A sharp integral inequality for closed spacelike submanifolds immersed in the de Sitter space

In this paper, we establish a sharp integral inequality for n -dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space S p n + p of index p , and we use it to characterize totally umbilical round spheres S n ( r ) , with r > 1 , of S 1 n + 1 ↪ S p n + p . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.