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In our previous paper in \cite{C}, we generalized the almost-Schur lemma ofDe Lellis and Topping for closed manifolds with nonnegative Rcci curvature toany closed manifolds. In this paper, we generalize the above results tosymmetric $(2,0)$-tensors and give the applications including $r$th meancurvatures of closed hypersurfaces in a space form and $k$ scalar curvaturesfor closed locally conformally flat manifolds.

An almost-Schur type lemma for symmetric $(2,0)$ tensors and applications