hermitean structure vs hermitean metric Define a hermitean structure $H$ on a complex linear space $V$ as $H: V\times V \to \mathbb{C}$ s.t.

i. $H(u,v)$ is $\mathbb{C}$-linear in $u$ for every $v \in V,$

ii. $H(u,v)=\overline{H(v,u)}$

iii. $H(u,u)&gt; 0\quad \forall u\neq 0.$

Define a hermitean metric on an almost complex manifold $M$ as $h: \mathbb{C}T_xM \times \mathbb{C}T_xM \to \mathbb{C}$ that depends smoothly on the point $x$ such that

i. $h(\overline{Z},\overline{W})=\overline{h(Z,W)} \quad \forall Z,W\in T_\mathbb{C}M$

ii. $h(Z,\overline{Z})&gt;0 \quad \forall Z&gt;0$

iii. $h(Z,W)=0 \quad \forall Z,W\in T^{1,0}M \quad \text{and} \quad \forall Z,W\in T^{0,1}M$

My question is: how do I prove that
$$h_{i\overline{j}}:=h\left(\frac{\partial}{\partial z^i},\frac{\partial}{\partial \overline{z}^j}\right)=\overline{h_{j\overline{i}}} \ ?$$
Using h.i and the symmetry of $h$ (h must be Riemannian metric itself) we have
the thesis.

Moreover, the correct relation is $h=\operatorname{Re}(H).$
Using h.i and the symmetry of $h$ (h must be Riemannian metric itself) we have
the thesis.

Moreover, the correct relation is $h=\operatorname{Re}(H).$