Homogeneous Lorentzian manifolds of a semisimple group

TL;DR

This paper classifies and describes the structure of minimal homogeneous Lorentzian manifolds for semisimple Lie groups, providing explicit lists and metrics for dimensions up to 11.

Contribution

It offers a comprehensive classification of minimal homogeneous Lorentzian manifolds for simple Lie groups, including explicit metric descriptions for low dimensions.

Findings

List of all minimal homogeneous Lorentzian manifolds up to dimension 11.

Description of invariant Lorentzian metrics on these manifolds.

Identification of conditions for the existence of such metrics.

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 $M$.