In this short note, we obtain an integral inequality for closed Riemannian manifolds with positive scalar curvature and give some rigidity characterization of the equality case, which generalizes the recent results of Catino which deal with the conformally flat case, and of Huang and Ma which deal with the harmonic curvature case. Moreover, we obtain an integral pinching condition with non-negative constant $$\sigma _2(A^{\tau })$$ σ 2 ( A τ ) , which can be seen as a complement to Bo and Sheng who considered conformally flat manifolds with constant quotient curvature of $$\sigma _k(A^{\tau })$$ σ k ( A τ ) .

中文翻译：

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关于正标量曲率黎曼流形刚性的注记

在这个简短的说明中，我们获得了具有正标量曲率的封闭黎曼流形的积分不等式，并给出了等式情况的一些刚性特征，它概括了处理共形平坦情况的 Catino 的最新结果，以及处理保形平面情况的 Huang 和 Ma 的最新结果与谐波曲率情况。此外，我们获得了一个具有非负常数 $$\sigma _2(A^{\tau })$$ σ 2 ( A τ ) 的积分捏合条件，这可以看作是对考虑共形平坦的 Bo 和 Sheng 的补充具有恒定商曲率 $$\sigma _k(A^{\tau })$$ σ k ( A τ ) 的流形。