Characterizations of complete spacelike submanifolds in the (n plus p)-dimensional anti-de Sitter space of index q

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