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 th...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1994, 1994
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| Edition: | 1st ed. 1994 |
| Series: | Universitext
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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