Geometries and Groups

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.

I. Forming geometrical intuition; statement of the main problem.- §1. Formulating the problem.- §2. Spherical geometry.- §3. Geometry on a cylinder.- §4. A world in which right and left are indistinguishable.- §5. A bounded world.- §6. What does it mean to specify a geometry?.- II. The theory of 2-dimensional locally Euclidean geometries.- §7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- §8. Classification of all uniformly discontinuous groups of motions of the plane.- §9. A new geometry.- §10. Classification of all 2-dimensional locally Euclidean geometries.- III. Generalisations and applications.- §11. 3-dimensional locally Euclidean geometries.- §12. Crystallographic groups and discrete groups.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- §13. Similarity of geometries.- §14. Geometries on the torus.- §15. The algebra of similarities: complex numbers.- §16. Lobachevsky geometry.- §17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.