Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector (Q910024)

We show that a complete space-like submanifold in a de Sitter space with parallel mean curvature vector is totally umbilical if the following are satisfied: 1) \(H^ 2\leq c\), when \(n=2\); 2) \(n^ 2H^ 2<4(n-1)c\), when \(n\geq 3\), where H is mean curvature. Conditions 1) and 2) are best possible.