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

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.