Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Abstract

Within the synthetic‐geometric framework of Lorentzian (pre‐)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399–447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as 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.

Alexandrov's Patchwork and the Bonnet-Myers Theorem for Lorentzian length spaces

“…Given the recent work of [BMS22, BS22] formulating curvature bounds on Lorentzian prelength spaces in terms of monotonicity and angles, it is worth re-examining how well the techniques utilized in the proof of [Har82] adapt to our setting. As [BS22] also introduces curvature comparison of Lorentzian pre-length spaces via hinges, the proof of the metric statement provided in [AKP19, Theorem 8.31] may also be a good point of ingress. Most promisingly, the first and third author of this work are currently part of a collaboration developing a Lorentzian description of (lower) curvature bounds using the so-called four point condition, see [BBI01, Proposition 10.1.1].…”

“…In metric geometry, there are several reformulations of curvature bounds expressed via classical triangle comparison, using angles, for example. Alternative versions also exist for timelike curvature bounds (see [BS22,BMS22]) and several of these characterizations will prove useful in our context. Before we state these explicitly, let us introduce some more terminology: Definition 2.13 (K-comparison angles and sign).…”

“…In particular, [KS18] develop comparison methods for synthetic Lorentzian geometry, in order to impose curvature bounds on Lorentzian pre-length spaces. Equivalent approaches have also been proposed by [BS22,BMS22], however none of these works relate curvature bounds enforced globally to those imposed on neighbourhoods. Furthermore, while the development of metric length spaces was guided by disciplines such as group theory and the study of partial differential equations, alongside its purely geometric origin, research into Lorentzian pre-length spaces has, for the most part, focused on their apparent necessity in general relativity.…”

“…Furthermore, while the development of metric length spaces was guided by disciplines such as group theory and the study of partial differential equations, alongside its purely geometric origin, research into Lorentzian pre-length spaces has, for the most part, focused on their apparent necessity in general relativity. This paper, continuing from the work of [BR22, Rot22,BS22], aims to develop suitable Lorentzian analogues to some of the fundamental results of metric length spaces, in order to facilitate the wider application of the Lorentzian length space framework. In particular, given their myriad of applications in the metric setting, we focus on the notion of spaces with global timelike curvature bounds.…”

Further observations on the definition of global hyperbolicity under low regularity

“…which they called 'K-global hyperbolicity'. This terminology was adopted in Cavalletti and Modino work on optimal transport over Lorentzian length spaces [7], and in posterior works using the same framework [5,6].…”