Editing Frame bundle (section)

==''G''-structures==
{{see also|G-structure}}

If a smooth manifold ''<math>M</math>'' comes with additional structure it is often natural to consider a subbundle of the full frame bundle of ''<math>M</math>'' which is adapted to the given structure. For example, if ''<math>M</math>'' is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of ''<math>M</math>''. The orthonormal frame bundle is just a reduction of the structure group of ''<math>F_{\mathrm{GL}}(M)</math>'' to the orthogonal group ''<math>\mathrm{O}(n)</math>''.

In general, if ''<math>M</math>'' is a smooth ''<math>n</math>''-manifold and ''<math>G</math>'' is a [[Lie subgroup]] of ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' we define a '''[[G-structure|''G''-structure]]''' on ''<math>M</math>'' to be a [[reduction of the structure group]] of ''<math>F_{\mathrm{GL}}(M)</math>'' to ''<math>G</math>''. Explicitly, this is a principal ''<math>G</math>''-bundle ''<math>F_{G}(M)</math>'' over ''<math>M</math>'' together with a ''<math>G</math>''-equivariant [[bundle map]]
:<math>{\mathrm F}_{G}(M) \to {\mathrm F}_{\mathrm{GL}}(M)</math>
over ''<math>M</math>''.

In this language, a Riemannian metric on ''<math>M</math>'' gives rise to an ''<math>\mathrm{O}(n)</math>''-structure on ''<math>M</math>''. The following are some other examples.

*Every [[orientability|oriented manifold]] has an oriented frame bundle which is just a ''<math>\mathrm{GL}^+(n,\mathbb{R})</math>''-structure on ''<math>M</math>''.
*A [[volume form]] on ''<math>M</math>'' determines a ''<math>\mathrm{SL}(n,\mathbb{R})</math>''-structure on ''<math>M</math>''.
*A ''<math>2n</math>''-dimensional [[symplectic manifold]] has a natural ''<math>\mathrm{Sp}(2n,\mathbb{R})</math>''-structure.
*A ''<math>2n</math>''-dimensional [[complex manifold|complex]] or [[almost complex manifold]] has a natural ''<math>\mathrm{GL}(n,\mathbb{C})</math>''-structure.
In many of these instances, a ''<math>G</math>''-structure on ''<math>M</math>'' uniquely determines the corresponding structure on ''<math>M</math>''. For example, a ''<math>\mathrm{SL}(n,\mathbb{R})</math>''-structure on ''<math>M</math>'' determines a volume form on ''<math>M</math>''. However, in some cases, such as for symplectic and complex manifolds, an added [[integrability condition]] is needed. A ''<math>\mathrm{Sp}(2n,\mathbb{R})</math>''-structure on ''<math>M</math>'' uniquely determines a [[nondegenerate form|nondegenerate]] [[2-form]] on ''<math>M</math>'', but for ''<math>M</math>'' to be symplectic, this 2-form must also be [[closed differential form|closed]].