A Hermitian metric on a complex vector space $V$ is a map $h:V\times V\longrightarrow \mathbb{C}$ satisfying

1. $h$ is $\mathbb{R}$-bilinear
2. $h(\alpha u,v)=\alpha h(u,v)$
3. $h(u,\alpha v)=\overline{\alpha} h(u,v)$
4. $h(v,u)=\overline{h(u,v)}$
5. $h(u,u)\in\mathbb{R}$, $h(u,u)>0$ if $u\neq 0$

$\forall u$, $v\in V$, $\alpha\in\mathbb{C}$.

A Hermitian metric on a complex manifold $M$ is a map $h:T^{1,0}M\times T^{1,0}M\longrightarrow\mathbb{C}$ that for each point is a Hermitian metric $h_p:T^{1,0}_p M\times T^{1,0}_p M\longrightarrow\mathbb{C}$.

$h$ is locally determined by the coefficients $h_{jk}=h\left(\dfrac{\partial}{\partial z^j},\dfrac{\partial}{\partial z^k}\right)=\alpha_{jk}+\mathrm{i} \beta_{jk}$; from the Hermiticity we have that $\alpha_{jk}=\alpha_{kj}$, $\beta_{jk}=-\beta_{kj}$.

Likewise, $h$ induces a tensor $\tilde{h}:TM\times TM\longrightarrow \mathbb{C}$ $J$-Hermitian, in the sense that $\tilde{h}(Ju,v)=\mathrm{i}\tilde{h}(u,v)$, $\tilde{h}(u,Jv)=-\mathrm{i}\tilde{h}(u,v)$, defined by $\tilde{h}(u,v)=h(\theta(u),\theta(v))$.

Therefore, even when $h$ cannot be described as a tensor (it is not $\mathbb{C}$-linear), $\tilde{h}$ is indeed a $\mathbb{C}$-valued tensor over $TM$:

From $h$ we also obtain a Riemannian metric $g:TM\times TM\longrightarrow\mathbb{R}$, $g=\text{Re}(\tilde{h})$ and a 2-form $\omega:TM\times TM\longrightarrow\mathbb{R}$, $\omega=-\text{Im}(\tilde{h})$, called Kähler form

and the following relations hold

which may be verified after proving them right in the basis elements