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

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

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

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

2. 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 $M\times \mathrm{GL}(n)$.

3. 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 $E$.