Reductive homogeneous Lorentzian manifolds

TL;DR

This paper classifies homogeneous Lorentzian manifolds with reductive symmetry groups, focusing on their subgroup structures and invariant metrics, especially for compact semisimple Lie groups, reducing complex cases to semisimple scenarios.

Contribution

It reduces the classification of such manifolds to semisimple Lie groups and provides explicit descriptions of subgroups and invariant metrics, including minimal admissible cases.

Findings

Classification of totally reducible subgroups of the Lorentz group into three types.

Explicit description of Type II and III homogeneous Lorentzian spaces.

Complete classification of minimal admissible homogeneous manifolds for compact semisimple Lie groups.

Abstract

We study homogeneous Lorentzian manifolds $M = G/L$ of a connected reductive Lie group $G$ modulo a connected reductive subgroup $L$, under the assumption that $M$ is (almost) $G$-effective and the isotropy representation is totally reducible. We show that the description of such manifolds reduces to the case of semisimple Lie groups $G$. Moreover, we prove that such a homogeneous space is reductive. We describe all totally reducible subgroups of the Lorentz group and divide them into three types. The subgroups of Type I are compact, while the subgroups of Type II and Type III are non-compact. The explicit description of the corresponding homogeneous Lorentzian spaces of Type II and III (under some mild assumption) is given. We also show that the description of Lorentz homogeneous manifolds $M = G/L$ of Type I, reduces to the description of subgroups $L$ such that $M=G/L$ is an admissible manifold, i.e., an effective homogeneous manifold that admits an invariant Lorentzian metric. Whenever the subgroup $L$ is a maximal subgroup with these properties, we call such a manifold minimal admissible. We classify all minimal admissible homogeneous manifolds $G/L$ of a compact semisimple Lie group $G$ and describe all invariant Lorentzian metrics on them.