Space-like submanifolds with parallel normalized mean curvature vector field in de Sitter space

Space-like submanifolds, with dimension greater than three and with negative definite normal bundle in a general de Sitter space, of any index, are studied. For the compact space-like submanifolds whose mean curvature has no zero and the corresponding normalized vector field is parallel, under natural boundedness assumptions on the lengths of the gradient of the length of the mean curvature and the covariant derivative of the second fundamental form, it is proved that they must be totally umbilical. As an application, two characterizations of totally umbilical space-like submanifolds in terms of the scalar curvature and the length of its second fundamental form are given. All the results extend the previous ones obtained by Liu for the case of space-like hypersurfaces in de Sitter space of index one. In addition, for the complete space-like submanifolds, whose normalized mean curvature vector field is parallel, two characterizations of totally umbilical space-like submanifolds and hyperbolic cylinders are obtained.