Unitary frame bundle

Let $\pi:E\to M$ be a complex vector bundle of rank $n$ over a smooth manifold and let $h$ be a Hermitian metric on $E$. The unitary frame bundle of $(E,h)$, denoted $\mathrm{U}(E)$, is the submanifold of the (complex) frame bundle $\mathrm{Fr}(E)$ consisting of frames that are unitary with respect to $h$:

The right action of $\mathrm{GL}(n,\mathbb C)$ on $\mathrm{Fr}(E)$ restricts to a free right action of the unitary group $\mathrm{U}(n)$ on $\mathrm{U}(E)$. With this action, $\mathrm{U}(E)\to M$ is a principal G-bundle with structure group $\mathrm{U}(n)$.

Equivalently, giving a Hermitian metric on $E$ is the same as specifying a reduction of the structure group from $\mathrm{GL}(n,\mathbb C)$ to $\mathrm{U}(n)$.

Examples

1. Trivial bundle. For $E=M\times\mathbb C^n$ with the standard Hermitian form, $\mathrm{U}(E)\cong M\times \mathrm{U}(n)$.

2. Complexified real bundle with metric. If $E_\mathbb R$ is a real rank-$n$ bundle with a bundle metric, then its complexification inherits a Hermitian metric, and the corresponding unitary frame bundle can be described as the complexification of the orthonormal frames.

3. Unitary frames for a complex tangent bundle. If a manifold carries additional structure making $TM$ into a complex rank-$n$ bundle with a Hermitian metric (e.g. in almost Hermitian geometry), then $\mathrm{U}(TM)$ is the unitary frame bundle used to define canonical connections.