Frame bundle of a rank-n vector bundle

Let $\pi:E\to M$ be a smooth real or complex vector bundle of rank $n$ over a smooth manifold. The frame bundle (or general linear frame bundle) of $E$ , denoted $\mathrm{Fr}(E)$ , is the manifold whose points are ordered bases (frames) of the fibers of $E$ :

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

where $\mathrm{Iso}_{\mathbb F}(\mathbb F^n,E_x)$ denotes the set of $\mathbb F$ -linear isomorphisms.

There is a natural projection

p:\mathrm{Fr}(E)\to M,\qquad p(u)=x \ \text{if}\ u:\mathbb F^n\to E_x,

and a free right action of the group $\mathrm{GL}(n,\mathbb F)$ given by precomposition of the frame:

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

A local trivialization $E|_U\cong U\times\mathbb F^n$ gives a frame-bundle chart $\mathrm{Fr}(E)|_U\cong U\times\mathrm{GL}(n,\mathbb F)$ , sending a frame to its coordinate matrix. These charts define its smooth structure. With this structure, $(\mathrm{Fr}(E),p)$ is a principal G-bundle with structure group $\mathrm{GL}(n,\mathbb F)$ .

A local frame $(e_1,\dots,e_n)$ over an open set $U$ determines a local section $U\to \mathrm{Fr}(E)$ by sending $x$ to the frame $(e_1(x),\dots,e_n(x))$ , and the corresponding changes of local section on overlaps are given by the usual transition matrices.

Examples

Frame bundle of the tangent bundle. $\mathrm{Fr}(TM)$ is the bundle of ordered bases of tangent spaces; it is the basic principal bundle underlying many constructions in differential geometry.

Trivial bundle. If $E\cong M\times \mathbb F^n$ , then $\mathrm{Fr}(E)\cong M\times \mathrm{GL}(n,\mathbb F)$ as a principal bundle (a canonical trivialization is obtained from the standard frame of $\mathbb F^n$ ).

Oriented frame bundle. If $E$ is an oriented real rank- $n$ bundle, the oriented frames form a subbundle $\mathrm{Fr}^+(E)\subset \mathrm{Fr}(E)$ which is a principal $\mathrm{GL}^+(n,\mathbb R)$ -bundle.