Null Hypersurfaces on Lorentzian Manifolds and Rigging Techniques

Abstract

We introduce the concept of rigging for a null hypersurface in a Lorentzian manifold, which allows us to induce all the necessary geometric objects in a null hypersurface and also to define a Riemannian metric on it, called rigged metric. This metric can be used as an auxiliary tool to study the null hypersurface. Its Levi-Civita connection is called rigged connection and, in general, it will not coincide with the induced connection on the null hypersurface. We show a necessary and sufficient condition for this to happen and we give some examples. Since both rigged connection as induced connection depend on the rigging, we investigate if they can coincide for a suitable choice of the rigging.