Editing Principal bundle (section)

==Use of the notion==

===Reduction of the structure group===
{{see also|Reduction of the structure group}}
Given a subgroup H of G one may consider the  bundle <math>P/H</math> whose fibers are homeomorphic to the [[coset space]] <math>G/H</math>.  If the new bundle admits a global section, then one says that the section is a '''reduction of the structure group from <math>G</math> to <math>H</math> '''.  The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of <math>P</math> that is a principal <math>H</math>-bundle. If <math>H</math> is the identity, then a section of <math>P</math> itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal <math>G</math>-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from <math>G</math> to <math>H</math>). For example:
[[File:Mobius frame bundle.png|thumb|The frame bundle <math>\mathcal{F}(E)</math> of the [[Möbius strip]] <math>E</math> is a non-trivial principal <math>\mathbb{Z}/2\mathbb{Z}</math>-bundle over the circle.]]
* A <math>2n</math>-dimensional real manifold admits an [[almost-complex structure]] if the [[frame bundle]] on the manifold, whose fibers are <math>GL(2n,\mathbb{R})</math>, can be reduced to the group <math>\mathrm{GL}(n,\mathbb{C}) \subseteq \mathrm{GL}(2n,\mathbb{R})</math>.
* An <math>n</math>-dimensional real manifold admits a <math>k</math>-plane field if the frame bundle can be reduced to the structure group <math>\mathrm{GL}(k,\mathbb{R}) \subseteq \mathrm{GL}(n,\mathbb{R})</math>.
* A manifold is [[orientable]] if and only if its frame bundle can be reduced to the [[special orthogonal group]], <math>\mathrm{SO}(n) \subseteq \mathrm{GL}(n,\mathbb{R})</math>.
* A manifold has [[spin structure]] if and only if its frame bundle can be further reduced from <math>\mathrm{SO}(n)</math> to <math>\mathrm{Spin}(n)</math> the [[Spin group]], which maps to <math>\mathrm{SO}(n)</math> as a double cover.

Also note: an <math>n</math>-dimensional manifold admits <math>n</math> vector fields that are linearly independent at each point if and only if its [[frame bundle]] admits a global section. In this case, the manifold is called [[parallelizable]].

===Associated vector bundles and frames===
{{See also| Frame bundle}}
If <math>P</math> is a principal <math>G</math>-bundle and <math>V</math> is a [[linear representation]] of <math>G</math>, then one can construct a vector bundle <math>E=P\times_G V</math> with fibre <math>V</math>, as the quotient of the product <math>P</math>&times;<math>V</math> by the diagonal action of <math>G</math>. This is a special case of the [[associated bundle]] construction, and <math>E</math> is called an [[associated vector bundle]] to <math>P</math>. If the representation of <math>G</math> on <math>V</math> is [[faithful representation|faithful]], so that <math>G</math> is a subgroup of the general linear group GL(<math>V</math>), then <math>E</math> is a <math>G</math>-bundle and <math>P</math> provides a reduction of structure group of the frame bundle of <math>E</math> from <math>GL(V)</math> to <math>G</math>. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.