Geometries on Surfaces (Encyclopedia of Mathematics and its Applications, Series Number 84)

One century after Hilbert constructed the first example of a non-classical affine plane, this book aims to summarize all the major results about geometries on surfaces. Acting both as a reference and a monograph, the authors have included detailed sections on what is known as well as outlining problems that remain to be solved. There are sections on classical geometries, methods for constructing non-classical geometries and classifications and characterizations of geometries. This work is related to a host of other fields including approximation, convexity, differential geometry topology and many more. This book will appeal to students, researchers and lecturers working in geometry or any one of the many associated areas outlined above. "The projective, Mobius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces that satisfy an axiom of joining. This book summarises all known major results and open problems related to these classical geometries and their close (non-classical) relatives." "Topics covered include: classical geometries; methods for constructing non-classical geometries; classifications and characterisations of geometries. This work is related to a host of other fields including interpolation theory, convexity, differential geometry, 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 referee for experts and as an accessible introduction for beginners, this book will interest anyone wishing to know more about incidence geometries and the way they interact."--Jacket The projective, Möbius, 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.

Both a reference and an introduction on the main results about topological geometries on surfaces.

Booknews

The authors' backgrounds are in incidence geometry, Polster at Monash University, and Steinke at University of Canterbury. Writing for beginners as well as those seasoned in the field, they cover classical geometries (e.g. the projective, Möbius, Lagurerre, and Minkowski planes over the real numbers); methods for constructing nonclassical geometries; and classifications and characterizations of geometries. They include all known major results and open problems, and they also cover the links between this material and fields such as interpolation theory, convexity, differential geometry, topology, the theory of Lie groups, among others. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Contents......Page 10 Preface......Page 18 1 Geometries for Pedestrians......Page 24 2 Flat Linear Spaces......Page 46 3 Spherical Circle Planes......Page 160 4 Toroidal Circle Planes......Page 235 5 Cylindrical Circle Planes......Page 312 6 Generalized Quadrangles......Page 383 7 Tubular Circle Planes......Page 418 Appendix 1 Tools and Techniques from Topology and Analysis......Page 452 Appendix 2 Lie Transformation Groups......Page 467 Bibliography......Page 481 Index......Page 506 In this book of geometry will usually consists of a nonempty point set and nonempty line set, where a line is a subset of the point set containing at least three points and every point is contained in at least three lines.