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I'm in the process of re-learning some differential geometry after a couple years of not doing math professionally and I'm finding myself stuck on what's probably a pretty simple point. I'm going to spell out where I think I am right now; please tell me if something is wrong. For simplicity, let's say $G$ is a Lie group and all the spaces we're talking about are smooth manifolds. Fix forever a manifold $F$ and a faithful action of $G$ on $F$.

Suppose we're given a principal $G$-bundle $P$ over some manifold $B$. We can form an associated "fiber bundle with fiber $F$ and structure group $G$" by taking $P\times_GF=(P\times F)/\sim$, where $(p.g,f)\sim(p,g.f)$.

Alternatively, you can see whether a fiber bundle $E$ arises in this way by asking whether there is a family of trivializations of $E$ for which the corresponding transition functions can be taken to land in $G$.

The question is this: is "having structure group $G$" just a property of fiber bundles, or is a "fiber bundle with structure group $G$" a fiber bundle with some extra structure? If the latter is true, is there a nice description of what this extra structure is that doesn't depend on a family of trivializations? Is it just a chosen isomorphism of $E$ with $P\times_GF$ for some $P$? If this is true, can there be two nonisomorphic principal bundles $P,P'$ for which $P\times_GF$ and $P'\times_GF$ (for the same action on $F$) are isomorphic as (ordinary, non-$G$) fiber bundles?

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  • $\begingroup$ To my knowledge there is always a structure group associated just by taking $\mathrm{Homeo}(F)$. $\endgroup$ Oct 7, 2018 at 0:53
  • $\begingroup$ You want to endow this with a good topology I guess (compact open in this case.) $\endgroup$ Oct 7, 2018 at 0:57
  • $\begingroup$ I'm not sure that you can have some "transition-free" description of the structure group, since the structure group is not unique. I'm a novice though, hopefully someone with a mature point of view can chime in. $\endgroup$ Oct 7, 2018 at 0:59

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The example you must have in mind is an $n$-dimensional manifold $M$ and its tangent bundle $TM$. $TM$ can be obtained from the bundle of frames and $\mathbb{R}^n$ where $Gl(n)$ acts naturally on $\mathbb{R}^n$.

There is a notion of $G$-structure defined on $M$ $G\subset Gl(n)$; this is equivalent to saying that $TM$ has a $G$-reduction; that is the coordinate change of $TM$ tak their values in $G$, a Riemannian manifold is an $O(n)$-reduction. Definitively, there can exists two $G$-structures defined on the same manifold which are not isomorphic.

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  • $\begingroup$ I see, I think this might address the confusion. So two different metrics on the same manifold will be associated to two principal $O(n)$-bundles which are very much not isomorphic, but which do give the same fiber bundle when I apply the associated bundle construction --- namely, the tangent bundle of the original manifold. Is it then just the case that the "additional structure" I'm looking for is just a choice of isomorphism of $E$ with $P\times_GF$? $\endgroup$ Oct 7, 2018 at 1:33

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