Following this note, today we talk about an almost-Schur lemma recently obtained by De Lellis and Topping, see here. If we denote by the traceless Ricci tensor, the main theorem of the paper is the following
Theorem. For any integer , if is a closed Riemannian manifold of dimension with nonnegative Ricci curvature, then
where is the average value of the scalar curvature over . Moreover equality holds if and only if is Einstein.
For a proof of the theorem, recall that the contracted second Bianchi identity tells us that
and hence that
Let be the unique solution to with . We may then compute
Now by integration by parts (i.e. the Bochner formula) we know that
and since the Ricci curvature is nonnegative, we have
which concludes the proof. For the equality case, we refer the reader to the original paper.
Using the identity mentioned in this note, one has the following estimate
and equality holds if and only if is Einstein.
Finally, they proved that the condition cannot be dropped as they constructed a counterexample for with . The case was also considered in the paper. Unfortunately, the case was left as an open question.