Frölicher–Nijenhuis bracket

Summary

The Frölicher–Nijenhuis bracket is a graded Lie bracket on differential forms with values in the tangent bundle of a smooth manifold. On vector fields, which are the forms of degree $0$, it is the usual Lie bracket of vector fields. It is defined through the derivations of the algebra of differential forms that commute with the exterior derivative. It expresses integrability conditions, such as those of almost complex structures and of connections, as the vanishing of a bracket.