Rigidity of Compact Pseudo-Riemannian Homogeneous Spaces for Solvable Lie Groups

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. The proofs rely on a combination of density properties for uniform subgroups of solvable Lie groups and the reduction theory of pseudo-Riemannian metric Lie groups.