Convex Functions and Geodesic Connectedness of Space-times

TL;DR

This paper investigates how convex functions influence the geodesic connectedness of space-times and semi-Riemannian manifolds, providing new geometric-topological proofs and criteria for the existence of convex functions.

Contribution

It introduces novel geometric-topological methods to establish geodesic connectedness in classes of space-times where previous techniques failed.

Findings

Null-disprisoning space-times with specific convex functions are geodesically connected.

Timelike convex hypersurfaces in Minkowski space are geodesically connected.

A new criterion for the existence of convex functions on semi-Riemannian manifolds.

Abstract

This paper explores the relation between convex functions and the geometry of space-times and semi-Riemannian manifolds (an investigation initiated by Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which known methods do not apply. For instance: A null-disprisoning space-time is geodesically connected if it supports a proper, nonnegative strictly convex function whose critical set is a point. Timelike strictly convex hypersurfaces of Minkowski space are geodesically connected. We also give a criterion for the existence of a convex function on a semi-Riemannian manifold. We compare our work with previously known results.