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