Reduction of the structure group

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In mathematics, in particular the theory of principal bundles, one can ask if a principal -bundle over a group "comes from" a subgroup of . This is called reduction of the structure group (to ), and makes sense for any map , which need not be an inclusion map (despite the terminology).

Definition

Formally, given a G-bundle B and a map HG (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[1] in G: given a map HG, 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 BB/H is a principal H-bundle over B/H. If σ : XB/H is a section, then the pullback bundle BH = σ−1B is a reduction of B.[2]

Examples

Examples for vector bundles, particularly the tangent bundle of a manifold:

Integrability

Many geometric structures are stronger than G-structures; they are G-structures with an integrability condition. Thus such a structure requires a reduction of the structure group (and can be obstructed, as below), but this is not sufficient. Examples include complex structure, symplectic structure (as opposed to almost complex structures and almost symplectic structures).

Another example is for a foliation, which requires a reduction of the tangent bundle to a block matrix subgroup, together with an integrability condition so that the Frobenius theorem applies.

See also

Notes

References

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