Let $\pi:E\to M$ be a smooth real vector bundle of finite rank over a smooth manifold $M$, let $g$ be a smooth fibre metric on $E$, and let $P:E\to E$ be a smooth vector bundle endomorphism covering $\operatorname{id}_M$. For each $p\in M$, denote by
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\begin{align*}
P_p:E_p\to E_p
\end{align*}
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the induced [linear map](/page/Linear%20Map) on the fibre over $p$. Then $P$ is the fibrewise $g$-[orthogonal projection](/theorems/437) onto a smooth subbundle $F\subset E$ if and only if, for every $p\in M$,