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Complete space-like submanifolds in a de Sitter space with parallel mean curvature vector. (English) Zbl 0695.53042
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.
Reviewer: Cheng Qingming

MSC:
53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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