The local geometry of compact homogeneous Lorentz spaces

TL;DR

This paper explores the local geometric structure of compact homogeneous Lorentz spaces with non-compact isometry groups, revealing they are all reductive and analyzing their curvature and isotropy representations.

Contribution

It provides a detailed analysis of the local geometry of these spaces, extending previous classifications and examining Ricci-flat cases.

Findings

All such spaces are reductive.

Ricci-flat spaces are either flat or have compact isometry groups.

Detailed analysis of isotropy representations and curvatures.

Abstract

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, Zeghib also described the geometric structure of the manifolds. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns out that they all are reductive. We investigate the isotropy representation and curvatures. In particular, we obtain that any Ricci-flat compact homogeneous Lorentz space is flat or has compact isometry group.