Compact spacelike surfaces in the 3-dimensional de Sitter space with non-degenerate second fundamental form

Abstract

In this paper, we establish several sufficient conditions for a compact spacelike surface with non-degenerate second fundamental form in the 3-dimensional de Sitter space to be spherical. With this aim, we develop a formula for these surfaces which involves the mean and Gaussian curvatures of the first fundamental form and the Gaussian curvature of the second fundamental form. By means of that formula, we prove, for instance, that the totally umbilical round spheres are the only compact spacelike surfaces such that the second fundamental form is non-degenerate and has constant Gaussian curvature.