# Hermitian metrics on the Associated Vector Bundle of a Principal $U(1)$-bundle

differential-geometryprincipal-bundlesvector-bundles

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?

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.