A synthetic Lorentzian Cartan-Hadamard theorem

TL;DR

This paper introduces a synthetic Lorentzian version of the Cartan-Hadamard theorem, extending classical geometric results to Lorentzian spaces using a novel notion of local concavity, and establishes existence, uniqueness, and globalization of timelike geodesics.

Contribution

It formulates and proves a Lorentzian Cartan-Hadamard theorem within synthetic geometry, generalizing prior smooth and metric space results to Lorentzian length spaces.

Findings

Existence and uniqueness of timelike geodesics under global hyperbolicity and future one-connectedness.

A globalization theorem for Lorentzian length spaces with curvature bounds.

Extension of classical convexity and geodesic results to synthetic Lorentzian geometry.

Abstract

We formulate and prove a synthetic Lorentzian Cartan-Hadamard theorem. This result both transfers the corresponding statement for locally convex metric spaces established by S. Alexander and R. Bishop to the Lorentzian setting, and simultaneously generalizes the smooth Lorentzian theorem discussed by J. Beem and P. Ehrlich to the recently established framework of synthetic Lorentzian geometry. Our approach is based on an appropriate notion of local concavity for Lorentzian (pre-)length spaces, which allows us to establish existence and uniqueness of timelike geodesics between any pair of timelike related points under the additional assumptions of global hyperbolicity and future one-connectedness. We also provide a globalization result for our notion of concavity in the setting of Lorentzian length spaces, and apply our results to obtain a globalization statement for nonnegative upper timelike curvature bounds.