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.- 3.1. First acquaintance.- 3.2. How to measure distances.- 3.3. The study of geometry on a cylinder.- §4. A world in which right and left are indistinguishable.- §5. A bounded world.- 5.1. Description of the geometry.- 5.2. Lines on the torus.- 5.3. Some applications.- §6. What does it mean to specify a geometry?.- 6.1. The definition of a geometry.- 6.2. Superposing geometries.- II. The theory of 2-dimensional locally Euclidean geometries.- §7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- 7.1. Definition of equivalence by means of motions.- 7.2. The geometry corresponding to a uniformly discontinuous group.- §8. Classification of all uniformly discontinuous groups of motions of the plane.- 8.1. Motions of the plane.- 8.2. Classification: generalities and groups of Type I and II.- 8.3. Classification: groups of Type III.- §9. A new geometry.- §10. Classification of all 2-dimensional locally Euclidean geometries.- 10.1. Constructions in an arbitrary geometry.- 10.2. Coverings.- 10.3. Construction of the covering.- 10.4. Construction of the group.- 10.5. Conclusion of the proof of Theorem 1.- III. Generalisations and applications.- §11. 3-dimensional locally Euclidean geometries.- 11.1. Motions of 3-space.- 11.2. Uniformly discontinuous groups in 3-space: generalities.- 11.3. Uniformly discontinuous groups in 3-space: classification.- 11.4. Orientability of the geometries.- §12. Crystallographic groups and discrete groups.- 12.1. Symmetry groups.- 12.2. Crystals and crystallographic groups.- 12.3. Crystallographic groups and geometries: discrete groups.- 12.4. A typical example: the geometry of the rectangle.- 12.5. Classification of all locally Cn or Dn geometries.- 12.6. On the proof of Theorems 1 and 2.- 12.7. Crystals and their molecules.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- §13. Similarity of geometries.- 13.1. When are two geometries defined by uniformly discontinuous groups the same?.- 13.2. Similarity of geometries.- §14. Geometries on the torus.- 14.1. Geometries on the torus and the modular figure.- 14.2. When do two pairs of vectors generate the same lattice?.- 14.3. Application to number theory.- §15. The algebra of similarities: complex numbers.- 15.1. The geometrical definition of complex numbers.- 15.2. Similarity of lattices and the modular group.- §16. Lobachevsky geometry.- 16.1. 'Motions'.- 16.2. 'Lines'.- 16.3. Distance.- 16.4. Construction of the geometry concluded.- §17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- 17.1. Discreteness of the modular group.- 17.2. The set of all geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.