Clifford–Klein form

Summary

Let $G/H$ be a homogeneous space of a Lie group $G$. A Clifford–Klein form is a quotient $\Gamma \backslash G / H$ by a discrete subgroup $\Gamma$ that acts properly discontinuously. When $\Gamma$ is torsion-free, the quotient is a manifold that looks locally like $G/H$. Compact space forms such as flat tori and closed hyperbolic manifolds are the classical examples. A central question is which homogeneous spaces admit compact Clifford–Klein forms.