Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.
(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)
Assume we have an isomorphism $f: E\otimes F \stackrel{\sim}{\longrightarrow} G$.
If $E$ and $G$ are equipped with Hermitian structures, that is we have $(E,h)$ and $(G,h'')$ where $h$ and $h''$ are positive definite Hermitian bilinear forms on $E$ resp. $G$.
Can we always find a Hermitian structure $h'\in Herm^{+}(F)$, depending on $h$ and $h''$, such that $f$ gets an isometry? That is: do we get an isometry:
$\tilde{f}: (E\otimes F,h\otimes h') \stackrel{\sim}{\longrightarrow} (G,h'')$?
Otherwise stated: can we solve the equation $h\otimes x=h''$ for $x\in Herm^{+}(F)$?
Edit: What if we only have the metric $h''$? Can we find $x,y$ with $x\otimes y\cong h''$?
There is the following situation:
We have a sheaf of $O_M$-algebras $R$ and two $R$-modules $S,T$ such that $R,S,T$ and $\mathcal{H}om_R(S,T)$ are locally free over $M$. Furthermore we have an isomorphism of sheaves $\mathcal{H}om_{O_M}(S,T)\cong \mathcal{H}om_{O_M}(R,O_M)\otimes_{O_M} \mathcal{H}om_R(S,T)$. Now we have metrics on the bundles associated to $S$ and $T$. Can we find metrics on the bundles associated to $\mathcal{H}om_{O_M}(R,O_M)$ $\mathcal{H}om_R(S,T)$ such that the isomorphism on the associated bundles gets an isometry? Can we do something better/easier in this special situation? Or is there no hope for anything in this situation?