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Geometries and Groups : Springer Series in Soviet Mathematics - Viacheslav V. Nikulin

Geometries and Groups

Springer Series in Soviet Mathematics

Paperback Published: 1st September 1994
ISBN: 9783540152811
Number Of Pages: 254

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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.

Contents: Forming geometrical intuition; statement of the main problem
The theory of 2-dimensional locally Euclidean geometries
Generalisations and applications
Geometries on the torus, complex numbers and Lobachevsky geometry
Historical remarks
List of notation
Index.
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540152811
ISBN-10: 3540152814
Series: Springer Series in Soviet Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 254
Published: 1st September 1994
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.98 x 16.0  x 1.6
Weight (kg): 0.41

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