On the geometry of flat pseudo-Riemannian homogeneous spaces

Abstract

Let M = ℝsn/Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ⊂ Iso(ℝsn) its fundamental group and G the Zariski closure of Γ in Iso(ℝsn). We show that the G-orbits in ℝsn are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ℝsn 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 ≥ 6. Moreover, we show that ℝsn is a trivial algebraic principal bundle G → M → ℝn−k. As a consquence, M is a trivial smooth bundle G/Γ → M → ℝn−k with compact fiber G/Γ.