Frame bundles and associated vector bundles

For a rank-$n$ $\mathbb{F}$-vector bundle $E\to X$, its frame bundle $\mathrm{Fr}(E)$ has fiber

$\mathrm{Fr}(E{)}_x=\{u:{\mathbb{F}}^n\stackrel{\cong }{\to }{E}_x\text{ linear}\}\mathrm{.}$

Its right action is precomposition, $u\cdot g=u\circ g$. A linear bundle chart identifies the frames with $U\times {\mathrm{GL}}_n(\mathbb{F})$ and the action with right multiplication, so $\mathrm{Fr}(E)$ is a principal bundle as in Principal g bundle and associated fiber bundle.

Conversely, for a right principal ${\mathrm{GL}}_n(\mathbb{F})$-bundle $P\to X$, let the group act on ${\mathbb{F}}^n$ on the left by its standard representation. The associated bundle

$P{\times }_{{\mathrm{GL}}_n(\mathbb{F})}{\mathbb{F}}^n$

is a locally trivial bundle with fiber ${\mathbb{F}}^n$ by Associated bundle is locally trivial and functorial under pullback. On the fiber over $x$, choose $p\in P_x$ and set $[p,v]+[p,w]=[p,v+w],a[p,v]=[p,av]\mathrm{.}$ Changing $p$ to $pg$ replaces $v,w$ by $g^{-1}v,g^{-1}w$, so these operations are well-defined because $g$ is linear. The associated local trivializations restrict to linear isomorphisms on fibers. Thus this is a rank-$n$ vector bundle in the sense of Real and complex topological vector bundles. Evaluation

$[u,v]⟼u(v)$

is well-defined because $(u\cdot g)(v)=u(gv)$, and it gives a canonical isomorphism $\mathrm{Fr}(E){\times }_{{\mathrm{GL}}_n(\mathbb{F})}{\mathbb{F}}^n\cong E$. These constructions commute with pullback. A linear chart numeration induces the same support-subordinate numeration on the frame bundle and conversely. For $n=0$ the structure group and every frame fiber are singletons.