Let $M$ be a smooth manifold (may be almost complex, almost Kahler, Kahler..). and Let $\phi : T^*M \rightarrow T^*M$ be a cotangent bundle automorphism. (the restriction of $\phi$ on the base $M$ is just the identity) Then it maps 1-forms to 1-forms. Moreover, $\phi$ can be canonically extended to the isomorphism of $\Lambda T^*M$, bundle of differential forms.

Question. Is there any criterion whether $\phi$ commutes with $d$ or not? ( I mean, I want to know when $\phi$ gives an isomorphism of de Rham cohomology ring $H^*(M)$)

identity. But there's still a question to answer there since there can be several of those. $\endgroup$4more comments