# Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow k(x)$, where $k(x)$ denotes the residue field at $x$. This defines another locally free sheaf $\mathcal{F}:=ker(f)$ of rank 2. We have the exact sequence $0\rightarrow \mathcal{F}\rightarrow \mathcal{E}\rightarrow k(x)\rightarrow 0$.

Now $\mathcal{E}$ and $\mathcal{F}$ induce vector bundles $E$ and $F$ of rank 2 on the Riemann surface $X(\mathbb{C})$.

If $h$ is a Hermitian metric on $E$, is there a way to (canonically?) induce a Hermitian metric $h'$ on $F$?

Unfortunately $F$ is not a subbundle of $E$, so i think one cannot simply use the restriction of $h$ to $F$, i.e. $h'=h_{F\hookrightarrow E}$, because the exact sequence does not give an exact sequence of vector bundles. Maybe one can somehow use the trivial Hermitian metric on $k(x)=\mathbb{C}$ and the metric $h$ to construct some Hermitian metric on $F$?

Or is there no canonical way to get a hermitian metric on $F$ just using $h$? Are there any other constructions that would give a reasonable Hermitian metric on $F$ depending on $h$?

Are the associated vector bundles $E$ and $F$ maybe too different although the locally free sheaves only differ at the one point $x$?

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