Orthonormal frame bundle

Let $\pi:E\to M$ be a real rank- $n$ vector bundle over a smooth manifold, equipped with a bundle metric $\langle\cdot,\cdot\rangle$ . Its orthonormal frame bundle is

\mathrm{O}(E):=\{(e_1,\dots,e_n)\in \mathrm{Fr}(E)\ :\ \langle e_i,e_j\rangle = \delta_{ij}\ \text{fiberwise}\}.

The right action of $\mathrm{GL}(n,\mathbb R)$ on $\mathrm{Fr}(E)$ restricts to a free transitive action of $\mathrm O(n)$ on each fiber of $\mathrm O(E)$ . Thus $\mathrm O(E)\to M$ is a principal $\mathrm O(n)$ -bundle and a reduction of the frame bundle's structure group.

Examples

If $E=TM$ with a Riemannian metric, then $\mathrm O(TM)$ is the bundle of orthonormal tangent frames.

For $E=M\times\mathbb R^n$ with its Euclidean metric, $\mathrm O(E)\cong M\times\mathrm O(n)$ .

If $n=1$ , then $\mathrm O(1)=\{\pm1\}$ , and $\mathrm O(E)\to M$ is the bundle of unit vectors, a double cover.