Homogeneous Lorentzian manifolds of a semisimple group

Abstract

We describe the structure of $d$-dimensional homogeneous Lorentzian $G$-manifolds $M=G/H$ of a semisimple Lie group $G$. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group $G$ acts properly, that is the stabilizer $H$ is compact. Then any homogeneous space $G/\bar H$ with a smaller group $\bar H \subset H$ admits an invariant Lorentzian metric. A homogeneous manifold $G/H$ with a connected compact stabilizer $H$ is called a minimal admissible manifold if it admits an invariant Lorentzian metric, but no homogeneous $G$-manifold $G/\tilde H$ with a larger connected compact stabilizer $\tilde H \supset H$ admits such a metric. We give a description of minimal homogeneous Lorentzian $n$-dimensional $G$-manifolds $M = G/H$ of a simple (compact or noncompact) Lie group $G$. For $n \leq 11$, we obtain a list of all such manifolds $M$ and describe invariant Lorentzian metrics on $M$.