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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.

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    $\begingroup$ 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 $\endgroup$ – YangMills Jun 12 at 23:50
  • $\begingroup$ In particular see Proposition 2 in the above paper which shows exactly what you can have as the torsion of $D$ $\endgroup$ – YangMills Jun 12 at 23:52
  • $\begingroup$ @YangMills thank you for the reference which I never noticed before and thanks for pointing out my mistakes. Also, I edit my question. $\endgroup$ – Ying Xie Jul 10 at 12:50
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    $\begingroup$ 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 $\endgroup$ – YangMills Jul 15 at 20:52
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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.

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