A review of Lorentzian synthetic theory of timelike Ricci curvature bounds

Cavalletti, Fabio; Mondino, Andrea

doi:10.1007/s10714-022-03004-4

Cited by 18 publications

(15 citation statements)

References 91 publications

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“…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].…”

Section: Lorentzian Length Spacesmentioning

confidence: 99%

Further observations on the definition of global hyperbolicity under low regularity

Minguzzi

2023

Class. Quantum Grav.

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The definitions of global hyperbolicity for closed cone structures and topological preordered spaces are known to coincide. In this work we clarify the connection with definitions of global hyperbolicity proposed in recent literature on Lorentzian length spaces and Lorentzian optimal transport, suggesting possible corrections for the terminology adopted in these works. It is found that in Kunzinger-Saemann's Lorentzian length spaces the definition of global hyperbolicity coincides with that valid for closed cone structures and, more generally, for topological preordered spaces: the causal relation is a closed order and the causally convex hull operation preserves compactness. In particular, it is independent of the metric, chronological relation or Lorentzian distance.

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Section: Lorentzian Length Spacesmentioning

confidence: 99%

Further observations on the definition of global hyperbolicity under low regularity

Minguzzi

2023

Class. Quantum Grav.

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“…Cavalletti and Mondino [ 17 ] took the field even further by introducing a synthetic notion of Ricci curvature bounds using techniques from optimal transport analogous to the Lott–Villani–Sturm theory of metric measure spaces with Ricci curvature bounded below [ 35 , 45 , 46 ]. They built on work by McCann [ 36 ] and Mondino–Suhr [ 39 ] who characterize timelike Ricci curvature bounds in terms of synthetic notions à la Lott–Villani–Sturm for smooth spacetimes.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Beran

Sämann

2023

Journal of London Math Soc

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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.

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“…It states that a globally hyperbolic n-dimensional Lorentzian manifold M with Ricci curvature bounded below by some (n − 1)K > 0 automatically has a diameter less than π √ K . In the case of synthetic Ricci curvature bounds, there is a synthetic Myers theorem (see [CMar,Prop. 5.10], [BM58, Thm.…”

Section: Introductionmentioning

confidence: 99%