Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds

TL;DR

This paper establishes a new equivalence between curvature bounds and triangle comparison conditions in semi-Riemannian spaces, extending Alexandrov geometry concepts to Lorentzian and related geometries.

Contribution

It proves the equivalence of curvature bounds with local triangle comparisons and introduces semi-Riemannian analogues of key Alexandrov lemmas, expanding geometric analysis tools.

Findings

Curvature bounds are equivalent to local triangle comparison conditions.

Semi-Riemannian analogues of Alexandrov lemmas are established.

Monotonicity persists even with changing model spaces.

Abstract

A semi-Riemannian manifold is said to satisfy $R\ge K$ (or $R\le K$) if spacelike sectional curvatures are $\ge K$ and timelike ones are $\le K$ (or the reverse). Such spaces are abundant, as warped product constructions show; they include, in particular, big bang Robertson-Walker spaces. By stability, there are many non-warped product examples. We prove the equivalence of this type of curvature bound with local triangle comparisons on the signed lengths of geodesics. Specifically, $R\ge K$ if and only if locally the signed length of the geodesic between two points on any geodesic triangle is at least that for the corresponding points of its model triangle in the Riemannian, Lorentz or anti-Riemannian plane of curvature $K$ (and the reverse for $R\le K$). The proof is by comparison of solutions of matrix Riccati equations for a modified shape operator that is smoothly defined along reparametrized geodesics (including null geodesics) radiating from a point. Also proved are semi-Riemannian analogues to the three basic Alexandrov triangle lemmas, namely, the realizability, hinge and straightening lemmas. These analogues are intuitively surprising, both in one of the quantities considered, and also in the fact that monotonicity statements persist even though the model space may change. Finally, the algebraic meaning of these curvature bounds is elucidated, for example by relating them to a curvature function on null sections.