Rigidity of compact pseudo-Riemannian homogeneous spaces for solvable Lie groups

TL;DR

This paper proves that compact pseudo-Riemannian homogeneous spaces with solvable isometry groups have a rigid structure, with the isometry group essentially determined by the solvable Lie group acting transitively.

Contribution

It establishes that such spaces have an almost free solvable Lie group action and that their metric derives from a bi-invariant metric on the group, showing a rigidity property.

Findings

The solvable Lie group acts almost freely on the manifold.

The metric on the manifold is induced by a bi-invariant metric on the group.

The identity component of the isometry group coincides with the acting group.

Abstract

Let $M$ be a compact connected pseudo-Riemannian manifold on which a solvable connected Lie group $G$ of isometries acts transitively. We show that $G$ acts almost freely on $M$ and that the metric on $M$ is induced by a bi-invariant pseudo-Riemannian metric on $G$. Furthermore, we show that the identity component of the isometry group of $M$ coincides with $G$.