Editing Almost complex manifold (section)

== Integrable almost complex structures ==
Every [[complex manifold]] is itself an almost complex manifold. In local holomorphic coordinates <math>z^\mu = x^\mu + i y^\mu</math> one can define the maps

:<math>J\frac{\partial}{\partial x^\mu} = \frac{\partial}{\partial y^\mu} \qquad J\frac{\partial}{\partial y^\mu} = -\frac{\partial}{\partial x^\mu}</math>

(just like a counterclockwise rotation of π/2) or

:<math>J\frac{\partial}{\partial z^\mu} = i\frac{\partial}{\partial z^\mu} \qquad J\frac{\partial}{\partial \bar{z}^\mu} = -i\frac{\partial}{\partial \bar{z}^\mu}.</math>

One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.

The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any given point ''p''. In general, however, it is not possible to find coordinates so that ''J'' takes the canonical form on an entire [[neighborhood (topology)|neighborhood]] of ''p''. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If ''M'' admits local holomorphic coordinates for ''J'' around every point then these patch together to form a [[Holomorphic function|holomorphic]] [[atlas (topology)|atlas]] for ''M'' giving it a complex structure, which moreover induces ''J''. ''J'' is then said to be '[[Frobenius theorem (differential topology)|integrable]]'. If ''J'' is induced by a complex structure, then it is induced by a unique complex structure.

Given any linear map ''A'' on each tangent space of ''M''; i.e., ''A'' is a tensor field of rank (1,&nbsp;1), then the '''Nijenhuis tensor''' is a tensor field of rank (1,2) given by

:<math> N_A(X,Y) = -A^2[X,Y]+A([AX,Y]+[X,AY]) -[AX,AY]. \, </math>

or, for the usual case of an almost complex structure ''A=J'' such that <math> J^2=-Id </math>,

:<math> N_J(X,Y) = [X,Y]+J([JX,Y]+[X,JY])-[JX,JY]. \, </math>

The individual expressions on the right depend on the choice of the smooth vector fields ''X'' and ''Y'', but the left side actually depends only on the pointwise values of ''X'' and ''Y'', which is why ''N''<sub>''A''</sub> is a tensor. This is also clear from the component formula

:<math> -(N_A)_{ij}^k=A_i^m\partial_m A^k_j -A_j^m\partial_mA^k_i-A^k_m(\partial_iA^m_j-\partial_jA^m_i).</math>

In terms of the [[Frölicher–Nijenhuis bracket]], which generalizes the Lie bracket of vector fields, the Nijenhuis tensor ''N<sub>A</sub>'' is just one-half of [''A'',&nbsp;''A''].

The '''Newlander–Nirenberg theorem''' states that an almost complex structure ''J'' is integrable if and only if ''N<sub>J</sub>''&nbsp;=&nbsp;0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.

There are several other criteria which are equivalent to the vanishing of the Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):

*The Lie bracket of any two (1,&nbsp;0)-vector fields is again of type (1,&nbsp;0)
*<math>d = \partial + \bar\partial</math>
*<math>\bar\partial^2=0.</math>

Any of these conditions implies the existence of a unique compatible complex structure.

The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether '''S'''<sup>6</sup> admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For [[real-analytic]] ''J'', the Newlander–Nirenberg theorem follows from the [[Frobenius theorem (differential topology)|Frobenius theorem]]; for ''C''<sup>∞</sup> (and less smooth) ''J'', analysis is required (with more difficult techniques as the regularity hypothesis weakens).