"Bundle of metrics" on a principal bundle? I've come across the term "bundle of metrics" on a principal bundle. In particular, my setting is that for $N\longrightarrow M$ a universal cover of a compact Riemann surface, $P\longrightarrow M$ a principal $GL(n,\mathbb{C})$-bundle, we can consider the "bundle of metrics" on the pullback of P along the universal covering map of M.
I wasn't able to find any definitions for this bundle. Would somebody be able to explain it to me?
 A: Let $P\to M$ be a principal $GL(n,\mathbb C)$ bundle over some manifold $M$. The quick (and mysterious) definition of "the bundle of metrics on $P$" is the fibre bundle (not prinicipal) $P/U(n)\to M$, with fiber $GL(n,\mathbb C)/U(n)$, where $U(n)\subset GL(n,\mathbb C)$ is the subgroup of unitary matrices. 
I will sketch the reason for this name. Let $E\to M$ be the complex vector bundle associated to $P$ by  the standard action of $GL(n,\mathbb C)$ on $\mathbb C^n$. Then there is a bijection between the following: 
(1) hermitian metrics on $E$; 
(2) sections of $P/U(n)\to M$. 
The map (1) $\to$ (2) is the following. There is a bijection between $P$ and the set of frames of $E$, ie linear isomorphisms between $\mathbb C^n$ and some fiber $E_x$ of $E$, $x\in M$.  With an hermitian metric on $E$, associate the set of unitary frames of $E$, ie linear isomorphisms $\mathbb C^n\to E_x$ which map the standard hermitian metric on $\mathbb C^n$ to the given hermitian metric on $E_x$. 
There is also a 3rd class of standard objects, in bijection with the the above two, which is useful to consider: 
(3) reductions of the structure group of $P$ from $GL(n,\mathbb C)$ to  $U(n).$
Note: the bijection of (2) with (3) is valid for any principal $G$ bundle and a subgroup $H$. In many cases, such as ours, the  reduction from $G$ to $H$ can be interpreted as some  geometric structure on a fibre bundle associated with some action of $G$. 
I hope you can fill-in the details and that it answers your question. 
