A Hermitian metric $H$ on a complex vector bundle $E$ is a smooth family of Hermitian products on each fiber $E_x$ satisfying
- $H(u,v)$ is $\mathbb{C}$-linear in $u$;
- $H(u,v) = \overline{H(v,u)}$ for all $u,v\in E_x$;
- $H(u,u)>0$ for all $u\neq 0$.
A Hermitian metric on a almost complex manifold $(M,J)$ is a Riemannian metric $h$ such that $h(X,Y) = h(JX,JY)$ for all $X,Y\in TM$, and we can extend it $\mathbb{C}$-linarly to the complexified tangent bundle $TM^{\mathbb{C}}$.
I am confused that it seems the Hermitian metric $H$ on $TM^{\mathbb{C}}$ as a complex vector bundle does not coincide with the (complexified) Hermitian metric $h$ on $(M,J)$, since $H$ is $\mathbb{C}$-aitilinear in the second variable while $h$ is not. So how to relate these two Hermitian metrics?