Geometries and Groups 3-540-15281-4, 9783540152811, 9783642615702, 3642615708, 0-387-15281-4

This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they ar

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English Pages 254 [257] Year 1994

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Geometries and Groups
3-540-15281-4, 9783540152811, 9783642615702, 3642615708, 0-387-15281-4

Author / Uploaded
Viacheslav V. Nikulin
Igor R. Shafarevich
M. Reid

Table of contents :
Content: 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 --
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