A conformal Hopf-Rinow theorem for semi-Riemannian spacetimes

Abstract

The famous Hopf-Rinow theorem states that a Riemannian manifold is metrically complete if and only if it is geodesically complete. The compact Clifton-Pohl torus fails to be geodesically complete, leading many mathematicians and Wikipedia to conclude that "the theorem does not generalize to Lorentzian manifolds". Recall now that in their 1931 paper Hopf and Rinow characterized metric completeness also by properness. Recently, the author and Garc\'{\i}a-Heveling obtained a Lorentzian completeness-compactness result with a similar flavor. In this manuscript, we extend our theorem to 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.Comment: 40 page