Complex Linear Brackets and Integrable Structures.

[Kreizhn]

Homework Statement

Let (M,J) be an almost complex manifold and [.,.] be the commutator bracket on vector fields. Show that if the map $v \mapsto [v,w]$ is complex linear then J is integrable.

The Attempt at a Solution

This question essentially just boils down to showing that the Nijenhuis tensor is zero. The curious thing is that when I'm computing the Nihenhuis tensor I always end up getting zero without having to use complex linearity of the bracket. Hence I must be making a trivial mistake, since certainly not all almost-complex manifolds are complex. Indeed, the version of the Nijenhuis tensor I am given is

$$N(v,w) = [Jv,Jw] - J[v,Jw] - J[Jv,w] - [v,w]$$
in which case my calculations simply reveal
$$\begin{align*} N(v,w) &= (JvJw-JwJv) - J(vJw - Jw v) - J(Jvw - wJv) - (vw - wv) \\ &= JvJw - Jw Jv - Jv Jw - wv +vw + Jw Jv - vw + wv \\ &= 0 \end{align*}$$

I figure I must be making a foolish mistake somewhere, but I cannot see it.

 

[Kreizhn]

Nevermind, got it. J is a section of the fibre bundle of almost complex structures over M and so is really only defined fibre-wise. Clearly J may act on the Lie brackets themselves, but is not distributive over the components (since they are not vector fields). The result is quite simple though with complex linearity added in, so the problem is resolved.