On curvature bounds in Lorentzian length spaces

TL;DR

This paper develops new notions of curvature bounds for Lorentzian length spaces, establishing the equivalence of various formulations and unifying causal and timelike curvature bounds under mild assumptions.

Contribution

It introduces novel convexity/concavity and four-point conditions for curvature bounds, unifying existing definitions in Lorentzian pre-length spaces.

Findings

Established equivalence of all known curvature bound formulations

Unified causal and timelike curvature bounds

Provided conditions applicable in non-intrinsic settings

Abstract

We introduce several new notions of (sectional) curvature bounds for Lorentzian pre-length spaces: On the one hand, we provide convexity/concavity conditions for the (modified) time separation function, and, on the other hand, we study four-point conditions, which are suitable also for the non-intrinsic setting. Via these concepts we are able to establish (under mild assumptions) the equivalence of all previously known formulations of curvature bounds. In particular, we obtain the equivalence of causal and timelike curvature bounds as introduced in Kunzinger and S\"amann (Ann. Glob. Anal. Geom. 54(3):399-447, 2018).