Reduction of The Structure Group - Definition

Definition

Formally, given a G-bundle B and a map H → G (which need not be an inclusion), a reduction of the structure group (from G to H) is an H-bundle such that the pushout is isomorphic to B.

Note that these do not always exist, nor if they exist are they unique.

As a concrete example, every even dimensional real vector space is the underlying real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits an almost complex structure if and only if it is the underlying real bundle of a complex vector bundle. This is a reduction along the inclusion GL(n,C) → GL(2n,R)

In terms of transition maps, a G-bundle can be reduced if and only if the transition maps can be taken to have values in H. Note that the term reduction is misleading: it suggests that H is a subgroup of G, which is often the case, but need not be (for example for spin structures): it's properly called a lifting.

More abstractly, "G-bundles over X" is a functor in G: given a map H → G, one gets a map from H-bundles to G-bundles by inducing (as above). Reduction of the structure group of a G-bundle B is choosing an H-bundle whose image is B.

The inducing map from H-bundles to G-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is orientable, and those that are orientable admit exactly two orientations.

If H is a Lie subgroup of G, then there is a natural one-to-one correspondence between reductions of a G-bundle B to H and global sections of the fiber bundle B/H obtained by quotienting B by the right action of H. Specifically, the fibration B → B/H is a principal H-bundle over B/H. If σ : X → B/H is a section, then the pullback bundle B_H = σ−1B is a reduction of B.