Transitive Actions

  • A. L. Onishchik
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 20)


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 xM. 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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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • A. L. Onishchik
    • 1
  1. 1.Yaroslavl UniversityYaroslavlRussia

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