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A Clifford–Klein space form is a double coset space $\Gamma \backslash G/H$, where $G$ is a Lie group, $H$ a closed subgroup of $G$, and $\Gamma$ a discrete subgroup of $G$ that acts properly discontinuously and freely on the homogeneous space $G/H$.
The classical space forms are the cosets of the n-sphere $S^n$ or the Cartesian space $\mathbb{R}^n$ or the hyperbolic space $\mathbb{H}^n$ by discrete subgroups of their isometry group acting properly discontinuously (Carmo 92, chapt 8).
The properly discontinuous free quotients of n-spheres by discrete groups of isometries are also called spherical space forms. The classification of these was raised as an open problem by Killing 1891 and a complete solution was finally compiled by (Wolf 74). For more on this see at group actions on n-spheres.
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Euclidean geometry |
hyperbolic geometry |
elliptic geometry |
Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen, Math. Ann. 39 (1891), 257–278
Toshiyki Koyabashi, Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds, (pdf)
Joseph Wolf, Spaces of constant curvature, Publish or Perish, Boston, Third ed., 1974
M. P. do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhaeuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty.
Ian Hambleton, Topological spherical space forms, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston (arXivL:1412.8187)
Last revised on August 15, 2018 at 17:18:48. See the history of this page for a list of all contributions to it.