  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.
