Given a $2m$-dimensional manifold $M$, an almost complex structure $J$ is equivalent to a $\text{GL}(m,\mathbb C)$-structure on $M$. 


I wonder why the intrinsic torsion of the $\text{GL}(m,\mathbb C)$-structure is equivalent to the Nijenhuis tensor of $J$. 


Can some one give a suggestion to calculate this or a reference with readable proof. 


Thanks in advance. 



Almost complex structure and intrinsic torsion
Not trying to beat M. G., I would rather provide an elementary explanation of what's essentially going on, trying to convince you that what you've asked is really something one might expect.


Let $\nabla$ be a connection preserving the field of endomorphisms $J$, i. e. $\nabla\_XJY = J\nabla\_XY$, and $T$ be its torsion. If we change the connection, if would differ by a vector valued 2-tensor $A$ satisfying $A(X,JY) = JA(X,Y)$. The torsion $T'$ of the changed connection $\nabla'$ would be given by $T'(X,Y) = T(X,Y) + A(X,Y) - A(Y,X)$.


Suppose we want to come up with an expression in terms of torsion, which would be invariant w. r. t. adding a $J$-linear in its second entry tensor $A$ to the connection. One needs to find a nontrivial relation satisfied by the skew symmetric vector valued 2-tensor $$\alpha(X,Y) = A(X,Y) - A(Y,X),$$ which would give rise to an expression in terms of torsion depending on the field $J$ alone. This is straightforward:


$$\alpha(X,Y) = A(X,Y) - A(Y,X) = -JA(X,JY) + JA(Y,JX) = \\ 
= -J(\alpha(X,JY) + A(JY,X) - \alpha(Y,JX)-A(JX,Y)) = \\
= -J(\alpha(X,JY) + \alpha(JX,Y)) + A(JX,JY) - A(JY,JX) =\\
= -J(\alpha(X,JY) + \alpha(JX,Y)) + \alpha(JX,JY).$$


Hence the tensor $\alpha$ satisfies the formula $$\alpha(X,Y) + J\alpha(X,JY) + J\alpha(JX,Y) - \alpha(JX,JY) = 0,$$


and the corresponding expression involving torsion


$$\nu(X,Y) = T(X,Y) + JT(X,JY) + JT(JX,Y) - T(JX,JY)$$ 


is independent on the choice of the connection.


Let's write the tensor $\nu$ in terms of the connection.


$$\nu(X,Y) = \nabla\_XY - \nabla\_YX - [X,Y] + \\
+ J\nabla\_XJY - J\nabla\_{JY}X - J[X,JY] + \\
+ J\nabla\_{JX}Y - J\nabla\_YJX - J[JX,Y] - \\
- \nabla\_{JX}JY + \nabla\_{JY}JX + [JX,JY].$$


The connection terms cancel out as $\nabla\_XY+J\nabla\_XJY = 0$, $J\nabla\_{JX}Y - \nabla\_{JX}JY = 0$, $-\nabla\_YX - J\nabla\_YJX = 0$ and $-J\nabla\_{JY}X + \nabla\_{JY}JX = 0$. What rests is $$\nu(X,Y) = [JX,JY] - J[X,JY] - J[JX,Y] - [X,Y],$$ i. e. $\nu$ is the Nijenhuis tensor.


