Frame Bundle

Frame Bundle

Let $E\to X$ be a real vector bundle of rank $k$. A frame at $x\in X$ is an ordered basis for $E_x$, viewed as a linear isomorphism $p:{\mathbb{R}}^k\to E_x$. Frame bundle of $E$ is $Fr(E)={\coprod }_{x\in X}\{(v_1,\dots ,v_k):v_i\in E_x,\{v_1,\dots ,v_k\}\text{ is an ordered basis of }{E}_x\}$ The group $GL(k,\mathbb{R})$ acts on the right by composition: $p\circ g:{\mathbb{R}}^k\to E_x$, as the action on $F_x$ is both free and transitive(it’s just the change of basis in linear algebra.), we get $Fr(E)$ is a principal $GL(k,\mathbb{R})$ bundle.