Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds

TL;DR

This paper introduces a normalized angle concept in Lorentzian pre-length spaces to establish comparison theorems and curvature bounds, extending classical geometric results to Lorentzian geometry with applications to curvature and variational formulas.

Contribution

It develops a new notion of normalized angle for Lorentzian pre-length spaces and proves Lorentzian comparison theorems, including a Lorentzian Toponogov theorem and convexity properties, with applications to curvature bounds.

Findings

Established Lorentzian Toponogov theorem

Proved Alexandrov convexity property in Lorentzian spaces

Derived a first variation formula for non-negatively curved spaces

Abstract

In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation Formula for non-negatively curved globally hyperbolic Lorentzian length spaces.