Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

TL;DR

This paper introduces hyperbolic angles in Lorentzian length spaces, providing new tools for analyzing timelike curvature bounds and improving geodesic non-branching results within this geometric framework.

Contribution

It defines hyperbolic angles and related concepts in Lorentzian length spaces, enabling characterization of timelike curvature bounds via angle monotonicity.

Findings

Characterization of timelike curvature bounds through angle monotonicity

Introduction of hyperbolic angles and exponential map in Lorentzian length spaces

Improved non-branching results for spaces with timelike curvature bounds

Abstract

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and S\"amann (Ann. Glob. Anal. Geom. 54(3):399--447, 2018) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts like timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature bounded below.