Simply connected indefinite homogeneous spaces of finite volume

TL;DR

This paper characterizes simply connected indefinite homogeneous spaces of finite volume, showing they are compact with specific algebraic structures, and explores the properties of their isometry groups depending on the metric index.

Contribution

It proves that such spaces are compact with an abelian solvable radical and a compact semisimple Levi factor, and analyzes the isometry group structure based on metric index.

Findings

The space is compact and has an abelian solvable radical.

The Levi factor is a compact semisimple Lie group.

For index less than three, the isometry group is compact.

Abstract

Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite volume with isometry group $G$. We show that $M$ is compact and that the solvable radical of $G$ is abelian and the Levi factor is a compact semisimple Lie group acting transitively on $M$. For metric index less than three, we find that the isometry group of $M$ is compact itself. Examples demonstrate that $G$ is not necessarily compact for higher indices. To prepare these results, we study Lie algebras with abelian solvable radical and a nil-invariant symmetric bilinear form. For these, we derive an orthogonal decomposition into three distinct types of metric Lie algebras.