Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group

Abstract

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 noncomplete examples, where the linear holonomy is non-abelian, starting in dimensions ≥ 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.

Holonomy groups of complete flat pseudo-Riemannian homogeneous spaces

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We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension ≥ 14. Due to an example constructed in a previous article [2], this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions ≤ 6.

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“…In an article [2] by Oliver Baues and the author, Wolf's unipotent representations for fundamental groups with abelian Hol(Γ) were generalized for groups with non-abelian linear holonomy. Also, it was shown that a (possibly incomplete) flat pseudo-Riemannian homogeneous manifold M with non-abelian linear holonomy group is of dimension dim M ≥ 8.…”

On the geometry of flat pseudo-Riemannian homogeneous spaces

“…Let B n,ω denote a simply connected Lie group with Lie algebra b n,ω . It is known that every compact flat pseudo-Riemannian homogeneous space M of split signature (k, k) can be realized as a quotient B a,ω /Γ, where a is an abelian Lie algebra (see Baues and Globke [2,Section 3]). An interesting open question along these lines is whether a compact flat pseudo-Riemannian homogeneous space M with split signature (k, k) and non-abelian fundamental group Γ can be realized as B n,ω /Γ, where n is some non-abelian 2-step nilpotent Lie algebra.…”

“…More recent studies by Oliver Baues and the author [1,2,5,6] investigated among other things the holonomy groups of these spaces. For some time only M with abelian linear holonomy group (given by the linear parts of Γ) were known, and it was unknown whether M with non-abelian linear holonomy existed.…”

“…For some time only M with abelian linear holonomy group (given by the linear parts of Γ) were known, and it was unknown whether M with non-abelian linear holonomy existed. For example, it was shown in [2] that for compact M the linear holonomy group is always abelian. The first example of a non-compact M with non-abelian linear holonomy in dimension n = 14 was given by the author in his thesis [5], see also [6].…”