I have difficulties finding an appropriate reference for the following question:
Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of $(M^{2n},g)$ $$ \operatorname{ric}(x,y) = g( \operatorname{Ric}X,Y) = \sum g(R(X,e_{k})Y,e_k) $$ where $e_{k}$'s denote a local orthonormal basis. Let us denote by $\operatorname{ric}^{-J}$ the $J$-anti-invariant part of $ \operatorname{ric}$: $$ \operatorname{ric}^{-J}(X,Y)=g(\operatorname{Ric}^{-J}X,Y )=\frac{1}{2}(\operatorname{ric}(X,Y)-\operatorname{ric}(JX,JY)). $$
Let $\widetilde{\rho}$ be the $(0,2)$-tensor field on $M$ defined by $$ \widetilde{\rho}(X,Y) = \omega(\widetilde{\operatorname{P}}X,Y) = \rho^{\star}(X,Y)-\frac{1}{2}\phi(X,Y), $$ where $\rho^{\star}$ is the Star-Ricci form, $\rho^{\star}=R(\omega)$ and $\phi$ is the $J$-invariant, semi-positive 2-form given by $$ \phi(X,Y)=(\nabla_{JX}\omega,\nabla_{Y}\omega); $$ here, $\nabla$ is the Levi-Civita connection of $(M,g)$ and $(\cdot,\cdot)$ is the pointwise inner product induced by the metric $g$ on various bundles of tensors and forms.
My Question is: what is the geometric meaning of $$ (\widetilde{\operatorname{P}} +\operatorname{Ric}^{-J},\operatorname{Ric})? $$ Is there any "closed relation" between $\phi$ (or ${\widetilde{\operatorname{P}}} +\operatorname{Ric}^{-J}$) and $\operatorname{Ric}$?
I am really not an expert in symplectic geometry, so Thanks in advance for any help or suggestion.
Reference:
[1] Apostolov, Vestislav, Draghici. "The curvature and the integrability of almost-Kähler manifolds: a survey." Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 35 (2003). 25-53.
[2] Kirchberg. "Some integrability conditions for almost Kähler manifolds." Journal of Geometry and Physics 49.1 (2004). 101-115.
[3] Moroianu. "From Kirchberg's inequality to the Goldberg conjecture." Dirac operators: yesterday and today. Beirut, Int. Press, Somerville, MA, (2005). 283-292
