Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group? Edit: My goal is to understand a bit better what the intuition behind compactness / non-compactness of the isometry of a closed Semi-Riemannian manifold is. For example are there topological restrictions on the underlying space? An example of the kind of statements I'm looking for is the following theorem (for Lorentzian manifolds): **Theorem:** Let $(M,g)$ be a closed, simply connected Lorentz manifold and assume $M$ and $g$ are analytic. Then the isometry group of $M$ is compact.