Let $\pi:E \to M$ be a smooth real vector bundle of rank $r$ over a smooth manifold $M$, and let $\operatorname{Fr}(E)$ denote the principal $GL_r(\mathbb{R})$-bundle of linear frames $\nu:\mathbb{R}^r \to E_p$ over points $p \in M$, with right action $\nu \cdot A := \nu \circ A$. Then smooth bundle metrics on $E$ are in bijection with principal $O(r)$-reductions $Q \subset \operatorname{Fr}(E)$, where $O(r) \subset GL_r(\mathbb{R})$ is defined with respect to the standard Euclidean [inner product](/page/Inner%20Product) on $\mathbb{R}^r$.