9783540356448

Summary

The first part of the book presents n-dimensional projective geometry over an arbitrary skew field; the real, the complex, and the quaternionic geometries are the central topics, finite geometries playing only a minor part. A detailed proof of the main theorem of projective geometry is followed by discussions concerning the cross ratio, Staudt's main theorem, duality, correlations, quadrics, and null systems. Polarities and null systems are classified for these geometries. Finally, changes of the geometric as well as the algebraic structures resulting from restrictions and extensions of the scalar domain are described, in particular, this includes the Hopf fibrations.The second part deals with the classical linear and projective groups and the associated geometries; it is based on the classification of polarities. The guiding principle for this is provided by F. Klein's Erlangen Program. The theory of vector spaces with scalar product, probably going back to E. Artin, is studied in detail including the theorem of E. Witt. After a general investigation of the projective geometry corresponding to a polarity, the elementary spherical, elliptic, and hyperbolic geometries are presented. For them complete systems of invariants for pairs of subspaces are established: stationary angles and distances. Moreover, based on the study of the associated symmetric linear endomorphisms, the symmetric bilinear forms and quadrics are classified for these geometries. Further topics of the book are Möbius geometry as well as elementary symplectic projective geometry. The last section contains a summary of selected results and problems from the geometry of transformation groups; in particular, the classification results for transitive actions of Lie group on spheres and projective spaces are described. The appendix collects brief accounts of some fundamental notions from algebra and topology with corresponding references to the literature.The book is intended to be a self-contained introduction into projective geometry for students and others interested in this subject.

Projective Geometry	p. 1
Projective Spaces	p. 2
Definitions and First Properties	p. 2
Plane Projective Geometry	p. 11
Homogeneous Coordinates	p. 15
Definition. Simplices	p. 15
Coordinate Transformations. The Projective Linear Group	p. 20
Inhomogeneous Projective Coordinates	p. 21
The Projective Linear Group over a Field	p. 22
Collineations	p. 24
Collinear Maps	p. 25
The Main Theorem of Projective Geometry	p. 32
The Group of Auto-Collineations	p. 35
Cross Ratio and Projective Maps	p. 40
The Group Aut P¹	p. 41
The Cross Ratio	p. 42
Projective Maps	p. 46
Harmonic Position	p. 51
Staudt's Main Theorem	p. 53
Projective Equivalence of Collinear Maps. Involutions	p. 55
Affine Geometry from the Projective Viewpoint	p. 59
Duality	p. 65
Duality in Plane Incidence Geometry	p. 66
Projective and Algebraci Duality	p. 67
Projective Pencil Geometries	p. 71
Dual Maps	p. 73
Correlations	p. 75
Definition. Canonical Correlation	p. 76
Correlative Maps	p. 77
F-Correspondences and ¿-Biforms	p. 80
Symmetric Auto-Correlative Maps	p. 84
Null Systems and Polar Maps	p. 84
Equivalence of Auto-Correlative Maps	p. 86
Classification of Null Systems	p. 87
Linear Line Complexes	p. 89
Polarities and Quadrics	p. 92
¿-Hermitean Biforms	p. 92
Classification of Polar Maps	p. 94
The Real Polar Maps	p. 96
The Complex Polar Maps	p. 96
The Quaternionic Polar Maps	p. 98
Quadrics	p. 101
Polar Maps of Projective Lines	p. 107
Tangents and Tangent Subspaces	p. 109
Dualization: Coquadrics	p. 111
Restrictions and Extensions of Scalars	p. 112
Hopf Fibrations	p. 112
Complex Structures	p. 115
Quaternionic Structures	p. 117
Projective Extensions	p. 119
The Projective K-Geometry of Pⁿ_L	p. 123
K-Classification of Projective L-Subspaces	p. 126
Extensions of Null Systems, Polar Maps, and Quadrics	p. 128
Cayley-Klein Geometries	p. 133
The Classical Groups	p. 133
The Linear and the Projective Groups	p. 134
The Projective Isotropy Group of a Correlation	p. 136
The Symplectic Groups	p. 137
The Orthogonal Groups	p. 139
The Unitary Groups	p. 141
The Quaternionic Skew Hermitean Polarities	p. 143
SL (n, K) is Generated by Transvections	p. 145
Vector Spaces with Scalar Product	p. 148
Vector, Projective, and Affine Geometries	p. 149
Subspaces	p. 150
E. Witt's Theorem	p. 151
Properties of Isotropic Subspaces	p. 151
The Proof of E. Witt's Theorem	p. 154
Transitivity Results	p. 157
Neutral Vector Spaces	p. 159
Tensors and Volume Functions	p. 161
The General Vector Product	p. 163
Adjoint Linear Maps	p. 165
Properties of the Root Subspaces	p. 172
The Projective Geometry of a Polarity	p. 176
The Quadric of a Polarity	p. 176
Efficiency	p. 179
Orthogonal Geometry. Reflections	p. 182
Invariants of Finite Configurations	p. 186
PG_n-Congruence of Finite Point Sequences	p. 186
Orbits of Points. Normalized Representatives	p. 189
Invariants of Points. Pairs	p. 191
Real Orthogonal Geometries	p. 196
Projective Orthogonal Geometries for Arbitrary Fields	p. 198
Plane Cone Sections	p. 203
Spherical and Elliptic Geometry	p. 207
Spherical as a Covering of Elliptic Geometry	p. 207
Distance and Angle	p. 211
The Law of Cosines and the Triangle Inequality	p. 217
Excess, Curvature, and Surface Area	p. 221
Spherical Trigonometry	p. 227
The Metric Geometry of Elliptic Space	p. 232
Angle between Subspaces and Distance of Great Spheres	p. 234
Quadrics	p. 238
Hyperbolic Geometry	p. 243
Models of Hyperbolic Space	p. 245
Distance and Angle	p. 256
Distance and Angles as Cross Rations	p. 260
The Hyperbolic Law of Cosines and Hyperbolic Metric	p. 266
Hyperbolic Trigonometry	p. 272
Hyperspheres, Equidistants, and Horospheres	p. 277
Stationary Angles	p. 283
Quadrics	p. 291
Pictures of Hyperbolic Quadrics	p. 309
Mobius Geometry	p. 314
Spheres in Mobius Space	p. 314
Pairs of Subspheres	p. 319
Cross Ratios and the Riemann Sphere	p. 331
Mobius Invariants and Euclidean Invariants	p. 341
Three-Dimensional Mobius Geometry	p. 344
Orbits, Cyclids of Dupin, and Loxodromes	p. 349
Projective Symplectic Geometry	p. 358
Symplectic Transvections	p. 359
Subspaces	p. 361
Triangles	p. 363
Skew Lines	p. 366
Symmetric Bilinear Forms and Quadrics	p. 374
Transformation Groups: Results and Problems	p. 385
Basic Notions from Algebra and Topology	p. 403
Notations	p. 403
Linear Algebra	p. 404
Transformation Groups	p. 404
Topology	p. 407
References	p. 417
Index	p. 423
Table of Contents provided by Ingram. All Rights Reserved.

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Projective And Cayley-klein Geometries

3540356444

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