# Hermitian metric as a section of the bundle $(E\otimes \bar{E})^*$

If $$M$$ is a Riemannian manifold and $$E \to M$$ a Riemannian bundle, then the Riemannian metric $$g$$ can be viewed as a section of the bundle $$\bigotimes^2 T^*M$$, which means that it is a $$(0,2)$$-tensor field.

If we instead consider a complex vector bundle $$E$$ over $$M$$ with a hermitian metric $$h$$, this wikipedia article claims that $$h$$ is a section of the bundle $$(E\otimes \bar{E})^*$$. Can someone here elaborate on why should $$h$$ be a section of this proposed bundle?

A Hermitian metric on a complex vector bundle $$E$$ over a smooth manifold $$M$$ is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be viewed as a smooth global section $$h$$ of the vector bundle $$(E \otimes \bar{E})^*$$ such that for every point $$p$$ in $$M$$, $$h_p(\eta, \bar{\zeta}) = \overline{h_p(\zeta, \bar{\eta})}$$ for all $$\zeta, \eta$$ in the fibre $$E_p$$ and $$h_p(\zeta, \bar{\zeta}) > 0$$ for all nonzero $$\zeta$$ in $$E_p$$.
A section of $$(E \otimes \bar{E})^*$$ alone is a sesquilinear form on $$E$$, where "sesquilinear form" in the linear algebra setting is what we call a linear map $$V \otimes \bar{V} \to \mathbb{C}$$ where $$V$$ is a complex vector space (equivalently, a bilinear map $$V \times \bar{V} \to \mathbb{C}$$). "Section" here encodes "smoothly varying," as with Riemannian metrics. The first condition in the quote above is then exactly the condition we pose on a sesquilinear form to make it into Hermitian, and the second encodes positive-definiteness.