Reducing a GL(n)-structure to O(n) using a bundle metric

Let $E\to M$ be a smooth real rank- $n$ vector bundle over a smooth manifold $M$ . Its frame bundle $\mathrm{Fr}(E)\to M$ is a principal $\mathrm{GL}(n,\mathbb R)$ -bundle.

A reduction of structure group from $\mathrm{GL}(n,\mathbb R)$ to a subgroup $H\subset \mathrm{GL}(n,\mathbb R)$ is, by definition, a principal $H$ -subbundle $Q\subset \mathrm{Fr}(E)$ such that

Q\times_H \mathbb R^n \cong E

as associated bundles.

Example (bundle metric gives an O(n)-reduction)

A bundle metric on $E$ is a smooth choice of inner product $\langle\cdot,\cdot\rangle_x$ on each fiber $E_x$ .

Given such a metric, define

O(E):=\{u\in \mathrm{Fr}(E): u:\mathbb R^n\to E_x \text{ is an isometry for the standard inner product}\}.

Then:

$O(E)\to M$ is a principal $\mathrm{O}(n)$ -bundle, where $\mathrm{O}(n)$ is a Lie group acting by postcomposition on frames.
The inclusion $O(E)\subset \mathrm{Fr}(E)$ is a reduction of the principal principal G-bundle $\mathrm{Fr}(E)\to M$ from $\mathrm{GL}(n)$ to $\mathrm{O}(n)$ .
Conversely, any principal $\mathrm{O}(n)$ -subbundle $Q\subset \mathrm{Fr}(E)$ determines a unique bundle metric by declaring frames in $Q$ to be orthonormal.

Specializing to $E=TM$ recovers the construction of the orthonormal frame bundle.

Examples

Riemannian manifolds. For $E=TM$ , choosing a Riemannian metric on $M$ produces the reduction $\mathrm{Fr}(TM)\supset O(TM)$ .

Trivial bundle with standard metric. For the trivial bundle $M\times\mathbb R^n$ , the standard Euclidean inner product yields $O(E)\cong M\times \mathrm{O}(n)$ as a reduction of $M\times \mathrm{GL}(n)$ .

Changing the metric changes the reduction. Two different bundle metrics on the same underlying bundle generally produce different $\mathrm{O}(n)$ -subbundles of $\mathrm{Fr}(E)$ , even though both reductions have the same associated bundle $E$ .