Equivalent conditions for reduction of structure group

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 $H\subset G$).

1. (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).

1. (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.

1. (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 $Q$.

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).

1. 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.

1. 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.