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 $\widehat{M}$ to which the $G$-action lifts so that $\widehat{M}$ 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 $\mathrm{rank}_{\mathbb{R}}(G)\geq 2$, then we prove that we can choose $\widehat{M}$ 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.