Reduction of structure group via H-valued transition functions

Let $\pi:P\to M$ be a principal G-bundle . Fix an open cover $\{U_i\}$ of $M$ and smooth local sections $s_i:U_i\to P$. Let

be the transition functions determined by the $s_i$, as in constructing transition functions from local sections .

Let $H\subset G$ be a Lie subgroup .

Construction (reduction from H-valued transitions)

Assume that for all $i,j$ and all $x\in U_i\cap U_j$ we have

Define a subset $Q\subset P$ by specifying it locally on each $U_i$:

Here $s_i(U_i)\cdot H$ means all points of the form $s_i(x)\cdot h$ with $x\in U_i$ and $h\in H$.

Compatibility on overlaps

On $U_i\cap U_j$, we have $s_j(x)=s_i(x)\cdot g_{ij}(x)$ and by assumption $g_{ij}(x)\in H$. Hence

so the locally defined subsets glue to a global smooth submanifold $Q\subset P$.

Result

The glued space $Q$ is a principal H-subbundle of $P$, i.e. a principal $H$-bundle over $M$ together with an $H$-equivariant inclusion $Q\hookrightarrow P$ covering the identity on $M$. Equivalently, $P$ is obtained from $Q$ by extending the structure group along $H\hookrightarrow G$:

This is a concrete realization of reduction of structure group , and it matches the criterion in reduction via H-valued transition functions .

Examples

1. Metric reduction $GL(n)\to O(n)$. If a rank-$n$ real vector bundle has a bundle metric , one can choose local orthonormal frames so that all transition matrices lie in $O(n)$. Applying the construction yields the orthonormal frame bundle, as in the orthonormal frame bundle reduction (see also reduction of GL structure to O using a bundle metric ).

2. Hermitian reduction $GL(n,\mathbb C)\to U(n)$. A Hermitian metric on a complex rank-$n$ vector bundle allows local unitary frames, so the transition functions are $U(n)$-valued. The resulting principal $U(n)$-subbundle is the unitary frame bundle reduction .

3. Oriented Riemannian reduction to $SO(n)$. If the bundle is both oriented (orientation ) and metric, then one can choose local oriented orthonormal frames, forcing transitions to lie in $SO(n)$. The construction produces the special orthonormal frame bundle .