Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group

Details Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group Flat Pseudo-Riemannian Homogeneous Spaces with Non-Abelian Holonomy Group

2010-09-17

arxiv.org/abs/1009.3383v1

Mathematics

Differential Geometry

Scientific paper

We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions $\geq 8$, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.

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