# Intrinsic description of fiber bundle with given structure group

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?

• To my knowledge there is always a structure group associated just by taking $\mathrm{Homeo}(F)$. Oct 7, 2018 at 0:53
• You want to endow this with a good topology I guess (compact open in this case.) Oct 7, 2018 at 0:57
• 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. Oct 7, 2018 at 0:59

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.
• 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 $P\times_GF$? Oct 7, 2018 at 1:33