What This Book Is All About

‘Geometries on surfaces’—what do you think of when you read such a title? Whatever it is will depend to a large extent on your background in mathematics. Our background is in incidence geometry, and, even if we were not the authors of this book, we would first think of examples such as the Euclidean plane and the geometry of circles on a sphere. These two geometries have a number of features in common. For example, the point sets of both geometries are surfaces, the lines or circles are curves that are nicely embedded in these surfaces, and both geometries satisfy an ‘axiom of joining’—in the Euclidean plane two points are contained in exactly one line and in the geometry on the sphere three points are contained in exactly one circle.

The Euclidean plane and the geometry of circles on a sphere are just two examples of a host of classical examples of geometries on surfaces. This book is about these classical geometries and their close relatives which live on the same surfaces, have the same kinds of lines, and satisfy the same axioms as their classical counterparts.

The history of our geometries on surfaces starts with Hilbert constructing a first example of a nonclassical R2-plane, that is, a close relative of the Euclidean plane. Today, one century of research later, our book tries to summarize all major results about geometries on surfaces.