New Characterizations of Compact Totally Umbilical Spacelike Surfaces in 4-Dimensional Lorentz–Minkowski Spacetime Through a Lightcone

Abstract

On each spacelike surface through the lightcone in 4-dimensional Lorentz–Minkowski spacetime, there exists an Artinian normal frame which contains the position vector field. In this way, a (globally defined) lightlike normal vector field, with nontrivial extrinsic meaning, is chosen on the surface. When the second fundamental form respect to that normal direction is non-degenerate, a new formula which relates the Gauss curvature of the induced metric and the Gauss curvature of this normal metric is obtained. Then, the totally umbilical round spheres are characterized as the only compact spacelike surfaces through the lightcone whose normal metric has constant Gauss curvature two. Such surfaces are also distinguished in terms of the Gauss–Kronecker curvature of that lightlike normal direction, of the area of the normal metric and of the first non-trivial eigenvalue of the Laplacian of the induced metric.