Editing Associated bundle (section)

===Examples of reduction===

Examples for [[vector bundle]]s include: the introduction of a ''metric'' resulting in reduction of the structure group from a [[general linear group]] <math>\mathrm{GL}(n)</math> to an [[orthogonal group]] <math>\mathrm{O}(n)</math>; and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group <math>\mathrm{GL}(2n, \mathbb{R})</math> to complex general linear group <math>\mathrm{GL}(n, \mathbb{C})</math>.

Another important case is finding a decomposition of a vector bundle <math>V</math> of rank <math>n</math> as a [[Whitney sum]] (direct sum) of sub-bundles of rank <math>k</math> and <math>n-k</math>, resulting in reduction of the structure group from <math>\mathrm{GL}(n, \mathbb{R})</math> to <math>\mathrm{GL}(k, \mathbb{R}) \times \mathrm{GL}(n-k, \mathbb{R})</math>.

One can also express the condition for a [[foliation]] to be defined as a reduction of the [[tangent bundle]] to a block matrix subgroup - but here the reduction is only a necessary condition, there being an ''integrability condition'' so that the [[Frobenius theorem (differential topology)|Frobenius theorem]] applies.