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