---
tag: R73WRFDXRY
kind: concept
name: Clifford–Klein form
msc: ["57S30", "22E40", "22F30"]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford–Klein_form
same_as:
  wikidata: Q5133088
  wikipedia: Clifford–Klein form
---

## Summary

Let $G/H$ be a [[homogeneous space|85TWR6E75P]] of a [[Lie group|7JBRGCVJRB]] $G$.
A Clifford–Klein form is a quotient $\Gamma \backslash G / H$ by a [[discrete subgroup|5M7D1BSHVZ]] $\Gamma$ that acts [[properly discontinuously|J5X5ED0B2Y]].
When $\Gamma$ is torsion-free, the quotient is a manifold that looks locally like $G/H$.
Compact [[space forms|1VM1AQT956]] such as flat tori and closed hyperbolic manifolds are the classical examples.
A central question is which homogeneous spaces admit compact Clifford–Klein forms.

## Explanations

### L1 human

Start with a very symmetric space, such as the plane or the hyperbolic plane.
Pick a discrete group of its symmetries that moves points far enough apart, and glue each point to all its images.
The glued space keeps the local geometry of the original space but can be compact.
For example, gluing the plane $\R^2$ along the translations by vectors of $\Z^2$ gives a flat torus.
Such glued spaces are called Clifford–Klein forms of the original space.