Nijenhuis tensor

Summary

Let $J$ be a field of endomorphisms of the tangent bundle. Its Nijenhuis tensor is a tensor field of type $(1,2)$ built from $J$ and the Lie bracket of vector fields. It is, up to a factor $2$, the Frölicher–Nijenhuis bracket of $J$ with itself. For an almost complex structure, it measures the obstruction to integrability. By the Newlander–Nirenberg theorem, an almost complex structure comes from a complex manifold structure exactly when its Nijenhuis tensor vanishes.