Integrability for Almost Complex Structure

Nijenhuis Tensor

Let $(M,J)$ be an almost complex manifold. Its Nijenhuis tensor $\mathcal{N}$ is:

\mathcal{N}(v,w) := [Jv, Jw] - J[v, Jw] - J[Jv, w] - [v, w],$$

where $v$ and $w$ are vector fields on $M$, $[\cdot ,\cdot ]$ is the usual bracket $[v,w]\cdot f:=v\cdot (w\cdot f)-w\cdot (v\cdot f),\text{for }f\in C^{\infty }(M),$ and $v\cdot f=df(v)$.