Abstract
Spacelike submanifolds usually appear in the study of questions related to causality in general relativity. In this paper, we study an n-dimensional spacelike submanifold in (n + p)-dimensional connected de Sitter space S^{n+p}_{q}(c) of index q (1leq q leq p) and of constant curvature c, and we obtain some integral inequalities of Simons type and rigidity theorems.
Highlights
During the last decades, the study of spacelike submanifolds in semi-Riemannian manifolds has got increasing interest motivated by their importance in problems related to Physics, such as the theory of general relativity
Under the assumption that the second fundamental form of M is locally timelike, Mariano [30] obtained some results of complete spacelike submanifold with parallel mean curvature vector in Sqn+p(c) (1 ≤ q < p)
We denote by ρ2 the nonnegative function ρ2 = S – nH2, where S and H are the norm square of the second fundamental form and the mean curvature vector of M, we see that ρ2 = 0 if and only if M is a totally umbilical spacelike submanifold
Summary
The study of spacelike submanifolds in semi-Riemannian manifolds has got increasing interest motivated by their importance in problems related to Physics, such as the theory of general relativity. [10] to complete spacelike submanifolds with parallel mean curvature vector fields in de Sitter space Spn+p.
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