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 q n + p (c) of index q (1 ≤ q ≤ p ) and of constant curvature c, and we obtain some integral inequalities of Simons type and rigidity theorems.