Isometric Splitting for Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds

We study the action of a connected noncompact simple Lie group G on a connected finite volume pseudo-Riemannian manifold M. Our main result provides a geometric splitting of the metric on such M that involves natural metrics on G. For suitable G such splitting is a warped product. For any G the splitting turns out to be a direct product in the ergodic case.