nLab Clifford-Klein space form

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.

Related entries

• locally Klein geometry

• Bertrand Russell, An Essay on the Foundations of Geometry

synthetic geometry
Euclidean geometry
hyperbolic geometry
elliptic geometry

References

• Wilhelm Killing, Ueber die Clifford-Klein’schen Raumformen, Math. Ann. 39 (1891), 257–278 (doi:10.1007/BF01206655)

• Heinz Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95, 313–339 (1926). (doi:10.1007/BF01206614, pdf)

• Toshiyki Koyabashi, Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds, (pdf)

• Joseph Wolf, Spaces of constant curvature, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 (doi:10.1090/chel/372)

• 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 (arXiv:1412.8187)