Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

TL;DR

This paper proves that the only divisible domain in the flat pseudo-Riemannian space of signature (2,2) is the entire space, establishing a new completeness result for closed flat pseudo-Riemannian manifolds in this setting.

Contribution

It introduces the first completeness theorem for closed flat pseudo-Riemannian manifolds of signature (2,2), extending beyond Euclidean and Lorentzian cases, and develops new geometric reduction techniques.

Findings

Only the entire space is divisible by a discrete isometry subgroup.

Established a geometric reduction for divisible domains in affine space.

Generalized the existence of syndetic hulls in certain semidirect product groups.

Abstract

Let $\mathbb{R}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of $\mathbb{R}^{2,2}$ is $\mathbb{R}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R \ltimes G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carri\`ere and Dal'bo.