Equivalent conditions for reduction of structure group

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$, and let $H\subset G$ be a Lie subgroup (see Lie subgroup ). A reduction of structure group to $H$ means, informally, that $P$ can be described using $H$ as the structure group instead of $G$ (see reduction of structure group ).

Theorem (TFAE: reduction to H)

The following are equivalent:

1. (Principal H-subbundle)
   There exists a principal H-subbundle $Q\subset P$ such that $Q\to M$ is a principal $H$-bundle and the inclusion $Q\hookrightarrow P$ is $H$-equivariant (with $H$ acting on $P$ through the inclusion $H\subset G$).

2. (Associated bundle model)
   There exists a principal $H$-bundle $Q\to M$ such that $P$ is isomorphic (as a principal $G$-bundle) to the extension of structure group

   $$P \cong Q\times_H G$$

   (compare extension of structure group ).

3. (H-valued transition functions)
   There exists a bundle atlas for $P$ whose transition functions take values in $H\subset G$. Equivalently, the cocycle of transition functions is represented by an $H$-valued cocycle (see principal bundle transition functions and reduction by cocycle ). This is the transition-function viewpoint used in constructing reductions via H-valued transition functions .

4. (Section of the coset bundle)
   Let $G$ act on the homogeneous space $G/H$ by left translation. Form the associated bundle

   $$P/H \;:=\; P\times_G (G/H),$$

   sometimes called the bundle of cosets (compare bundle of orbits in the special case of homogeneous fibers). Then $P$ admits a reduction of structure group to $H$ if and only if $P/H\to M$ admits a smooth global section.

In (4), given a reduction $Q\subset P$, the corresponding section of $P/H$ sends $x\in M$ to the coset represented by any $q\in Q_x$. Conversely, a section selects an $H$-orbit in each fiber of $P$, and its preimage defines the reduced subbundle $Q$.

Examples

1. Riemannian metric reduces GL(n) to O(n).
   Let $E\to M$ be a rank-$n$ real vector bundle with frame bundle $\mathrm{Fr}(E)$ (see frame bundle ). A bundle metric on $E$ is equivalent to a reduction of $\mathrm{Fr}(E)$ from $GL(n)$ to $O(n)$, whose reduced bundle is the orthonormal frame bundle (see orthonormal frame bundle reduction and metric reduction example ).

2. Orientation reduces GL(n) to GL+(n).
   An orientation of a real vector bundle is equivalent to a reduction of the structure group from $GL(n)$ to the identity component $GL^+(n)$. In transition-function terms, this means one can choose local frames so that all transition matrices have positive determinant.

3. Unitary to special unitary reduction.
   For a complex Hermitian vector bundle, the unitary frame bundle gives a reduction to $U(n)$ (see unitary frame bundle reduction ). A further reduction to $SU(n)$ corresponds to choosing a trivialization of the determinant bundle compatible with the Hermitian structure, producing the special unitary frame bundle reduction .