Some Remarks on Nijenhuis Bracket, Formality, and K\"ahler Manifolds

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $\delta$ such that $d\delta$-lemma holds. We first show that such derivation $\delta$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $\delta$ (which is a natural, essentially necessary condition to get a $d\delta$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $d\delta$-lemma holds: we consider the cases when $R$ is self adjoint with respect to a Riemannian metric or compatible with an almost symplectic structure. Finally, we show that if $R$ is scew-symmetric with respect to a Riemannian metric, has constant determinant, and if its Nijenhujis torsion vanishes, then the orthogonal component of $R$ in its polar decomposition is a complex structure compatible with the metric, which gives us a new characterization of K\"ahler structures