The Focus Of This Book And Its Geometric Notions Is On Real Vector Spaces X That Are Finite Or Infinite Inner Product Spaces Of Arbitrary Dimension Greater Than Or Equal To 2. It Characterizes Both Euclidean And Hyperbolic Geometry With Respect To Natural Properties Of (general) Translations And General Distances Of X. Also For These Spaces X, It Studies The Sphere Geometries Of Möbius And Lie As Well As Geometries Where Lorentz Transformations Play The Key Role. Proofs Of Newer Theorems Characterizing Isometries And Lorentz Transformations Under Mild Hypotheses Are Included, Such As 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. New To This Third Edition Is A Chapter Dealing With A Simple And Great Idea Of Leibniz That Allows Us To Characterize, For These Same Spaces X, Hyperplanes Of Euclidean, Hyperbolic Geometry, Or Spherical Geometry, The Geometries Of Lorentz-minkowski And De Sitter, And This Through Finite Or Infinite Dimensions Greater Than 1. Another New And Fundamental Result In This Edition Concerns The Representation Of Hyperbolic Motions, Their Form And Their Transformations. Further We Show That The Geometry (p,g) Of Segments Based On X Is Isomorphic To The Hyperbolic Geometry Over X. Here P Collects All X In X Of Norm Less Than One, G Is Defined To Be The Group Of Bijections Of P Transforming Segments Of P Onto Segments. The Only Prerequisites For Reading This Book Are Basic Linear Algebra And Basic 2- And 3-dimensional Real Geometry. This Implies That Mathematicians Who Have Not So Far Been Especially Interested In Geometry Could Study And Understand Some Of The Great Ideas Of Classical Geometries In Modern And General Contexts. 1. Translation Groups -- 2. Euclidean And Hyperbolic Geometry -- 3. Sphere Geometries Of Möbius And Lie -- 4. Lorentz Transformations -- 5. Projective Mappings, Isomorphism Theorems -- 6. Planes Of Leibniz, Lines Of Weierstrass, V Aria -- A. Notation And Symbols -- B. Bibliography. Walter Benz. Includes Bibliographical References And Index.

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Classical geometries in modern contexts : geometry of real inner product spaces / monograph

Adobe Acrobat 8.13

Benz, Walter

Springer Basel : Imprint: Birkhäuser

Springer Nature Switzerland AG

Birkhauser

Springer Nature, Basel, 2012

Switzerland, Switzerland

3rd ed, New York, ©2012

3rd ed, Dordrecht, 2012

Aug 13, 2012

Aug 14, 2012

Kolxo3 -- 10

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Source title: Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces Third Edition

