Newlander–Nirenberg theorem

Statement

Let $J$ be a smooth almost-complex structure on a smooth manifold $M$, and let $N_J$ be its Nijenhuis tensor. The Newlander–Nirenberg theorem states

Equivalently, vanishing of the Nijenhuis tensor guarantees local complex coordinates whose standard multiplication by $i$ induces $J$.

The theorem is local: it produces a complex-manifold structure, not one global coordinate chart. The statement here is in the $C^\infty$ category; finite- and low-regularity versions require their own hypotheses. In real dimension two, every smooth almost-complex structure satisfies the criterion and is therefore integrable.