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

g_{ij}:U_i\cap U_j \to G

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

Let $H\subseteq G$ be a closed embedded 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

g_{ij}(x)\in H.

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

Q|_{U_i} := s_i(U_i)\cdot H \;\subset\; P|_{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

s_j(x)\cdot H = s_i(x)\cdot g_{ij}(x)\cdot H = s_i(x)\cdot H,

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 $H\hookrightarrow G$ :

Q\times_H G \;\cong\; P.

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

Examples

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).

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.

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.