the Newlander-Nirenberg theorem

Suppose someone hands you a manifold with a smoothly chosen quarter-turn J in each tangent space and asks: can I find local complex coordinates z^1, ..., z^n in which J is simply multiplication by i? In other words, is this soft almost-complex structure really the shadow of an actual complex manifold? The Newlander-Nirenberg theorem answers this completely, and the answer is a clean yes-or-no test computed from J alone.

The theorem states: an almost-complex structure J on a smooth manifold is integrable — that is, it arises from a holomorphic atlas making the manifold a complex manifold — if and only if its Nijenhuis tensor N_J vanishes identically. Equivalently, the (1,0) tangent distribution (the +i-eigenspace of J) must be closed under the Lie bracket, a Frobenius-type involutivity condition. When N_J = 0, the theorem guarantees the existence of holomorphic coordinate charts; the proof is genuinely hard analysis (solving an overdetermined first-order PDE system, requiring careful estimates), which is why it is far deeper than the smooth Frobenius theorem it generalizes. In the real-analytic category the result is older and easier (Frobenius/Cauchy-Kowalevski); the smooth (C-infinity) case is the celebrated 1957 Newlander-Nirenberg achievement.

Its significance is that it converts a hard global question — 'does this manifold admit a complex structure compatible with J?' — into a pointwise tensor computation. It also explains a dimension miracle: in real dimension 2, N_J is automatically zero, so every almost-complex structure on a surface is integrable, and a choice of conformal structure is the same as a choice of complex structure (each oriented Riemannian surface is a Riemann surface). A caution: the theorem decides integrability of J, but it does not by itself produce a metric or any Kähler property — those are extra layers stacked on top of mere complex-manifold structure.