A note on rigidity of Riemannian manifolds with positive scalar curvature

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 σ2(Aτ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _2(A^{\tau })$$\end{document}, which can be seen as a complement to Bo and Sheng who considered conformally flat manifolds with constant quotient curvature of σk(Aτ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _k(A^{\tau })$$\end{document}.