Geometry II

1. Front Matter

   Pages i-ix

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2. Geometry of Spaces of Constant Curvature

   • D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov

   Pages 1-138

3. Discrete Groups of Motions of Spaces of Constant Curvature

   • E. B. Vinberg, O. V. Shvartsman

   Pages 139-248

4. Back Matter

   Pages 249-256

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Spaces of constant curvature, i.e. Euclidean space, the sphere, and Loba­ chevskij space, occupy a special place in geometry. They are most accessible to our geometric intuition, making it possible to develop elementary geometry in a way very similar to that used to create the geometry we learned at school. However, since its basic notions can be interpreted in different ways, this geometry can be applied to objects other than the conventional physical space, the original source of our geometric intuition. Euclidean geometry has for a long time been deeply rooted in the human mind. The same is true of spherical geometry, since a sphere can naturally be embedded into a Euclidean space. Lobachevskij geometry, which in the first fifty years after its discovery had been regarded only as a logically feasible by-product appearing in the investigation of the foundations of geometry, has even now, despite the fact that it has found its use in numerous applications, preserved a kind of exotic and even romantic element. This may probably be explained by the permanent cultural and historical impact which the proof of the independence of the Fifth Postulate had on human thought.

• Crystallographic Groups
• Diskrete Bewegungsgruppen
• Fuchsian Groups
• Fuchssche Gruppen
• Hyperbolic Geometry
• Hyperbolische Geometrie
• Kristallographische Gruppen
• Lobachevsky Space
• Lobatschewskischer Raum
• Reflection Groups
• Riemannian geometry
• Räume konstanter Krümmung
• Volume

• Department of Mathematics, Moscow University, Moscow, Russia

  E. B. Vinberg

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