Geometries on Surfaces

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Hardcover | 3 October 2001

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514 Pages

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16.4 x 24.2 x 3.5

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The projective, Moebius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical) relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will interest anyone wishing to know more about point-line geometries and the way they interact.

Industry Reviews

'The main objective of the book, to give an intuitive and fairly complete picture of the wealth of geometries living on surfaces and of the beauty of the subject, has been accomplished in an excellent way. The text provides an easily accessible and well-motivated introduction to topological geometry.' Zentralblatt fur Mathematik und ihre Grenzgebiete Mathematics Abstracts

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Preface	p. xvii
Geometries for Pedestrians	p. 1
Geometries of Points and Lines	p. 1
Geometries on Surfaces	p. 9
Definitions of Frequently Used Terms	p. 20
Flat Linear Spaces	p. 23
Models of the Classical Flat Projective Plane	p. 24
Convexity Theory	p. 28
Continuity of Geometric Operations and the Line Space	p. 36
Isomorphisms, Automorphism Groups, and Polarities	p. 44
Topological Planes and Flat Linear Spaces	p. 53
Classification with Respect to the Group Dimension	p. 56
Constructions	p. 63
Planes with Special Properties	p. 103
Other Invariants and Characterizations	p. 109
Related Geometries	p. 116
Open Problems	p. 134
Spherical Circle Planes	p. 137
Models of the Classical Flat Mobius Plane	p. 137
Derived Planes and Topological Properties	p. 144
Constructions	p. 154
Groups of Automorphisms and Groups of Projectivities	p. 169
The Hering Types	p. 185
Characterizations of the Classical Plane	p. 195
Planes with Special Properties	p. 200
Subgeometries and Lie Geometries	p. 202
Open Problems	p. 208
Toroidal Circle Planes	p. 212
Models of the Classical Flat Minkowski Plane	p. 213
Derived Planes and Topological Properties	p. 228
Constructions	p. 237
Automorphism Groups and Groups of Projectivities	p. 255
The Klein-Kroll Types	p. 268
Characterizations of the Classical Plane	p. 275
Planes with Special Properties	p. 281
Subgeometries and Lie Geometries	p. 283
Open Problems	p. 287
Cylindrical Circle Planes	p. 289
Models of the Classical Flat Laguerre Plane	p. 290
Derived Planes and Topological Properties	p. 299
Constructions	p. 304
Automorphism Groups and Groups of Projectivities	p. 329
The Kleinewillinghofer Types	p. 342
Characterizations of the Classical Plane	p. 351
Planes with Special Properties	p. 352
Subgeometries and Lie Geometries	p. 353
Open Problems	p. 358
Generalized Quadrangles	p. 360
The Classical Antiregular 3-Dimensional Quadrangle	p. 361
Basic Properties	p. 364
From Circle Planes to Generalized Quadrangles and Back	p. 365
Flat Minkowski Planes	p. 369
Sisters of Circle Planes	p. 371
Flat Biaffine Planes and Flat Homology Semibiplanes	p. 375
Different Ways to Cut and Paste	p. 387
The Apollonius Problem	p. 388
Tubular Circle Planes	p. 395
Unisolvent Sets of Functions	p. 396
Nested (Ph)unisolvent Sets and Their Circle Planes	p. 406
Convexity and Cut-and-Paste Constructions	p. 420
Open Problems	p. 427
Tools and Techniques from Topology and Analysis	p. 429
Lie Transformation Groups	p. 444
Bibliography	p. 458
Index	p. 483
Table of Contents provided by Syndetics. All Rights Reserved.

Geometries on Surfaces

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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