Hello!

Let $E\rightarrow M$ be a holomorphic vector bundle. We denote by $\mathcal{E}$ its sheaf of holomorphic sections and by $\mathcal{O}$ the sheaf of holomorphic functions on $M$. We also denote by $\nabla$ the unique flat $T^{1,0}M$-connection on $E$ (it seems that this is a standard fact from complex geometry that such a connection always exists on a holomorphic vector bundle). Let $\langle\cdot,\cdot\rangle$ be a smoothly varying $\mathbb{C}$-valued symmetric and non degenerate $\mathbb{C}$-bilinear form on the fibers on $E$ (this give us a metric).

I would like to show that this metric induces a homomorphism of sheaves of $\mathcal{O}$-modules $\mathcal{E}\otimes_{\mathcal{O}}\mathcal{E}\rightarrow\mathcal{O}$ (that is the metric becomes $\mathcal{O}$-bilinear) is equivalent to the fact that the connection $\nabla$ is metric, that is: $$X\cdot\langle\phi,\psi\rangle=\langle\nabla_X\phi,\psi\rangle+\langle\phi,\nabla_X\psi\rangle$$ for any smooth sections $\phi$ and $\psi$ of $E$ and any anti-holomorphic vector field $X$ on $M$ (I mean a section of $T^{0,1}M$).

Note that here, **we do not assume that the metric is hermitian**. This is a lemma stated without proof in a article I am reading. Since it comes without proof it is certainly a standard or simple fact. Can somebody give me a hint or a reference?

Please tell me if I am saying something wrong too.

Thank you!