Hermitian metrics on the Associated Vector Bundle of a Principal $U(1)$-bundle If a complex line bundle $L$ over some manifold $M$ has a Hermitian metric, then $M$ has a frame bundle, which is a principal $U(1)$-bundle over $M$.
Now reverse this, if we have a principal $U(1)$-bundle over $M$, we then have an associated line bundle $L$. Does this induce a Hermitian metric on $L$? Is it canonical?
 A: Yes it does. More generally, let $\overline{\mathbb{C}}^n$  be
${\mathbb{C}}^n$ with the conjugate complex structure, then the standard hermitian metric on $\mathbb C^n$ is an element $h_0$ of the vector space of sesquilinear forms  $S(n) = (\mathbb C^n)^*\otimes_{\mathbb C} (\overline{\mathbb{C}}^n)^*$.
$U(n)$ acts on $S(n)$  by $A\cdot s(X,Y) = s(A^{-1}X, A^{-1}Y)$ and, most importantly,  fixes $h_0$.
Consider now a principal $U(n)$ bundle $P$, then we have the associated vector bundles $E = P\times_{U(n)} \mathbb C^n$ and $T= P\times_{U(n)}S(n)$.
Since $h_0$ is fixed by $U(n)$, it defines a section of $T$, this is defined by $h_0$ in a local frame of  $T$ induced by a local section of $P$,  and the definition globalizes because $U(n)$ fixes $h_0$ (just check how things transform using the transformation functions of $T$ induced by $P$, see also this question Associated bundles).
$h$ gives an hermitian metric on $E$ because $T$ is naturally isomorphic to the bundle of sesquilinear forms of $E$.
Moreover, $h$ will also be parallel with respect to any connection on $T$ induced by a connection on $P$, see for example the question Tensor fields defining $G$-structure are parallel.
