# nLab Clifford-Klein space form

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# Contents

## Idea

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.

## References

• 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.