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  "body": "\n## Statement\n\nLet $(M, J)$ be an [[almost complex manifold|9SYWRGVTRS]] of dimension $2n$, with $J$ smooth.\nLet $N_J$ be the Nijenhuis tensor of $J$.\nThen the following are equivalent.\n\n1. $N_J = 0$ on $M$.\n2. $M$ has an atlas of charts $\\varphi \\colon U \\to \\C^n$ whose transition maps are holomorphic and such that $d\\varphi \\circ J = i \\cdot d\\varphi$ on each $U$.\n\nIn that case $J$ is the almost complex structure of a unique complex manifold structure on $M$.\n\n## Remarks\n\nThe implication from 2 to 1 is a direct computation: in holomorphic charts $J$ has constant coefficients, so all the brackets in $N_J$ cancel.\nThe hard direction is from 1 to 2.\nWhen $J$ is real analytic, consider the bundle of $(0,1)$-vectors.\nThe condition $N_J = 0$ says that this bundle is closed under the bracket, and a complex form of the [[Frobenius theorem|5Q4BHBFXJY]] gives the charts.\nFor smooth $J$ one must solve a nonlinear system of partial differential equations of $\\bar\\partial$ type, which is the analytic heart of the proof.\n\nIn real dimension $2$ the tensor $N_J$ always vanishes, because $N_J(X, JX) = 0$ and $N_J$ is antisymmetric.\nSo every almost complex structure on a surface is integrable.\nIn particular, an oriented surface with a [[Riemannian metric|Q61VFHCH2J]], with $J$ the rotation by a right angle, is a Riemann surface.\n\n## Explanations\n\n### L1 human\n\nAn almost complex structure lets one multiply tangent vectors by $i$, but it may not come from complex coordinates.\nThe theorem gives a test: a certain tensor built from $J$ and brackets of vector fields must vanish.\nWhen it vanishes, complex coordinates exist near every point.\n\n### L2 human\n\nThe theorem turns the question \"is $M$ a complex manifold\" into the vanishing of a tensor, a condition that can be checked locally.\nIt is used to show that Kähler and many other geometric structures defined by tensors are really complex manifolds.\nOn surfaces it contains the existence of isothermal coordinates.\nIn higher dimension many almost complex manifolds have non-integrable structures; on $S^6$ the octonionic structure is not integrable, and whether $S^6$ admits any complex structure is an open problem.\n",
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