0 Cover 1
0 front 2
Classical Geometries in Modern Contexts 4
Geometry of Real Inner Product Spaces 4
Contents 6
Preface 10
Preface to the Second Edition 14
Preface to the Third Edition 16
1 20
Chapter 1 Translation Groups 20
1.1 Real inner product spaces 20
1.2 Examples 21
1.3 Isomorphic, non-isomorphic spaces 22
1.4 Inequality of Cauchy–Schwarz 23
1.5 Orthogonal mappings 24
1.6 A characterization of orthogonal mappings 26
1.7 Translation groups, axis, kernel 29
1.8 Separable translation groups 33
1.9 Geometry of a group of permutations 35
1.10 Euclidean, hyperbolic geometry 39
1.11 A common characterization 40
1.12 Other directions, a counterexample 53
2 56
Chapter 2 Euclidean and Hyperbolic Geometry 56
2.1 Metric spaces 56
2.2 The lines of L.M. Blumenthal 57
2.3 The lines of Karl Menger 62
2.4 Another definition of lines 64
2.5 Balls, hyperplanes, subspaces 65
2.6 A special quasi–hyperplane 69
2.7 Orthogonality, equidistant surfaces 70
2.8 A parametric representation 73
2.9 Ends, parallelity, measures of angles 75
2.10 Angles of parallelism, horocycles 80
2.11 Geometrical subspaces 82
2.12 The Cayley–Klein model 85
2.13 Hyperplanes under translations 89
2.14 Lines under translations 91
2.15 Hyperbolic coordinates 93
2.16 All isometries of (X, eucl), (X, hyp) 94
2.17 Isometries preserving a direction 96
2.18 A characterization of translations 97
2.19 Different representations of isometries 98
2.20 A characterization of isometries 99
2.21 A counterexample 104
2.22 An extension problem 105
2.23 A mapping which cannot be extended 110
3 112
Chapter 3 Sphere Geometries of Möbius and Lie 112
3.1 Möbius balls, inversions 112
3.2 An application to integral equations 115
3.3 A fundamental theorem 117
3.4 Involutions 121
3.5 Orthogonality 126
3.6 Möbius circles, MN- and MN-spheres 130
3.7 Stereographic projection 139
3.8 Poincaré’s model of hyperbolic geometry 142
3.9 Spears, Laguerre cycles, contact 152
3.10 Separation, cyclographic projection 158
3.11 Pencils and bundles 163
3.12 Lie cycles, groups Lie (X), Lag (X) 169
3.13 Lie cycle coordinates, Lie quadric 173
3.14 Lorentz boosts 178
3.15 M(X) as part of Lie (X) 186
3.16 A characterization of Lag (X) 189
3.17 Characterization of the Lorentz group 191
3.18 Another fundamental theorem 192
4 194
Chapter 4 Lorentz Transformations 194
4.1 Two characterization theorems 194
4.2 Causal automorphisms 196
4.3 Relativistic addition 200
4.4 Lightlike, timelike, spacelike lines 203
4.5 Light cones, lightlike hyperplanes 205
4.6 Characterization of some classes of hyperplanes 210
4.7 L (Z) as subgroup of Lie (X) 212
4.8 A characterization of LM-distances 213
4.9 Einstein’s cylindrical world 216
4.10 Lines, null–lines, subspaces 219
4.11 2-point invariants of (C (Z), MC(Z)) 221
4.12 De Sitter’s world 224
4.13 2-point invariants of (Σ(Z), MΣ(Z)) 224
4.14 Elliptic and spherical distances 227
4.15 Points 229
4.16 Isometries 231
4.17 Distance functions of X0 234
4.18 Subspaces, balls 236
4.19 Periodic lines 237
4.20 Hyperbolic geometry revisited 241
5 249
Chapter 5 δ–Projective Mappings, Isomorphism Theorems 249
5.1 δ–linearity 249
5.2 All δ–affine mappings of (X, δ) 251
5.3 δ–projective hyperplanes 253
5.4 Extensions of δ–affine mappings 254
5.5 All δ–projective mappings 257
5.6 δ–dualities 258
5.7 The δ–projective Cayley–Klein model 260
5.8 M–transformations from X' onto V' 265
5.9 Isomorphic Möbius sphere geometries 267
5.10 Isomorphic Euclidean geometries 270
5.11 Isomorphic hyperbolic geometries 274
5.12 A mixed case 279
6 283
Chapter 6 Planes of Leibniz, Lines of Weierstrass, Varia 283
6.1 L–hyperplanes of metric spaces 283
6.2 Hyperbolic hyperplanes as L–hyperplanes 285
6.3 Elliptic and spherical hyperplanes 288
6.4 Hyperplanes of Lorentz–Minkowski geometry 290
6.5 De Sitter’s world as substructure of (Z, L(Z)) 291
6.6 Hyperplanes of de Sitter’s world 293
6.7 Hyperbolic lines as defined by K. Weierstrass 297
6.8 Orthogonal lines in metric spaces 300
6.9 Lines orthogonal to hyperplanes 304
6.10 A fundamental representation of motions 304
6.11 Dimension–free hyperbolic geometry under mild hypotheses 307
6.12 All 2–point invariants of (P,G) 311
back 314
Appendix A Notation and symbols 314
Appendix B Bibliography 316
Index 321

The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies the sphere geometries of Möbius and Lie as well as geometries where Lorentz transformations play the key role. Proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses are included, such as 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. New to this third edition is a chapter dealing with a simple and great idea of Leibniz that allows us to characterize, for these same spaces X, hyperplanes of euclidean, hyperbolic geometry, or spherical geometry, the geometries of Lorentz-Minkowski and de Sitter, and this through finite or infinite dimensions greater than 1. Another new and fundamental result in this edition concerns the representation of hyperbolic motions, their form and their transformations. Further we show that the geometry (P, G) of segments based on X is isomorphic to the hyperbolic geometry over X. Here P collects all x in X of norm less than one, G is defined to be the group of bijections of P transforming segments of P onto segments. The only prerequisites for reading this book are basic linear algebra and basic 2- and 3-dimensional real geometry. This implies that mathematicians who have not so far been especially interested in geometry could study and understand some of the great ideas of classical geometries in modern and general contexts

The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies the sphere geometries of Mobius and Lie as well as geometries where Lorentz transformations play the key role. Proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses are included, such as for insta

Front Matter....Pages i-xvii
Translation Groups....Pages 1-36
Euclidean and Hyperbolic Geometry....Pages 37-92
Sphere Geometries of Möbius and Lie....Pages 93-174
Lorentz Transformations....Pages 175-229
δ –Projective Mappings, Isomorphism Theorems....Pages 231-264
Planes of Leibniz, Lines of Weierstrass, Varia....Pages 265-295
Back Matter....Pages 297-309

2012-12-29