Definition of isomorphism classes of vector bundles with reduced structure group

Question

I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq GL(n)$. For concreteness, let us work in the smooth category with $G$ a Lie group. This idea of classifying vector bundles with reduced structure group up to isomorphism seems to be an often-mentioned concept, and is closely related to the homotopy classes of maps $X\rightarrow BG$, and to the first Cech cohomology $H^1(X,G)$, but I have so far not seen it defined using just concepts from fibre bundles.

My attempt at the definition is to say that $E$ and $F$ are isomorphic if their underlying principal $G$-bundles are isomorphic, but I'm not sure how to formulate a definition without principal bundles.

The definition in https://en.wikipedia.org/wiki/Fiber_bundle#Bundle_maps does not apply to this case because $G$ may not act on the total space of $E$ and $F$. (See the first answer in https://mathoverflow.net/questions/92813/vector-bundles-vs-principal-g-bundles).

Remark 3.43 in https://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf also hints at the connection to principal bundles, but again a definition of what it means for two Hilbert bundles $H_1$ and $H_2$, each with a reduction of its structure group to a subgroup of $GL(H)$ (where $H$ is the fibre), to be isomorphic, is not made explicit.

Any thoughts would be appreciated.

Answer 1

There is such a definition in Section I.2 of the book "The topology of fibre bundles" by N. Steenrod. It is essentially Definition 2.5.

In fact, it is a bit more general. He defines an map of fibre bundles from $ p \colon E \to X$ to $ p' \colon E' \to X'$ having the same fibre $F$ and structure group $G$. It is a couple of maps $ \bar{h} \colon E \to E'$ and $h \colon X \to X'$ such that:

1) $p'\bar{h} = hp$

2) $\bar{h}$ restricts to a homeomorphism in each fibre.

3) There exist open covers $\{ U_{\alpha} \}$ and $\{ U'_{\beta} \}$ where the bundles trivialize, the transition functions lie in $G$ and for each $x \in U_{\alpha} \cap h^{-1}(U'_{\beta})$, the homeomorphism $$ F \to p^{-1}(x) \to p'^{-1}(h(x)) \to F $$ given by the restrictions of the corresponding trivialization of $E$, the map $\bar{h}$ and the trivialization of $E'$, gives an element of $G$. Moreover the resulting maps $U_{\alpha} \cap h^{-1}(U'_{\beta}) \to G$ are continuous.

And then an isomorphism corresponds to one of these maps where $h$ is the identity.