In mathematics, a Clifford–Klein form is a double coset space where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G/H.
According to Moritz Epple, the Clifford-Klein forms began when W. K. Clifford used quaternions to twist their space.
"Every twist possessed a space-filling family of invariant lines", the Clifford parallels.
They formed "a particular structure embedded in elliptic 3-space", the Clifford surface, which demonstrated that "the same local geometry may be tied to spaces that are globally different."
Wilhelm Killing thought that for free mobility of rigid bodies there are four spaces: Euclidean, hyperbolic, elliptic and spherical.