Abstract: For a Lie group G and a closed subgoup H, Cartan geometries of type (G,H) can be viewed as curved analogs of the homogeneous space G/H. Using a characterization of Lie transformation groups due to Palais, one shows that for any such geometry the group of automorphisms is a finite dimensional Lie group, whose dimension is bounded above by the dimension of G.

In the case of parabolic geometries, the group G is semisimple and H is a parabolic subgroup. In the lecture I will explain how purely algebraic methods lead to surprising results on the possible dimensions of automorphism groups of parabolic geometries.

I will illustrate the results in the case of 3-dimensional CR manifolds. In particular, I will prove a result due to E. Cartan which shows that, up to coverings, any homogenous 3-dimensional CR structure which is not locally flat, is a left invariant structure on a Lie group.