Rigging Technique for 1-Lightlike Submanifolds and Preferred Rigged Connections

Abstract

We extend the rigging technique to null submanifolds of a Lorentzian manifold, which allows us to construct two connections: the induced connection and the rigged connection. We study the relationship between them after changing the rigging and the existence and uniqueness of a preferred rigged connection, i.e., a rigged connection which coincides with the induced connection. As an application, we derive some integral formulas for compact null hypersurfaces with a preferred rigged connection and we obtain some consequences. For instance, we show that if the null mean curvature is zero, then the transverse null mean curvature vector field is orthogonal to any Killing vector field at some point.