A Toponogov globalisation result for Lorentzian length spaces

TL;DR

This paper extends Toponogov's Globalisation Theorem to Lorentzian length spaces with curvature bounds, providing new tools for synthetic Lorentzian geometry and applications like Lorentzian Bonnet--Myers and Splitting Theorems.

Contribution

It introduces a Lorentzian analogue of Toponogov's theorem using a 'cat's cradle' construction, advancing the synthetic geometric framework for Lorentzian spaces.

Findings

Established a Lorentzian version of Toponogov's Globalisation Theorem.

Proved a lemma on triangle subdivision in spaces with local curvature bounds.

Presented applications including Lorentzian Bonnet--Myers and Splitting Theorems.

Abstract

In the synthetic geometric setting introduced by Kunzinger and S\"amann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet--Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov--Hausdorff convergence.