Special unitary frame bundle

Let $E\to M$ be a rank- $n$ complex vector bundle with a Hermitian metric, and let

\det(E):=\Lambda^nE.

Choose a unit-norm nowhere-vanishing section $\Omega\in\Gamma(\det(E))$ . The special unitary frame bundle determined by $\Omega$ is

\mathrm{SU}(E,\Omega) = \left\{(e_1,\ldots,e_n)\in\mathrm U(E): e_1\wedge\cdots\wedge e_n=\Omega_{\pi(e_1,\ldots,e_n)} \right\},

where $\mathrm U(E)$ is the unitary frame bundle. It is a principal $\mathrm{SU}(n)$ -subbundle of $\mathrm U(E)$ .

Existence and equivalent data

The Hermitian metric on $E$ induces one on the determinant line bundle $\det(E)$ . The following data are equivalent:

a reduction of $\mathrm U(E)$ from $\mathrm U(n)$ to $\mathrm{SU}(n)$ ;
a unit-norm trivializing section $\Omega$ of $\det(E)$ ;
a trivialization of $\det(E)$ as a Hermitian line bundle.

Consequently, an $\mathrm{SU}(n)$ -reduction exists exactly when $\det(E)$ is trivial as a complex line bundle. The reduction is not determined by $E$ and its metric alone: different unit determinant sections generally give different reductions.

There is no intrinsic condition that a unitary frame have “determinant $1$ ” until such a section is chosen. Intrinsically, the determinant homomorphism gives the associated principal bundle

\mathrm U(E)\times_{\det}\mathrm U(1)\cong \mathrm U(\det E),

and $\Omega$ selects its identity frame in every fiber.

Examples

Trivial bundle. For $E=M\times\mathbb C^n$ and $\Omega=\mathbf e_1\wedge\cdots\wedge\mathbf e_n$ , $\mathrm{SU}(E,\Omega)\cong M\times\mathrm{SU}(n)$ .
Rank one. When $n=1$ , an $\mathrm{SU}(1)$ -reduction exists exactly when $E$ is trivial. A chosen unit section $\Omega$ identifies the reduced bundle with $M$ , but this identification depends on $\Omega$ .