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

TL;DR

This paper investigates codimension-two spacelike submanifolds in Lorentzian spacetimes with umbilical lightlike normals, revealing their geometric constraints, topological restrictions, and their relation to lightlike hypersurfaces, with implications for general relativity.

Contribution

It provides a detailed geometric characterization of such submanifolds, including a factorization theorem and conformal relations, advancing understanding of lightlike structures in Lorentzian geometry.

Findings

Submanifolds are contained in totally umbilical lightlike hypersurfaces.

Sharp topological restrictions are derived for compact cases.

Conformal invariance of the parallel rescaling of lightlike normals is established.

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: any such submanifold is contained in a lightlike hypersurface, which is totally umbilical whenever the lightlike normal direction is umbilical. 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.