Editing G-structure on a manifold (section)

=== 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.)