# Geometries and Groups

Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-viii
2. Viacheslav V. Nikulin, Igor R. Shafarevich
Pages 1-52
3. Viacheslav V. Nikulin, Igor R. Shafarevich
Pages 53-120
4. Viacheslav V. Nikulin, Igor R. Shafarevich
Pages 121-186
5. Viacheslav V. Nikulin, Igor R. Shafarevich
Pages 187-244
6. Back Matter
Pages 245-254

### Introduction

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.

### Keywords

Lattice Mathematica Non-Euclidean Geometry Symmetry group addition algebra boundary element method field form mathematics philosophy set similarity story symmetry

#### Authors and affiliations

1. 1.Steklov Mathematical InstituteMoscowRussia

### Bibliographic information

• Book Title Geometries and Groups
• Authors Viacheslav V. Nikulin
Igor R. Shafarevich
• Series Title Universitext
• DOI https://doi.org/10.1007/978-3-642-61570-2
• Copyright Information Springer-Verlag Berlin Heidelberg 1994
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Softcover ISBN 978-3-540-15281-1
• eBook ISBN 978-3-642-61570-2
• Series ISSN 0172-5939
• Series E-ISSN 2191-6675
• Edition Number 1
• Number of Pages VIII, 254
• Number of Illustrations 1 b/w illustrations, 0 illustrations in colour