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 $M$ . The frame bundle of $E$ is

\mathrm{Fr}(E):=\bigsqcup_{x\in M}\mathrm{Iso}(\mathbb R^n,E_x),

the set of linear isomorphisms $u:\mathbb R^n\to E_x$ over all $x\in M$ . Such an isomorphism is a frame, or ordered basis, of $E_x$ .

Principal-bundle structure

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

u\cdot A := u\circ A,\qquad A\in \mathrm{GL}(n,\mathbb R),

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

\mathrm{Fr}(E)|_U \cong U\times \mathrm{GL}(n,\mathbb R).

Examples

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

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.

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