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 ℝ^n+p_q of index q , with 1≤ q≤ 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 ℝ^n+p_q be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in 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 ℝ^n+1_1 .