Let $\pi: E \to M$ be a smooth real vector bundle of rank $n$ over a smooth manifold $M$, and let $\operatorname{Fr}(E)$ denote its frame bundle, viewed as a principal $GL_n(\mathbb{R})$-bundle whose fibre over $x \in M$ consists of linear isomorphisms $u: \mathbb{R}^n \to E_x$. Then smooth fibre metrics on $E$ are in natural bijection with smooth principal $O(n)$-subbundles $Q \subset \operatorname{Fr}(E)$ whose extension of structure group to $GL_n(\mathbb{R})$ is $\operatorname{Fr}(E)$.

Explicitly, a smooth fibre metric $g$ determines the $O(n)$-reduction

\begin{align*} \operatorname{Fr}_g(E) := \{u \in \operatorname{Fr}(E) : u: (\mathbb{R}^n, \langle \cdot,\cdot\rangle_0) \to (E_{\pi(u)}, g_{\pi(u)}) \text{ is an isometry}\}, \end{align*}

where $\langle \cdot,\cdot\rangle_0$ is the standard Euclidean [inner product](/page/Inner%20Product) on $\mathbb{R}^n$. Conversely, an $O(n)$-reduction $Q \subset \operatorname{Fr}(E)$ determines the fibre metric $g^Q$ defined by declaring every frame $u \in Q_x$ to be an isometry $\mathbb{R}^n \to E_x$. These two constructions are inverse to one another.