We study the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere \({\mathbb {S}}^{n+1}\). In particular, we establish a sharp integral inequality for the behavior of the norm of the total umbilicity tensor, equality characterizing the totally umbilical hypersurfaces, and a certain family of standard tori of the form \({\mathbb {S}}^1(\sqrt{1-r^2})\times {\mathbb {S}}^{n-1}(r)\). Moreover, under an appropriate constraint on the total umbilicity tensor, we are able to extend this result for any integer k, with \(2\le k\le n-1\), equality characterizing the totally umbilical hypersurfaces and a certain family of standard product of spheres of the form \({\mathbb {S}}^k(\sqrt{1-r^2})\times {\mathbb {S}}^{n-k}(r)\).

中文翻译：

---

欧氏球面中具有恒定标量曲率的紧致超曲面的积分不等式

我们研究了等量浸入单位欧几里得球体\（{\ mathbb {S}} ^ {n + 1} \）中的具有恒定标量曲率的紧致定向超曲面的刚度。特别是，我们为总脐带张量的范数的行为建立了尖锐的积分不等式，其特征在于完全脐带超曲面的相等性，以及形式为\（{\ mathbb {S}} ^ 1（ \ sqrt {1-r ^ 2}）\次{\ mathbb {S}} ^ {n-1}（r）\）。而且，在适当的总脐度张量约束下，我们可以将这个结果扩展到任何整数k，其中\（2 \ le k \ le n-1 \），等式表征了全部脐带超曲面和的某些族形式球的标准产品\（{\ mathbb {S}} ^ k（\ sqrt {1-r ^ 2}）\次{\ mathbb {S}} ^ {nk}（r）\）。