Editing G-structure on a manifold (section)

== ''G''-structures ==
Every [[vector bundle]] of dimension <math>n</math> has a canonical <math>GL(n)</math>-bundle, the [[frame bundle]]. In particular, every [[Differentiable manifold|smooth manifold]] has a canonical vector bundle, the [[tangent bundle]]. For a Lie group <math>G</math> and a group homomorphism <math>\phi\colon G \to GL(n)</math>, a <math>G</math>-structure is a reduction of the structure group of the frame bundle to <math>G</math>.

=== Examples ===
The following examples are defined for [[Real vector bundle|real vector bundles]], particularly the [[tangent bundle]] of a [[manifold|smooth manifold]].
{| class="wikitable"
!Group homomorphism
!Group <math>G</math>
!<math>G</math>-structure
!Obstruction
|-
|<math>GL^+(n) < GL(n)</math>
|[[General linear group#real case|General linear group of positive determinant]]
|[[Orientation (manifold)|Orientation]]
|Bundle must be orientable
|-
|<math>SL(n) < GL(n)</math>
|[[Special linear group]]
|[[Volume form]]
|Bundle must be orientable (<math>SL \to GL^+</math> is a [[deformation retract]])
|-
|<math>SL^{\pm}(n) < GL(n)</math>
|Determinant <math>\pm 1</math>
|Pseudo-[[volume form]]
|Always possible
|-
|<math>O(n) < GL(n)</math>
|[[Orthogonal group]]
|[[Riemannian metric]]
|Always possible (<math>O(n)</math> is the [[maximal compact subgroup]], so the inclusion is a deformation retract)
|-
|<math>O(1,n-1) < GL(n)</math>
|[[Indefinite orthogonal group]]
|[[Pseudo Riemannian metric|Pseudo-Riemannian metric]]
|Topological obstruction<ref>It is a [[gravitational field]] in [[gauge gravitation theory]] ({{Cite journal|last1=Sardanashvily|first1=G.|year=2006|title=Gauge gravitation theory from the geometric viewpoint|journal=International Journal of Geometric Methods in Modern Physics|volume=3|issue=1|pages=v–xx|arxiv=gr-qc/0512115|bibcode=2005gr.qc....12115S}})</ref>
|-
|<math>GL(n,\mathbf{C}) < GL(2n,\mathbf{R})</math>
|[[Complex general linear group]]
|[[almost complex manifold|Almost complex structure]]
|Topological obstruction
|-
|<math>GL(n,\mathbf{H})\cdot Sp(1) < GL(4n,\mathbf{R})</math>
|
* <math>GL(n,\mathbf{H})</math>: [[Quaternion|quaternionic]] general linear group acting on <math>\mathbf{H}^n \cong \mathbf{R}^{4n}</math> from the left
* <math>Sp(1)=Spin(3)</math>: group of unit quaternions acting on <math>\mathbf{H}^n</math> from the right
|almost quaternionic structure<ref name=":0">{{harvnb|Besse|1987|loc=§14.61}}</ref>
|Topological obstruction<ref name=":0" />
|-
|<math>GL(k) \times GL(n-k) < GL(n)</math>
|[[General linear group]]
|Decomposition as a [[Whitney sum]] (direct sum) of sub-bundles of rank <math>k</math> and <math>n-k</math>.
|Topological obstruction
|}
Some <math>G</math>-structures are defined in terms of others: Given a Riemannian metric on an oriented manifold, a <math>G</math>-structure for the 2-fold [[covering space|cover]] <math>\mbox{Spin}(n) \to \mbox{SO}(n)</math> is a [[spin manifold|spin structure]]. (Note that the group homomorphism here is ''not'' an inclusion.)

=== Principal bundles ===
Although the theory of [[principal bundle]]s plays an important role in the study of ''G''-structures, the two notions are different.  A ''G''-structure is a principal subbundle of the [[frame bundle#Tangent frame bundle|tangent frame bundle]], but the fact that the ''G''-structure bundle ''consists of tangent frames'' is regarded as part of the data.  For example, consider two Riemannian metrics on '''R'''<sup>''n''</sup>.  The associated O(''n'')-structures are isomorphic if and only if the metrics are isometric.  But, since '''R'''<sup>''n''</sup> is contractible, the underlying O(''n'')-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces are trivial bundles.

This fundamental difference between the two theories can be captured by giving an additional piece of data on the underlying ''G''-bundle of a ''G''-structure: the '''[[solder form]]'''.  The solder form is what ties the underlying principal bundle of the ''G''-structure to the local geometry of the manifold itself by specifying a canonical isomorphism of the tangent bundle of ''M'' to an [[associated bundle|associated vector bundle]].  Although the solder form is not a [[connection form]], it can sometimes be regarded as a precursor to one.

In detail, suppose that ''Q'' is the principal bundle of a ''G''-structure.  If ''Q'' is realized as a reduction of the frame bundle of ''M'', then the solder form is given by the [[pullback (differential geometry)|pullback]] of the [[frame bundle#Solder form|tautological form of the frame bundle]] along the inclusion.  Abstractly, if one regards ''Q'' as a principal bundle independently of its realization as a reduction of the frame bundle, then the solder form consists of a representation &rho; of ''G'' on '''R'''<sup>n</sup> and an isomorphism of bundles &theta; : ''TM'' &rarr; ''Q'' &times;<sub>&rho;</sub> '''R'''<sup>n</sup>.