On the Geometry of Flat Pseudo-Riemannian Homogeneous Spaces

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and \Gamma\subset\Iso(\RR^n_s) its fundamental group. We show that $M$ is a trivial fiber bundle G/\Gamma\to M\to\RR^{n-k}, where $G$ is the Zariski closure of $\Gamma$ in \Iso(\RR^n_s). Moreover, we show that the $G$-orbits in \RR^n_s are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $\Gamma$) has linear abelian 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 $\dim G\geq 6$.Comment: 20 pages, 1 figure, additional acknowledgmen

A Supplement to the Classification of Flat Homogeneous Spaces of Signature (m,2)

Duncan and Ihrig (1993) gave a classification of the flat homogeneous spaces of metric signature (m,2), provided that a certain condition on the development image of these spaces holds. In this note we show that this condition can be dropped, so that Duncan and Ihrig's classification is in fact the full classification for signature (m,2).Comment: shortened, one reference added, fixed typo

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

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

Holonomy Groups of Complete Flat Pseudo-Riemannian Homogeneous Spaces

We show that a complete flat pseudo-Riemannian homogeneous manifold with non-abelian linear holonomy is of dimension at least 14. Due to an example constructed in a previous article by Oliver Baues and the author, this is a sharp bound. Also, we give a structure theory for the fundamental groups of complete flat pseudo-Riemannian manifolds in dimensions less than 7. Finally, we observe that every finitely generated torsion-free 2-step nilpotent group can be realized as the fundamental group of a complete flat pseudo-Riemannian manifold with abelian linear holonomy.Comment: 16 page

Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let be a real finite‐dimensional Lie algebra equipped with a symmetric bilinear form ⟨·,·⟩ . We assume that ⟨·,·⟩ is nil‐invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of is an infinitesimal isometry for ⟨·,·⟩ . Among these Lie algebras are the isometry Lie algebras of pseudo‐Riemannian manifolds of finite volume. We prove a strong invariance property for nil‐invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non‐compact type of act by infinitesimal isometries for ⟨·,·⟩ . Moreover, we study properties of the kernel of ⟨·,·⟩ and the totally isotropic ideals in in relation to the index of ⟨·,·⟩ . Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most 2. Examples show that the theory becomes significantly more complicated for index greater than 2. We apply our results to study simply connected pseudo‐Riemannian homogeneous spaces of finite volume