Construction: Frame bundle Fr(E) of a vector bundle E

Let $\pi:E\to M$ be a smooth real vector bundle of rank $n$ over a smooth manifold $M$.

Construction (frame bundle)

The frame bundle of $E$ is

the set of linear isomorphisms $u:\mathbb R^n\to E_x$ for varying $x$. An element of $\mathrm{Fr}(E)$ is called a frame (or ordered basis) of the fiber $E_x$.

There is a smooth projection $\mathrm{Fr}(E)\to M$ sending $u$ to its basepoint $x$. The general linear group $\mathrm{GL}(n,\mathbb R)$ acts on the right by

making $\mathrm{Fr}(E)\to M$ into a principal G-bundle with $G=\mathrm{GL}(n,\mathbb R)$.

Local trivializations of $E$ give local trivializations of $\mathrm{Fr}(E)$: if $E|_{U}\cong U\times\mathbb R^n$, then

Examples

1. Tangent bundle. For $E=TM$, the tangent bundle of $M$, $\mathrm{Fr}(TM)$ is the usual frame bundle of the manifold.

2. Trivial bundle. If $E\cong M\times\mathbb R^n$, then $\mathrm{Fr}(E)\cong M\times \mathrm{GL}(n,\mathbb R)$ as principal bundles.

3. Oriented frames. If $E$ is oriented, the subspace of positively oriented frames forms a principal $\mathrm{GL}^+(n,\mathbb R)$-subbundle of $\mathrm{Fr}(E)$.