= G-structure on a vector bundle
{c}
{title2=$P\subseteq\operatorname{Fr}(E),\quad G\subseteq GL(m,\mathbb R)$}
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>.
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