Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space ({mathbb{R}^{n+p}_{q}}) of index q, with ({1leq qleq p}). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of ({mathbb{R}^{n+p}_{q}}) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in ({mathbb{R}^{n+p}_{p}}) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space ({mathbb{R}^{n+1}_{1}}).