the Nijenhuis tensor
/ NYE-en-howss /
You have installed a quarter-turn rotation J on every tangent space of a manifold (an almost-complex structure). Natural question: does this collection of rotations actually fit together into a genuine notion of holomorphic coordinates, or is it only a point-by-point illusion of complexness? The Nijenhuis tensor is the single gadget that measures the obstruction. It is a tensor built from J and the Lie bracket of vector fields, and it vanishes exactly when the almost-complex structure is the honest complex structure of a complex manifold.
Precisely, for an almost-complex structure J the Nijenhuis tensor N_J is the map taking two vector fields X, Y to N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y], where [.,.] is the Lie bracket. Although it is written with brackets, N_J turns out to be tensorial — its value at a point depends only on the values of X and Y at that point, not on their derivatives — so it really is a tensor (of type (1,2), antisymmetric in X, Y). The deep meaning: complexified tangent vectors split into a +i-eigenspace and a -i-eigenspace of J (the (1,0) and (0,1) parts), and N_J measures precisely the failure of the (1,0) eigenspace distribution to be closed under Lie bracket — i.e. the failure of the Frobenius integrability condition for the holomorphic directions.
Its importance is concentrated in one theorem: the Newlander-Nirenberg theorem says J is integrable (comes from a complex atlas) if and only if N_J is identically zero. In real dimension 2 the Nijenhuis tensor automatically vanishes by antisymmetry, which is why every almost-complex surface is a complex curve and every oriented Riemannian surface is a Riemann surface. A frequent confusion: N_J = 0 is the condition for a complex structure, not for a Kähler structure — Kähler additionally demands that an associated 2-form be closed, a strictly stronger and metric-dependent requirement.
The standard almost-complex structure J_0 on R^{2n} (multiplication by i) has constant coefficients, so all the Lie brackets [JX, JY], [JX, Y], [X, JY], [X, Y] vanish on coordinate fields; hence N_{J_0} = 0 and R^{2n} = C^n is genuinely complex. By contrast, the natural J on S^6 from octonion multiplication gives a non-vanishing N_J, certifying it as merely almost-complex, not complex.
Vanishing N_J certifies integrability; the octonionic J on S^6 has N_J nonzero, so S^6 is only almost-complex.
Despite being written with Lie brackets, N_J is genuinely a tensor (pointwise, derivative-free). And N_J = 0 gives a complex structure only — it says nothing about a metric, let alone the Kähler condition.