  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 $G$.

Differential Geometry