Nijenhuis tensor of an almost-complex structure

Definition

Let $J$ be an almost-complex structure on a smooth manifold $M$. Its Nijenhuis tensor is the vector-valued two-form

for smooth vector fields $X,Y$.

Although the formula uses Lie brackets, its value at a point depends only on the values of $X$ and $Y$ there, so $N_J\in\Omega^2(M;TM)$. After complexifying the tangent bundle, $N_J=0$ exactly when the $+i$-eigenbundle of $J$ is closed under Lie brackets.

The Newlander–Nirenberg theorem identifies the vanishing of $N_J$ with integrability in the smooth category.