On the Geometry of Flat Pseudo-Riemannian Homogeneous Spaces

TL;DR

This paper investigates the structure of complete flat pseudo-Riemannian homogeneous manifolds, showing they form trivial fiber bundles with specific properties of their isometry groups and orbit structures, including conditions on holonomy and subgroup dimensions.

Contribution

It establishes the fiber bundle structure of such manifolds, characterizes the geometry of their orbits, and identifies conditions on the isometry group and holonomy for non-abelian cases.

Findings

Manifolds are trivial fiber bundles with base ^{n-k}

G-orbits are affinely diffeomorphic to G with the (0)-connection

Non-degenerate metric on G-orbits implies G has linear abelian holonomy

Abstract

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 $\dim G\geq 6$.