Lie Groups and Lie Algebras I pp 121-149 | Cite as

# Transitive Actions

## Abstract

**1.1. Definitions and Examples.** Recall that any transitive action of a group *G* on a set *M* is isomorphic to an action of this group by left translations on the set of all cosets *G/H* where *H = G* _{ x } is the stabilizer of any point *x* ∈ *M.* In the differentiable case (see 1.4 of Chap. 1) *G/H* is an analytic homogeneous space of the Lie group *G* and the isomorphism is a diffeomorphism. The *G-* space *G/H* is called a *group model* (or a *Klein model)* of the homogeneous space *M.* A group model depends on the choice of the point *x* so that the subgroup *H* is defined up to conjugation in *G.* By means of a group model any property of the homogeneous space M can be expressed in terms of the group *G* and a subgroup *H.* We shall now illustrate this method on the simplest examples.

## Keywords

Homogeneous Space Maximal Compact Subgroup Riemannian Structure Transitive Action Homogeneous Vector Bundle## Preview

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