PseudoRiemannian geometry and actions of simple Lie groups

Abstract

Let G be a connected noncompact simple Lie group acting isometrically on a connected compact pseudoRiemannian manifold M. Denote with n 0 and m 0 the dimension of the maximal null subspaces tangent to G and M, respectively. Then we always have n 0 ⩽ m 0 . Our main result states that, if n 0 = m 0 , then the G-action is, up to a finite covering, an algebraic action. We use this to obtain a complete characterization of a large family of G-actions, thus providing a partial positive answer to the conjecture proposed in Zimmer's program for pseudoRiemannian manifolds. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 341 (2005).