## Compact pseudo-Riemannian homogeneous spaces

A pseudo-Riemannian homogeneous space $$M$$ of finite volume can be presented as $$M=G/H$$, where $$G$$ is a Lie group acting transitively and isometrically on $$M$$, and $$H$$ is a closed subgroup of $$G$$.
The condition that $$G$$ acts isometrically and thus preserves a finite measure on $$M$$ leads to strong algebraic restrictions on $$G$$. In the special case where $$G$$ has no compact semisimple normal subgroups, it turns out that the isotropy subgroup $$H$$ is a lattice, and that the metric on $$M$$ comes from a bi-invariant metric on $$G$$.
This result allows us to recover Zeghib’s classification of Lorentzian compact homogeneous spaces, and to move towards a classification for metric index $$2$$.