The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

Beran, Tobias; Ohanyan, Argam; Rott, Felix; Solis, Didier A.

doi:10.48550/arxiv.2209.14724

Cited by 3 publications

(3 citation statements)

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“…We also introduced a diversity of metric tools, and thereby furthering the development of the theory considerably. Moreover, these tools and methods have further applications in Lorentzian geometry and general relativity: The most striking example is the recently established splitting theorem for Lorentzian length spaces with non-negative curvature in [9]. The splitting theorem can be understood as a rigidity statement of the classical Hawking-Penrose singularity theorem [28] and as such is of fundamental importance in Einstein's theory of gravity.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Beran

Sämann

2023

Journal of London Math Soc

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“…For instance, Toponogov type triangle comparison [6], or the existence of hyperbolic angles and exponential maps [10]. Applications of these results include gluing and amalgamation [9], as well as a splitting theorem that generalizes the landmark result for Lorentzian manifolds with non-positive timelike sectional curvature [11].…”

Section: Introductionmentioning

confidence: 99%

On the space of compact diamonds of Lorentzian length spaces

Barrera¹,

de²,

Solis³

2023

Preprint

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In this work we introduce the taxicab and uniform products for Lorentzian pre-length spaces. We further use these concepts to endow the space D(R ×T X) of causal diamonds with a Lorentzian length space structure, closely relating its causal properties with its geometry as a metric space furnished with its associated Hausdorff distance. Among the general results, we show that this space is geodesic and globally hyperbolic for complete length space (X, d).

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“…Some recent developments include Toponogov type triangle comparison [4], the existence of hyperbolic angles and exponential maps [9], gluing and amalgamation [8], Bonnet-Myers type results [7] as well as a splitting theorem that generalizes the landmark result for Lorentzian manifolds with non-positive timelike sectional curvature [10].…”

Section: Introductionmentioning

confidence: 99%