Conditions on a null hypersurface of a Lorentzian manifold to be a null cone

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We prove that a totally umbilic null hypersurface whose Ricci curvature vanishes on null tangent vectors is contained in a null cone. For this, we use the rigging technique to induce a Riemannian metric on the null hypersurface, which is locally decomposed as a warped product with a one-dimensional base. This allows us to prove that all null geodesics in the null hypersurface pass through a common point.