Integrable almost-complex structure

An almost-complex structure $J$ on a smooth $2n$-manifold is integrable if every point has local coordinates in $\mathbb C^n$ whose ordinary multiplication by $i$ induces $J$. Such charts have holomorphic transition maps and therefore make the manifold a complex manifold.

By the Newlander–Nirenberg theorem, integrability is equivalent to vanishing of the Nijenhuis tensor. The condition is automatic in real dimension two but is a genuine differential constraint in higher dimensions.