Editing G-structure on a manifold (section)

{{DISPLAYTITLE:''G''-structure on a manifold}}
{{Use American English|date = March 2019}}
{{Short description|Structure group sub-bundle on a tangent frame bundle}}
In [[differential geometry]], a '''''G''-structure''' on an ''n''-[[manifold]] ''M'', for a given [[structure group]]<ref>Which is a [[Lie group]] <math>G \to GL(n,\mathbf{R})</math> mapping to the [[general linear group]] <math>GL(n,\mathbf{R})</math>. This is often but not always a [[Lie subgroup]]; for instance, for a [[spin structure]] the map is a [[covering space]] onto its image.</ref> ''G'', is a principal ''G''-[[subbundle]] of the [[frame bundle#Tangent frame bundle|tangent frame bundle]] F''M'' (or GL(''M'')) of ''M''.

The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are [[tensor field]]s. For example, for the [[orthogonal group]], an O(''n'')-structure defines a [[Riemannian metric]], and for the [[special linear group]] an SL(''n'','''R''')-structure is the same as a [[volume form]].  For the [[trivial group]], an {''e''}-structure consists of an [[parallelizable manifold|absolute parallelism]] of the manifold.

Generalising this idea to arbitrary [[principal bundle]]s on topological spaces, one can ask if a principal <math>G</math>-bundle over a [[group (mathematics)|group]] <math>G</math> "comes from" a [[subgroup]] <math>H</math> of <math>G</math>. This is called '''reduction of the structure group''' (to <math>H</math>).

Several structures on manifolds, such as a [[Complex manifold|complex structure]], a [[symplectic structure]], or a [[Kähler manifold|Kähler structure]], are ''G''-structures with an additional [[integrability condition]].