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Complete space-like submanifolds with constant scalar curvature in a de Sitter space. (English) Zbl 1071.53034

A de Sitter space \(S_p^{n+p}(c)\) is an \((n+p)\)-dimensional connected complete pseudo-Riemannian manifold of index \(p\) with constant curvature \(c>0\). Complete space-like submanifolds \(M^n\) (\(n\geqslant3\)) with constant normalized scalar curvature and with parallel normalized mean curvature vector field in \(S_p^{n+p}(c)\) are studied. It is proven that under certain restrictions for the square of the second fundamental form, the submanifold is totally umbilical, or \(n=3\) and \(M^3\) is a hyperbolic cylinder in \(S_1^4\). The result is an extension of a theorem proven by X. Liu [Manuscr. Math. 105, No.3, 367–377 (2001; Zbl 1002.53043)].

MSC:

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

Citations:

Zbl 1002.53043
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