Editing G-structure on a manifold (section)

=== Properties and examples ===
Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism <math>\phi</math> is an important part of the data.

As a concrete example, every even-dimensional real [[vector space]] is isomorphic to 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 isomorphic to the underlying real bundle of a complex vector bundle. This is then a reduction along the inclusion ''GL''(''n'','''C''') → ''GL''(2''n'','''R''')

In terms of [[transition map]]s, 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 structure]]s): it's properly called a [[Homotopy lifting property|lifting]].

More abstractly, "''G''-bundles over ''X''" is a [[functor]]<ref>Indeed, it is a [[bifunctor]] in ''G'' and ''X''.</ref> in ''G'': Given a Lie group homomorphism ''H'' → ''G'', one gets a map from ''H''-bundles to ''G''-bundles by [[Induced representation|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 closed 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''<sub>H</sub> = σ<sup>−1</sup>''B'' is a reduction of ''B''.<ref>In [[classical field theory]], such a section <math>\sigma</math> describes a classical [[Higgs field (classical)|Higgs field]] ({{cite journal|last1=Sardanashvily|first1=G.|year=2006|title=Geometry of Classical Higgs Fields|journal=International Journal of Geometric Methods in Modern Physics|volume=03|pages=139–148|arxiv=hep-th/0510168|doi=10.1142/S0219887806001065}}).
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