On the geometry of flat pseudo-Riemannian homogeneous spaces |
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Authors: | Wolfgang Globke |
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Institution: | 1. School of Mathematical Sciences, The University of Adelaide, Adelaide, SA, 5005, Australia
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Abstract: | Let M = ? s n /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ? Iso(? s n ) its fundamental group and G the Zariski closure of Γ in Iso(? s n ). We show that the G-orbits in ? s n are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ? 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 ? s n is a trivial algebraic principal bundle G → M → ? n?k . As a consquence, M is a trivial smooth bundle G/Γ → M → ? n?k with compact fiber G/Γ. |
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