\begin{align*}
0 \longrightarrow E' \xrightarrow{i} E \xrightarrow{p} E'' \longrightarrow 0
\end{align*}
latex_env
admin
be a short exact sequence of smooth real vector bundles over $M$, where $i:E'\to E$ and $p:E\to E''$ are smooth vector bundle morphisms over $\operatorname{id}_M$. Suppose $E$ is equipped with a smooth bundle metric $\langle\cdot,\cdot\rangle_E$. Then there exists a smooth subbundle $F\subset E$ such that