Global hyperbolicity in higher signatures

TL;DR

This paper generalizes the concept of global hyperbolicity to higher signature pseudo-Riemannian spaces, proving key properties like compactness of causal diamonds and existence of solutions to geometric problems, with applications to holonomy representations.

Contribution

It extends global hyperbolicity to (p, q) signatures with p ≤ q ≤ 2, establishing foundational properties and linking to holonomy representations in higher signature spaces.

Findings

Proved compactness of causal diamonds in higher signature globally hyperbolic spaces.

Established existence of solutions to Plateau problems in these spaces.

Characterized GH-regular representations as holonomies of globally hyperbolic models.

Abstract

We provide a generalization of global hyperbolicity in pseudo-Riemannian spaces of signature (p, q) for p ___ q ___ 2. We then prove the compactness of causal diamonds in globally hyperbolic spaces and deduce the existence of solutions to a Plateau problem for this class of spaces. Finally, caracterize GH-regular representations in SO(p,q+1) as holonomies of some globally hyperbolic spaces modeled on H^{p,q}.