On the geometry of flat pseudo-Riemannian homogeneous spaces

Let M = a"e (s) (n) /I" be a complete flat pseudo-Riemannian homogeneous manifold, I" aS, Iso(a"e (s) (n) ) its fundamental group and G the Zariski closure of I" in Iso(a"e (s) (n) ). We show that the G-orbits in a"e (s) (n) are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on a"e (s) (n) to the G-orbits is nondegenerate, then M has abelian linear 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 a parts per thousand yen 6. Moreover, we show that a"e (s) (n) is a trivial algebraic principal bundle G -> M -> a"e (n-k) . As a consquence, M is a trivial smooth bundle G/I" -> M -> a"e (n-k) with compact fiber G/Gamma.