Orthonormal frame bundle

It is an example of a extension and reduction of a principal bundle.

Suppose $P$ is the frame bundle $F M$ of a manifold $M$ . In this case, $G = G L (n)$ , and consider $H = O (n)$ . The orbit space $P / H = F M / O (n)$ is, I guess, also a bundle over $M$ . The fibre $(P / H)_{x}$ consists of classes of basis of $T_{x} M$ where the equivalence of basis is given by the existence of an orthogonal transformation between them.
Now, if we have a section $σ$ of the bundle $P / H$ we are assigning in a smooth way a class of bases of $T_{x} M$ for every $x \in M$ . We have, then, a new bundle principal bundle $Q = ⨆_{x \in M} {x} \times σ (x)$ with group $O (n)$ . It can be shown that is a reduction of the frame bundle.
On the other hand, every class $σ (x)$ determines a bilinear form in $T_{x} M$ , indeed, a metric (see this). So we have a Riemannian metric on $M$ . On the contrary, a Riemannian metric $g$ let us specify a section $σ$ of $P / H$ .

By the way, as we know (see here), every manifold can be equipped with such a metric, and so we always have the required section.

This example is a particular case of a G-structure.