When is an almost Hermitian manifold almost Kähler?

Let $$(M, J, h)$$ be an almost Hermitian manifold, where $$J$$ is an almost complex structure and $$h$$ is a Hermitian metric. Let $$D$$ be the unique $$h$$-connection compatible with $$J$$, i.e. $$Dh = 0$$ and $$(0,1)$$-part of $$D$$ coincides with the Cauchy-Riemann operator $$\overline{\partial}_J$$. Let $$\tau$$ be the torsion of $$D$$. If we decompose our connection $$D$$ into (1,0) and (0,1) parts $$D = D' + D''$$, then the torsion $$\tau$$ of $$D$$ will also be decomposed $$\tau = \tau' + \tau''$$. It is not hard to see that $$\tau' = N$$, the Nijenhuis tensor for $$J$$, which is exactly the obstruction for an integrable complex structure. What about the other part $$\tau''$$? Is there any geometric meaning?

So far, I find that it will be an obstruction for being almost Kähler, i.e. $$d\omega=0$$, where $$\Im h:=\omega$$. I mean, the following holds: $$d\omega=0 \Rightarrow \tau''=0$$.

My question is about the converse. Is it true that $$\tau''=0 \Rightarrow d\omega=0$$?

By the way, if $$M$$ itself a Hermitian manifold, it is well-known that $$\tau''=\tau$$, which will be the exact obstruction for being Kähler.

• This question starts from a wrong premise: it is not true that there is a unique linear connection $D$ compatible with $h$ and $J$! This is well-known material, see e.g. this paper of Gauduchon: verbit.ru/MATH/Gauduchon_Hermitian_Operators01.pdf Jun 12 '19 at 23:50
• In particular see Proposition 2 in the above paper which shows exactly what you can have as the torsion of $D$ Jun 12 '19 at 23:52
• @YangMills thank you for the reference which I never noticed before and thanks for pointing out my mistakes. Also, I edit my question. Jul 10 '19 at 12:50
• The answer to the edited question is well-known: $\tau''=0$ is equivalent to $(d\omega)^{(2,1)}=0$, which is a condition known as quasi-Kahler. See e.g. Lemma 2.4 here arxiv.org/pdf/math/0703773.pdf Jul 15 '19 at 20:52

Gray and Hervella (Annali di Matematica pura ed applicata (1980) 123: 35) showed that (in dimension $$\geq 6$$) there are sixteen classes of almost hermitian manifolds, of which almost Kähler is one.
Let $$(M,g,J)$$ be an almost hermitian manifold of dimension $$\geq 4$$ and let $$\omega(X,Y) := g(JX, Y)$$. Then $$(M,g,J)$$ is almost Kähler if and only if $$d\omega = 0$$. See Table I (for dimension $$\geq 6$$) and Table II (for dimension $$=4$$) in the paper.