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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?

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As P is a principal bundle, the fibre of P is isomorphic to $GL(n,\mathbb{C})$. This isn't a vector space, though. I know what a hermitian metric on P is, namely a reduction to $SU(n)$ - this comes from what happens on frame bundles, where the choice of a hermitian metric on a complex vector bundle $E$ gives a reduction of Frame(E) to SU(n). I don't see how a "bundle of reductions" works out. –  David Hornshaw May 7 at 9:45

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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.

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How would one go about to show the bijection between (2) and (3)? –  David Hornshaw May 7 at 16:49
    
Its basically the definition of reduction. See any textbook, eg Kobayashi Nomitzu. –  Gil Bor May 7 at 16:55
    
I think what confused me was that I didn't know how to do this with local sections. So (2) should be global sections of P/H, then its true. Thank you for the answer, I've accepted it. –  David Hornshaw May 8 at 5:40
    
You are welcome. Yes, by sections of $P/H\to M$ I means global sections. –  Gil Bor May 9 at 4:15

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