Characterizations of complete spacelike submanifolds in the $$\mathbf{(n+p)}$$ ( n + p ) -dimensional anti-de Sitter space of index $$\mathbf{q}$$ q

Abstract

Our purpose in this paper is to study the geometry of n-dimensional complete spacelike submanifolds immersed in the $$(n+p)$$ -dimensional anti-de Sitter space $$\mathbb {H}^{n+p}_{q}$$ of index q, with $$1\le q\le p$$ . Under suitable constraints on the Ricci curvature and the second fundamental form, we show that a complete maximal spacelike submanifold of $$\mathbb {H}^{n+p}_{q}$$ must be totally geodesic. Furthermore, we establish sufficient conditions to guarantee that a complete spacelike submanifold with nonzero parallel mean curvature vector in $$\mathbb {H}^{n+p}_{p}$$ must be pseudo-umbilical, which means that its mean curvature vector is an umbilical direction.