An Embedding Theorem for Automorphism Groups of Cartan Geometries

An embedding theorem for automorphism groups of Cartan geometries

The main motivation of this paper arises from classical questions about actions of Lie groups preserving a geometric structure: which Lie groups can act on a manifold preserving a given structure, and which cannot? Which algebraic properties of the acting group have strong implications on the geometry of the manifold, such as local homogeneity, or on the dynamics of the action? These questions ...

Automorphism Groups of Parabolic Geometries

We show that elementary algebraic techniques lead to surprising results on automorphism groups of Cartan geometries and especially parabolic geometries. The example of three–dimensional CR structures is discussed in detail.

On Automorphism Groups of Parabolic Geometries

A Frobenius theorem for Cartan geometries, with applications

The classical result on local orbits in geometric manifolds is Singer’s homogeneity theorem for Riemannian manifolds [1]: given a Riemannian manifold M , there exists k, depending on dimM , such that if every x, y ∈ M are related by an infinitesimal isometry of order k, thenM is locally homogeneous. An open subset U ⊆ M of a geometric manifold is locally homogeneous if for every x, x ∈ U , ther...

NILPOTENCY AND SOLUBILITY OF GROUPS RELATIVE TO AN AUTOMORPHISM

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...