Compact spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime with a non-degenerate lightlike normal direction

TL;DR

This paper characterizes compact spacelike surfaces in four-dimensional Lorentz-Minkowski spacetime that pass through the lightcone, showing they must be totally umbilical round spheres under certain conditions involving the lightlike normal vector and curvature relations.

Contribution

It introduces new formulas relating Gauss curvatures and characterizes totally umbilical spheres via the $ ext{eta}$-second fundamental form and curvature conditions.

Findings

Surfaces with non-degenerate $ ext{eta}$-second fundamental form are totally umbilical spheres.

A new curvature relation formula is established for these surfaces.

Characterizations involve Gauss-Kronecker curvature and area of the $ ext{eta}$-second fundamental form.

Abstract

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $\eta$. Several sufficient assumptions on such a surface with non-degenerate $\eta$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With this aim, a new formula which relates the Gauss curvatures of the induced metric and of the $\eta$-second fundamental form is developed. Then, totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone such that its $\eta$-second fundamental form is non-degenerate and has constant Gauss curvature two. Another characterizations of totally umbilical round spheres in terms of the Gauss-Kronecker curvature of $\eta$ and the area of the $\eta$-second fundamental form are also given.