Spacelike surfaces which admit a nondegenerate null normal section in a Lorentzian space form

Abstract

In this paper, we develop a formula for spacelike surfaces in a 4- dimensional Lorentzian space form which involves its mean curvature vector eld and the Gauss curvature of the induced metric and the Gauss curvature of the second fundamental form associated to a non- degenerate null normal section. By means of this formula, we stablish several su cient conditions for compact spacelike surfaces with con- stant Gauss curvature to have a null umbilical direction. As another application, we give a new proof of the Liebmann rigidity theorem in Euclidean, hemispherical and hyperbolic spaces, and in the De Sitter spacetime.