Riemannian flow techniques on totally geodesic null hypersurfaces
Manuel Gutiérrez, Raymond A. Hounnonkpe
TL;DR
This paper develops a framework in which a rigged null hypersurface carries a Riemannian flow structure that is bundle-like with respect to the rigged metric, enabling a precise link between transverse (screen) geometry and ambient spacetime curvature. By proving that the transverse curvature equals the ambient curvature in general and remains constant under constant-curvature ambient spaces, the authors translate curvature conditions into statements about the flow and its dynamics, including criteria for isometric flows when the rigging is closed. They further analyze when rigged-metric geodesics coincide with ambient geodesics, establishing equivalence conditions and completeness results, particularly in static or closed-rigging settings. Finally, they address the existence of periodic geodesics in compact Lorentzian manifolds, showing under various curvature and causality hypotheses that infinite periodic null geodesics (and sometimes spatial geodesics) must exist, thereby illuminating the interplay between curvature, causality, and global geodesic structure.
Abstract
We study the influence of the existence of totally geodesic null hypersurface on the properties of a Lorentzian manifold. By coupling the rigging technique with the existence of a null foliation we prove the existence of a Riemann flow structure which allows us to use powerful results to show how curvature conditions on the spacetime restricts its causal structure. We also study the existence of periodic null or spacelike geodesic.