Let $(M,g,J)$ be an **almost Kähler manifold** and let $\rho$ denote its *Ricci form*
$$
\rho(X,Y) = \operatorname{ric}^{\mbox{c}}(JX,Y)
$$
where $\operatorname{ric}^{\mbox{c}}$ is the $J$-invariant part of the Ricci tensor of the Riemannian manifold $(M,g)$.

If $\rho(X,JY)=\operatorname{ric}(X,Y)$ (i.e. $(M,g,J)$ has *$J$-invariant Ricci tensor*), is it true that $\rho$ is a closed $2$-form?

A second question is:

what are well known results from the theory of Kähler manifolds valid in almost Kahler manifolds with $J$-invariant Ricci tensor?