G-structure - Integrability Conditions

Integrability Conditions

Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with 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 structure, or an 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 two-form ω on M that is non-degenerate (which is an -structure, or almost symplectic structure), together with the extra condition that dω = 0; this latter is called an integrability condition.

Similarly, foliations correspond to G-structures coming from block matrices, together with integrability conditions so that the Frobenius theorem applies.