Geometries and Groups

by Viacheslav V. Nikulin and Igor R. Shafarevich

This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.

• Format: Paperback
• Language: English
• ISBN: 3540152814
• ISBN13: 9783540152811
• Release Date: November 1987
• Publisher: Springer
• Length: 254 Pages
• Weight: 0.91 lbs.
• Dimensions: 0.6" x 6.3" x 9.4"

Customer Reviews

Interesting material, perhaps a bit loosely held together

The book begins gently with some background labelled "forming intuition" (although defining a geometry as a point set with a metric feels more like forming formalism to me). Then we get to the main theme of the book: isometry groups. They help us classify locally Euclidean geometries in two dimensions. The presentation is very elementary and explicit with details, and therefore quite tedious (we are already by page 120). Next we do the same thing in 3 dimensions, which is especially interesting, the authors argue, since the universe in which we live is three-dimensional. But it is hard to imagine these potential universes, except to say that there are 18 types and decide which of them are bounded or orientable. Perhaps sensing our dissapointment, the authors seem to say: well, yes, but at least these ideas pay off in physics as you can see here in our next chapter on the marvelously interesting theory of crystallography. We are not too impressed: Crystals?! Who cares about ****ing crystals? Anyway, after that the authors decide that it is interesting to study different locally Euclidean geometries on the torus. This leads to the modular group, and now we should be convinced that it is interesting to look for a geometry to accommodate the modular group as a group of motions. Lo and behold, hyperbolic geometry falls out, and the book ends triumphantly since we only wanted hyperbolic geometry in order to understand the modular group and torus geometries. Apparently, our previous interest in the true geometry of the universe, repeatedly appealed to in the discussion of locally Euclidean geometries, is gone without a trace.