Characterizations of Spacelike Submanifolds with Constant Scalar Curvature in the de Sitter Space

Abstract

In this paper, we deal with complete spacelike submanifolds $$M^n$$ immersed in the de Sitter space $$\mathbb S_p^{n+p}$$ of index p with parallel normalized mean curvature vector and constant scalar curvature R. Imposing a suitable restriction on the values of R, we apply a maximum principle for the so-called Cheng–Yau operator L, which enables us to show that either such a submanifold must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only the submanifold is isometric to a hyperbolic cylinder of the ambient space. In particular, when $$n=2$$ this provides a nice characterization of the totally umbilical spacelike surfaces of $$\mathbb {S}^{2+p}_p$$ with codimension $$p\ge 2$$ . Furthermore, we also study the case in which these spacelike submanifold are L-parabolic.