G-structure on fiber bundles

I was studying about fiber bundles with a $$G$$-structure, and I arrive to the definition (below all the spaces are smooth manifolds):

Given a topological group $$G$$ a $$G$$-structure on fiber bundle $$(E,M,F)$$ with projection $$\pi:E\rightarrow M$$ and typical fiber $$G$$ (or a $$G$$-bundle) consists (roughly) on:

1) A faithful action of $$G$$ on the typical fiber $$F$$.

2) A bundle atlas (collection of open trivialization neighborhoods which cover the base space $$M$$) $$\{U_\alpha , \phi_\alpha \}_{\alpha \in \mathcal{A}}$$ with $$\phi_\alpha:\pi^{-1}(U_\alpha) \rightarrow U_\alpha \times F$$. It is usually called $$G$$-atlas.

3) For all $$\alpha,\beta \in \mathcal{A}$$ there exists a smooth map $$g_{\alpha,\beta}:U_\alpha \cap U_\beta \rightarrow G$$, called the transition function from $$\phi_\alpha$$ to $$\phi_\beta$$, such $$g_{\alpha,\beta}(p)=\phi_\beta \circ \phi_\alpha^{-1}(p)$$.

The first thing that bothers me is that in the above definition is important which is the $$G$$-atlas you first consider. I used to think that in differential geometry, given any atlas of a manifold $$M$$, if you change it by a bigger atlas (adding new coordinates charts compatible with de previous one) or by a smaller atlas (keeping a subset of charts which they still covers $$M$$), all the relevant topological and differential structure should not change (please let me known if this is incorrect). But, this is what not happen in the definition of the G-structure of a fiber bundle.

After all, the same fiber bundle may be equipped with many different G-structures, depending on the $$G$$-atlas you considered. E.g. I'm thinking a globally trivial vector bundle, with can be equipped by a trivial group structure according to the trivial group $$G=\{1\}$$ or by the $$GL(\mathbb{R^n})$$-structure.

Then, the two main questions are:

1) There is an "intrinsic" definition of G-structure on a fiber bundle such not depends on the $$G$$-atlas, i.e. some kind of "coordinate free" definition? If I'm well understanding, I hope such definition can't exist.

2) Given any fiber bundle $$(E,M,F)$$ (without a priori G-structure) and consider some bundle atlas of such fiber bundle. Now I can construct all the transition functions $$t_{\alpha\ \beta}=\phi_\alpha \circ \phi_\beta^{-1} : F \rightarrow F$$. Does the collection of such transition functions $${t_{\alpha\ \beta}}$$ define a G-structure on $$F$$ ? It may not define a topological group, but I expect that the algebraic group properties of the definition of G-structure on a fiber bundle must be satisfied. In other words, every fiber bundle admits a $$G$$-structure, but it usually depends on the bundle atlas considered.

Thanks,

D

What you are getting at, I think, is the idea of reduction of structure group. This is a very nice aspect of bundles actually. For example, if we only allowed some minimal group to act, then we could not consider unoriented vector bundles with structure group $$O(n)$$, and oriented bundles with structure group $$SO(n)$$ at the same time.

There is however a coordinate free definition of $$G$$-principal bundles, and then through those bundles a coordinate free definition of $$G$$ bundles.

One way to define a $$G$$ principal bundle is as a homotopy class of map $$f:M\to BG$$. Then pull back $$EG$$ along $$f$$. To get a general bundle with fiber a $$G$$-space $$F$$, simply define $$E=f^*(EG)\times_G F$$.

As for your second question, yes this gives a fibre bundle with structure group $$Aut(F)$$.

If you suppose that $$E$$ is a principal $$H$$-bundle and $$G\subset H$$, a $$G$$-structure is defined by a global section of $$E/H$$.

• Did you mean E/G? E/H isn’t even a bundle. Commented Mar 27 at 14:33