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(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 nonbranching result for spaces with timelike curvature bounded below.

A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

“…Synthetic lower bounds on the timelike sectional curvature in terms of comparison triangles (á la Alexandrov) for Lorentzian synthetic spaces have been proposed ans studied in [58] (see also the more recent [10]). It is a natural open problem whether such synthetic lower bounds on the timelike sectional curvature imply the synthetic lower bounds on the timelike Ricci curvature surveyed here.…”

“…Lorentzian (pre)-length spaces give a natural framework to develop a theory of optimal transport and of synthetic curvature bounds. Synthetic timelike sectional curvature bounds for Lorentzian (pre)-length spaces have been investigated in [58] (see also the more recent [10]). Synthetic timelike Ricci curvature lower bounds in Lorentzian (pre)-lenght spaces, which constitute the subject of the present survey, have been developed by the authors in [24] (inspired by the aforementioned smooth characterisations obtained in [71,75]).…”

Causal completions as Lorentzian pre-length spaces

“…Now we need to prove Î+ (𝐼 − (𝑡 2 , 𝑥 2 )) ∩ Î− (𝐼 − (𝑡 2 , 𝑥 1 )) = ∅. 9 Here 𝜕 B 𝑀 and 𝜕 𝐶 𝑀 are the Busemann and (metric) Cauchy boundaries or 𝑀 . Refer to [3,4,17] for a detailed account on the structure and topology of V .…”

“…Their notion of causal space lays at the foundations of the theory of Lorentzian pre-length spaces first introduced by Kunzinger and Sämman [28]. The purpose of this work is to present the future (or past) causal completion of a globally hyperbolic spacetime as a Lorentzian pre-length space, thus adding an interesting source of examples to this rapidly growing field [21,11,8,9,6,22,29].…”

A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

“…Synthetic timelike sectional curvature bounds for Lorentzian (pre)-length spaces have been investigated in [59] (see also the more recent [10]). Synthetic timelike Ricci curvature lower bounds in Lorentzian (pre)-lenght spaces, which constitute the subject of the present survey, have been developed by the authors in [25] (inspired by the aforementioned smooth characterisations obtained in [72,76]).…”