A survey on sufficient conditions for geodesic completeness of nondegenerate hypersurfaces in Lorentzian geometry

Abstract

This is a survey of the principal results about the geodesic completeness of nondegenerate hypersurfaces in Lorentzian manifolds from a structural point of view. Some of these results retain their validity in the case of semi-Riemannian submanifolds in semi-Euclidean spaces, as well.

Global Hyperbolicity through the Eyes of the Null Distance

“…Lemma 3.5. Let g R and g W be two conformally equivalent Riemannian metrics on M as in (22), and suppose K is a compact set in M. Then there exist constants…”

“…Proof. The first statement about the lengths follows immediately from the Definition (22), and we may pick c K := max x∈K (x). The first inequality for the distances is trivial and the second inequality follows from [16, Thm.…”

“…Nonetheless, even the physically undesired assumption of compactness does not guarantee geodesic completeness (Clifton-Pohl torus). Altogether these properties render any Hopf-Rinow type statement for spacetimes virtually a lost case (see the early works of Busemann [17], Beem [6]; and [7,18,22,26] for work on completeness of spacelike submanifolds). We reopen the case and characterize global hyperbolicity in an entirely new way.…”