Geometry II

Spaces of Constant Curvature

  • E. B. Vinberg

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 29)

Table of contents

  1. Front Matter
    Pages i-ix
  2. D. V. Alekseevskij, E. B. Vinberg, A. S. Solodovnikov
    Pages 1-138
  3. E. B. Vinberg, O. V. Shvartsman
    Pages 139-248
  4. Back Matter
    Pages 249-256

About this book

Introduction

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.

Keywords

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

Editors and affiliations

  • E. B. Vinberg
    • 1
  1. 1.Department of MathematicsMoscow UniversityMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-02901-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1993
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08086-9
  • Online ISBN 978-3-662-02901-5
  • Series Print ISSN 0938-0396
  • About this book