Geometries and Groups

by Viacheslav V. Nikulin and Igor R. Shafarevich

Book Overview

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.... This description may be from another edition of this product.

• 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.