Induced Riemannian structures on null hypersurfaces

TL;DR

This paper introduces a method to induce Riemannian metrics on null hypersurfaces within Lorentzian manifolds using a fixed transverse vector field, enabling new and existing results to be derived through Riemannian techniques.

Contribution

It constructs a Riemannian metric on null hypersurfaces from a transverse vector field and explores its relationship with the ambient Lorentzian structure, providing new tools for null hypersurface analysis.

Findings

New results on null hypersurfaces derived using Riemannian methods

Established relationships between Lorentzian and Riemannian structures on hypersurfaces

Provided a framework for analyzing null hypersurfaces via induced Riemannian metrics

Abstract

Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $\zeta$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $\zeta$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.