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
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.
- Homogeneous Space
- Maximal Compact Subgroup
- Riemannian Structure
- Transitive Action
- Homogeneous Vector Bundle
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