Strict Myers rigidity for Lorentzian Ricci comparison (smooth and synthetic)

Establish a strict Lorentzian Myers rigidity theorem, both in the smooth category of Lorentzian manifolds with timelike Ricci curvature bounded below by a positive constant and in the synthetic setting of Lorentzian length spaces with synthetic Ricci lower bounds, showing that if the upper bound on timelike diameter is achieved then the space is (up to the natural scaling) a higher-dimensional anti-de Sitter spacetime.

Background

In the Lorentzian setting there exist Myers-type theorems that bound timelike diameter under lower bounds on timelike Ricci curvature, both in smooth spacetimes and in synthetic frameworks. However, rigidity statements—classifying the spaces that achieve the bound—are more delicate.

The paper proves a Bonnet–Myers rigidity theorem under synthetic timelike sectional curvature bounds (yielding warped-product structures), but notes that an analogous strict rigidity result under Ricci comparison, in both smooth and synthetic settings, is not presently available. Such a result would characterize equality cases by identifying the space with anti-de Sitter geometry.