Nomizu-Smyth type results in the hyperbolic and De Sitter spaces via Gauss-Kronecker curvature (Q2051036)

The authors deal with extrinsic geometry of hypersurfaces \(\Sigma\) of either the hyperbolic space \(\mathbb{H}^{n+1}\) or of the de Sitter space \(\mathbb{S}^{n+1}_1\). In the second case, they show that such \(\Sigma\) is totally umbilical whenever \(\Sigma\) is complete, spacelike, its Gauss map sends \(\Sigma\) into a totally umbilical hypersurface of \(\mathbb{H}^{n+1}\) and its Gauss-Kronecker curvature does not vanish at some points of \(\Sigma\). The result in the first case is similar.