Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces

Abstract

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 $\cos$.

arXiv commentClearer introduction, added smooth statement. 44 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:2209.14724