Special orthonormal frame bundle

Let $\pi:E\to M$ be a real vector bundle of rank $n$ over a smooth manifold , equipped with a bundle metric and an orientation . The special orthonormal frame bundle, denoted $\mathrm{SO}(E)$, is the subbundle of the orthonormal frame bundle consisting of orthonormal frames that are oriented:

The right action of the group $\mathrm{O}(n)$ on $\mathrm{O}(E)$ restricts to a free right action of the special orthogonal group $\mathrm{SO}(n)$, which is a Lie group . With this action, $\mathrm{SO}(E)\to M$ is a principal bundle with structure group $\mathrm{SO}(n)$.

Equivalently, $\mathrm{SO}(E)$ is the reduction of the full frame bundle to $\mathrm{SO}(n)$ determined jointly by the metric and the orientation.

Examples

1. Oriented Riemannian manifolds. If $E=TM$ and $M$ is oriented and Riemannian, then $\mathrm{SO}(TM)$ is the usual bundle of oriented orthonormal tangent frames used in defining spin structures and Levi-Civita connections.

2. Trivial oriented bundle. For $E=M\times\mathbb R^n$ with the standard metric and the standard orientation, $\mathrm{SO}(E)\cong M\times \mathrm{SO}(n)$.

3. Rank-one case. If $n=1$, then $\mathrm{SO}(1)=\{1\}$, and $\mathrm{SO}(E)\to M$ is canonically isomorphic to $M$ whenever $E$ is oriented and metrized (there is a unique oriented unit vector in each fiber).