Nomizu-Smyth type results in the hyperbolic and de Sitter spaces via Gauss-Kronecker curvature

In this paper, our purpose is to establish new Nomizu-Smyth type results concerning complete Riemannian immersions in the hyperbolic and de Sitter spaces. Assuming that the image of the Gauss map of such an immersion lies in a totally umbilical hypersurface and supposing the existence of a point of the immersion where the Gauss-Kronecker curvature does not vanish, we prove that the immersion must be a totally umbilical hypersurface of the ambient space.