Groups of Automorphisms of Cartan Geometries
Uri Bader
University of Chicago
March 26, 2007
Abstract: Here is a partial
list of results of a similar nature:
- By a famous theorem of Arzela and Ascoli the group of isometries of a
compact Riemannian manifold is compact.
- A conjecture of Lichnerowitz (proved in the 80's by Lelong-Ferrand)
asserts that a compact Riemannian manifold has a compact group of
conformal automorphisms, unless this manifold is the standard sphere
(then the conformal group is SO(n,1)).
- Zimmer proved in the 80's that a non-compact simple Lie-group that
acts on a compact Lorentzian manifold must be SL(2,R).
All of the above concern with groups of automorphisms of compact
manifold endowed with a Cartan geometry - (roughly) a manifold
infinitesimally modeled over a homogeneous space.
In a joint work with Karin Melnick and Charles Frances we study the
relation between the group of automorpisms of a manifold with a Cartan
geomotry, and the structure group.
In my talk I will recall/explain what a Cartan geometry is, and what it
means that a Cartan geometry is flat (the latter also known as a
(G,X)-structure). I will then state some new theorems, and explain the
method of their proofs.