Editing G-structure on a manifold (section)

== Integrability conditions and flat ''G''-structures ==
Several structures on manifolds, such as a complex structure, a [[symplectic structure]], or a [[Kähler manifold|Kähler structure]], are ''G''-structures (and thus can be obstructed), but need to satisfy an additional [[integrability condition]]. Without the corresponding integrability condition, the structure is instead called an "almost" structure, as in an [[almost complex structure]], an [[almost symplectic manifold|almost symplectic structure]], or an [[almost Kähler manifold|almost Kähler structure]].

Specifically, a [[symplectic manifold]] structure is a stronger concept than a ''G''-structure for the [[symplectic group]]. A symplectic structure on a manifold is a [[2-form]] ''&omega;'' on ''M'' that is non-degenerate (which is an <math>Sp</math>-structure, or almost symplectic structure), ''together with'' the extra condition that d''&omega;'' = 0; this latter is called an [[integrability condition]].

Similarly, [[foliation]]s correspond to ''G''-structures coming from [[block matrix|block matrices]], together with integrability conditions so that the [[Frobenius theorem (differential topology)|Frobenius theorem]] applies.

A '''flat ''G''-structure''' is a ''G''-structure ''P'' having a global section (''V''<sub>1</sub>,...,''V''<sub>n</sub>) consisting of [[Lie derivative|commuting vector fields]].  A ''G''-structure is '''integrable''' (or ''locally flat'') if it is locally isomorphic to a flat ''G''-structure.