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I'm in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any two vector fields $X,Y$ one has $ J[X,Y]=J[X,JY]+J[JX,Y]+[X,Y] $ I tried applying all the definitions of commutator I actually know, but I can't manage it...

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Edit: Am I using the wrong definition of commutator? Can you please confirm me that $[X,Y]$ is defined for $X=X_i\partial_i$, $Y=Y_j\partial_j$ to be the vector field $ \sum_{i=1}^N\Big(\sum_{j=1}^N X_j\frac{\partial Y_i}{\partial x_j}-Y_j\frac{\partial X_i}{\partial x_j}\Big)\partial_i $

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    I've checked in several sources and they all agree with "my" version, but $[X,Y] = YX - XY$ in those sources, which is $-$ what you write.2011-07-22

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"Kobayashi-Nomizu's Foundations states the result in the form of theorem 2.5 page 124, and they define the Nijenhuis tensor in "my" way, upto a factor 2"

It's in the volume II, anyway from this book follow that [JX, JY]= [X, Y]+ J[X, JY]+ J[JX, Y] (for a complex strucure J)

in your expression above on the left you write the term J[X, Y] instead [JX, JY].

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    @tetrapharmakon: replace $X$ by $JX$ then you get a minus sign on the left and you're reduced to the previous identity.2011-07-22