On the Geometry of Flat Pseudo-Riemannian Homogeneous Spaces

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $\mathrm{\Gamma }\subset \text{Iso}({\text{RR}}_s^n)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G\mathrm{/}\mathrm{\Gamma }\to M\to {\text{RR}}^{n-k}$, where $G$ is the Zariski closure of $\mathrm{\Gamma }$ in $\text{Iso}({\text{RR}}_s^n)$. Moreover, we show that the $G$-orbits in ${\text{RR}}_s^n$ are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $\mathrm{\Gamma }$) has linear abelian holonomy. If additionally $G$ is not abelian, then $G$ contains a certain subgroup of dimension 6. In particular, for non-abelian $G$ orbits with non-degenerate metric can appear only if $\dim G\ge 6$.