 Frame of a vector bundle

A frame over an open set $U$ is a list of smooth sections $e_1,\dots ,e_r$ of a rank-$r$ vector bundle whose values form a basis in every fibre over $U$. It is equivalent to a vector bundle trivialization by $(x,v)\mapsto {\sum }_a{e}_a(x)v^a$. Two local frames differ by a smooth map $g:U\to \mathrm{GL}(r)$, written $e^{\prime }=eg$.