\begin{align*}
0 \longrightarrow F \xrightarrow{\iota} E \xrightarrow{\pi} Q \longrightarrow 0
\end{align*}
latex_env
admin
be a short exact sequence of smooth real vector bundles over $M$, where $\iota: F \to E$ and $\pi: E \to Q$ are smooth vector-bundle morphisms over $\operatorname{id}_M$. Suppose that $E$ is equipped with a smooth bundle metric $g$, meaning a smooth assignment of inner products