A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes

TL;DR

This paper extends the Hopf-Rinow theorem to semi-Riemannian spacetimes, establishing a conformal and properness-based characterization of completeness in a broader geometric context.

Contribution

It generalizes the classical Hopf-Rinow theorem to proper cone structures and $(n- u, u)$-spacetimes, broadening the understanding of completeness in semi-Riemannian geometry.

Findings

Generalization of the Hopf-Rinow theorem to semi-Riemannian manifolds.

Extension of null distance and properness concepts to new spacetime classes.

Implication that classical metric completeness results hold in broader settings.

Abstract

The famous Hopf-Rinow Theorem states, amongst others, that a Riemannian manifold is metrically complete if and only if it is geodesically complete. The Clifton-Pohl torus fails to be geodesically complete proving that this theorem cannot be generalized to compact Lorentzian manifolds. On the other hand, Hopf and Rinow characterized metric completeness also by properness. Garc\'ia-Heveling and the author recently obtained a Lorentzian completeness-compactness result for open manifolds with a similar flavor. In this manuscript, we extend the null distance used in this approach and our theorem to proper cone structures and to a new class of semi-Riemannian manifolds, dubbed $(n-\nu,\nu)$-spacetimes. Moreover, we demonstrate that our result implies, and hence generalizes, the metric part of the Hopf-Rinow Theorem.