# Contents

## Definition

Given a topological group or algebraic group or Lie group, etc., $G$, a homogeneous $G$-space is a topological space or scheme, or smooth manifold etc. with transitive $G$-action.

A principal homogeneous $G$-space is the total space of a $G$-torsor over a point.

There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.

## Examples

A special case of homogeneous spaces are coset spaces arising from the quotient $G/H$ of a group $G$ by a subgroup. For the case of Lie groups this is also called Klein geometry.

## Properties

Under weak topological conditions (cf. Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces), every topological homogeneous space $M$ is isomorphic to a coset space $G/H$ for a closed subgroup $H\subset G$ (the stabilizer of a fixed point in $X$).

Revised on October 31, 2013 08:24:05 by Urs Schreiber (145.116.130.141)