Almost complex structures and integrability

Let M be a smooth real manifold of even dimension and let J be a smooth real bundle endomorphism whose square is minus identity. Complexifying the tangent bundle produces the plus-i and minus-i eigenbundles, called the tangent bundles of types one-zero and zero-one. Their direct sum is the full complexified tangent bundle, and complex conjugation exchanges them. This construction needs only the algebraic identity and smoothness. It does not yet supply coordinate vector fields with the desired eigenvalues.

Conversely, a smooth complex subbundle complementary to its conjugate defines J by assigning eigenvalue minus i to that subbundle and plus i to its conjugate. The resulting J preserves the underlying real tangent bundle. This description is useful for building examples: prescribe candidate zero-one fields, verify linear independence together with their conjugates, then test whether their brackets remain in the zero-one span. Complementarity tests the pointwise structure, while the bracket test addresses its integrability.

Use the Nijenhuis convention displayed below for smooth real vector fields. Its sign and scalar normalization vary between references, but its vanishing does not. It is tensorial: derivative terms involving multiplication of an argument by a function cancel. Vanishing is equivalent to closure of the zero-one eigenbundle under Lie brackets. In genuine holomorphic coordinates, coordinate zero-one fields commute and brackets of their smooth combinations remain of the same type, proving necessity.

The smooth Newlander–Nirenberg theorem states the converse: on a smooth manifold with a smooth, meaning infinitely differentiable, almost complex structure, vanishing Nijenhuis tensor gives local holomorphic coordinates inducing J. This is a deep local existence theorem, invoked here rather than proved. Sharper finite-regularity formulations exist, but no claim about the minimal regularity threshold is needed or made. A continuous J cannot be inserted into the displayed classical bracket formula without further interpretation, and algebraic vanishing at a single point is not the required neighborhood condition.

A constant J on a real vector space is integrable: choose a real-linear identification taking it to standard multiplication by i. Its constant eigenfields commute, and its Nijenhuis tensor vanishes. The identifying real matrix need not commute with the standard J; commuting is required when the source and target already carry the same standard structure. Pulling the structure back along a general real change of coordinates instead changes J itself. Confusing these two operations would wrongly reject many integrable constant structures.

In real dimension two, the zero-one eigenbundle has complex rank one, and brackets of its sections automatically remain in that line. A smooth almost complex structure there is therefore integrable by the theorem. Higher dimensions allow distinct zero-one fields whose bracket acquires a one-zero component. The practice example constructs exactly that obstruction in complex two-space. Checking that bracket is more informative than checking the determinant of a coordinate matrix, because a determinant contains no derivative information about how an eigendistribution changes from point to point.