G-structure on a vector bundle ($P\subseteq \mathrm{Fr}(E),G\subseteq GL(m,\mathbb{R})$)

A G-structure on a real rank-$m$ vector bundle is a principal $G$ reduction of its frame bundle. Equivalently it has local frames whose transition matrices take values in $G$, with the reduction consisting of their right $G$-orbits. An orthogonal structure on a real vector bundle is equivalent to a fiber metric; a special orthogonal reduction is equivalent to that metric together with an orientation of a vector bundle.