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We prove a synthetic Bonnet–Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$: It states that the space necessarily is a warped product with warping function $\cos: (-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}_+$. From this, one also sees that a globally hyperbolic spacetime with curvature bounded above by K < 0 and an open distance realizer of length $L=\frac{\pi}{\sqrt{|K|}}$ is a warped product with warping function cos.

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In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-Émery setting as well in the Ollivier-Ricci curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet-Myers theorem. The proofs in the different settings require different techniques.

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A Bonnet–Myers rigidity theorem for globally hyperbolic Lorentzian length spaces

2 - Curvature Calculations for Antitrees

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2 results

A Bonnet–Myers rigidity theorem for globally hyperbolic Lorentzian length spaces

2 - Curvature Calculations for Antitrees

Summary