Compact spacelike surfaces in the 3-dimensional de Sitter space with non-degenerate second fundamental form. (Q1405277)

The authors study the geometry of a compact spacelike surface \(M\) in the de Sitter space \(S^3_1\). It is known that the totally umbilical round spheres are the only compact spacelike surfaces with constant Gaussian curvatute \(K < 1\). Using this result, they prove the following: For a compact spacelike surface \(M\) in \(S^3_1\) with Gaussian curvature \(K < 1\), the following assersions are equivalent: (i) \(K\) is constant (ii) The Gaussian curvature of the second fundamental form is constant. (iii) \(M\) is a totally umbilical round sphere.