Reduction of The Structure Group - Integrability

Integrability

Many geometric structures are stronger than G-structures; they are G-structures with an integrability condition. Thus such a structure requires a reduction of the structure group (and can be obstructed, as below), but this is not sufficient. Examples include complex structure, symplectic structure (as opposed to almost complex structures and almost symplectic structures).

Another example is for a foliation, which requires a reduction of the tangent bundle to a block matrix subgroup, together with an integrability condition so that the Frobenius theorem applies.