Codimension-Two Spacelike Submanifolds with Umbilical Lightlike Normal Sections and Their Relationship to Lightlike Hypersurfaces

Abstract

We study codimension-two spacelike submanifolds in Lorentzian spacetimes that admit umbilical lightlike normal directions. We show that such submanifolds are subject to strong geometric and topological constraints, establishing explicit relationships between extrinsic geometry, mean curvature, and shear-isotropy. In the compact case, we obtain sharp restrictions on their topology. We precisely characterize when the umbilical lightlike normal vector field can be rescaled to be parallel, in terms of the curvature tensor of the ambient spacetime, and prove that this property is conformally invariant. Our main result is a factorization theorem: an embedded codimension-two spacelike submanifold carrying a lightlike normal vector field is contained, through the normal exponential construction, in a lightlike hypersurface endowed with a geodesic lightlike extension and an integrable screen distribution. The generated hypersurface is totally umbilical precisely when every leaf of this screen distribution is umbilical along the restriction of the lightlike generator. In particular, an umbilical lightlike normal section generates a totally umbilical lightlike hypersurface whenever the ambient spacetime is locally conformally flat. We also provide explicit conformal relations between the induced metrics on the family of spacelike leaves generated by the lightlike normal flow, with consequences for isometry, parallelism of lightlike normal directions, volume evolution, and variational properties. These results yield a detailed geometric framework relevant to the mathematical study of horizons and lightlike structures in general relativity.