Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Abstract

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n_0 \leq m_0. We study G-actions that satisfy the condition n_0 = m_0. With no rank restrictions on G, we prove that M has a finite covering \hat{M} to which the G-action lifts so that \hat{M} is G-equivariantly diffeomorphic to an action on a double coset K\backslash L/\Gamma, as considered in Zimmer's program, with G normal in L (Theorem A). If G has finite center and \rank_\R(G)\geq 2, then we prove that we can choose \hat{M} for which L is semisimple and \Gamma is an irreducible lattice (Theorem B). We also prove that our condition n_0 = m_0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmer's program.