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.
Publisher: Cambridge University Press
Number of pages: 514
Weight: 890 g
Dimensions: 234 x 156 x 29 mm
'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
"This book mainly concerns geometries with point sets that form some sort of surface (e.g. a plane, a sphere, a torus, a cylinder) and line sets of a general nature that validate certain standard sets of incidence axioms.... Altogether well designed, quite suitable for browsing, and accessible to undergraduates, this book will form a vital supplement to courses in the foundations of geometry. General readers; lower-division undergraduates through faculty." Choice
"...a carefully written introduction to the subject of topological geometries on surfaces, with many illustrations, motivations, and examples." Mathematical Reviews