Let $M$ be a smooth manifold, let $\pi:E\to M$ be a smooth complex vector bundle of complex rank $r$, and let $h$ be a smooth Hermitian metric on $E$. Let $\operatorname{Fr}_{\mathbb C}(E)\to M$ denote the complex frame bundle, whose fibre over $p\in M$ is the set of complex-linear isomorphisms $u:\mathbb C^r\to E_p$, with right action $u\cdot A:=u\circ A$ for $A\in GL(r,\mathbb C)$. Define

\begin{align*} \operatorname{Fr}_U(E,h)_p:=\{u\in \operatorname{Fr}_{\mathbb C}(E)_p: h_p(u z,u w)=(z,w)_{\mathbb C^r}\text{ for all }z,w\in \mathbb C^r\}, \end{align*}

where $(\cdot,\cdot)_{\mathbb C^r}$ is the standard Hermitian [inner product](/page/Inner%20Product) on $\mathbb C^r$. Then

\begin{align*} \operatorname{Fr}_U(E,h):=\bigcup_{p\in M}\operatorname{Fr}_U(E,h)_p \end{align*}

is a smooth principal $U(r)$-subbundle of $\operatorname{Fr}_{\mathbb C}(E)\to M$.