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 \(\mathbb S_p^{n+p}\) of index p, and we use it to characterize totally umbilical round spheres \(\mathbb S^n(r)\), with \(r>1\), of \(\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}\). 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.

在本文中，我们建立了具有恒定标量曲率的n维封闭空间类子流形的尖锐积分不等式，并将其浸入索引p的de Sitter空间\（\ mathbb S_p ^ {n + p} \）中的平行归一化平均曲率矢量场的，我们用它来表征全脐圆球体\（\ mathbb小号^ N（R）\） ，用\（R> 1 \），的\（\ mathbb S_1 ^ {N + 1} \ hookrightarrow \ mathbb S_P ^ {n + p} \）。我们的方法是基于对Cheng-Yau算子的适当较低估计，该算子作用于此类空子流形的无痕第二基本形式的平方范数。