p. 107
Möbius circles, MN- and MN-spheres	p. 111
Stereographic projection	p. 120
Poincaré's model of hyperbolic geometry	p. 123
Spears, Laguerre cycles, contact	p. 133
Separation, cyclographic projection	p. 139
Pencils and bundles	p. 144
Lie cycles, groups Lie (X), Lag (X)	p. 150
Lie cycle coordinates, Lie quadric	p. 154
Lorentz boosts	p. 159
<$>{\op M}<$>(X) as part of Lie (X)	p. 167
A characterization of Lag (X)	p. 170
Characterization of the Lorentz group	p. 172
Another fundamental theorem	p. 173
Lorentz Transformations	p. 175
Two characterization theorems	p. 175
Causal automorphisms	p. 177
Relativistic addition	p. 181
Lightlike, timelike, spacelike lines	p. 184
Light cones, lightlike hyperplanes	p. 186
Characterization of some hyperplanes	p. 191
<$>{\op L}<$> (Z) as subgroup of Lie (X)	p. 193
A characterization of LM-distances	p. 194
Einstein's cylindrical world	p. 197
Lines, null-lines, subspaces	p. 200
2-point invariants of (C(Z), MC (Z))	p. 202
De Sitter's world	p. 205
2-point invariants of (¿(Z), M¿(Z))	p. 205
Elliptic and spherical distances	p. 208
Points	p. 210
Isometries	p. 212
Distance functions of X0	p. 215
Subspaces, balls	p. 217
Periodic lines	p. 218
Hyperbolic geometry revisited	p. 222
¿-Projective Mappings, Isomorphism Theorems	p. 231
¿-linearity	p. 231
All ¿-affine mappings of (X, ¿)	p. 233
¿-projective hyperplanes	p. 235
Extensions of ¿-affine mappings	p. 236
All ¿-projective mappings	p. 239
¿-dualities	p. 240
The ¿-projective Cayley-Klein model	p. 242
M-transformations from X′ onto V′	p. 247
Isomorphic Möbius sphere geometries	p. 249
Isomorphic Euclidean geometries	p. 252
Isomorphic hyperbolic geometries	p. 256
A mixed case	p. 261
Notation and symbols	p. 265
Bibliography	p. 267
Index	p. 273
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