I am stuck while reading what appears to be a very basic differential geometric argument about existence of torsion-free symplictic connections from here: https://arxiv.org/pdf/math/0511194.pdf It is at the bottom of the page 1, and is about the following. Let $\nabla^0$ be a torsion-free connection, and $\omega$ a symplectic from on a manifold $M$. Define the tensor $N(X,Y)$ as the one the satisfies $$ \nabla^0_X \omega(Y,Z) = \omega(N(X,Y),Z) $$ Then it is claimed that since $\omega$ is closed, the cyclic sum $$ \omega(N(X,Y),Z) + \omega(N(Y,Z),X) + \omega(N(Z,X),Y) $$ vanishes.
Why is this true?
Use the definition of the covariant derivative, the antisymmetry of $\omega$, and $\nabla_{X}Y - \nabla_{Y}X = [X, Y]$ (Lie bracket): \begin{align} \begin{split} \omega(N(X, Y), Z) &+ \omega(N(Y, Z), X) + \omega(N(Z, X), Y)\\ & = (\nabla_{X}\omega)(Y, Z) + (\nabla_{Y}\omega)(Z, X) + (\nabla_{Z}\omega)(X, Y)\\ &= X\omega(Y, Z) + Y\Omega(Z, X) + Z \omega(X, Y) \\&\quad- \omega(\nabla_{X}Y, Z) - \omega(Y, \nabla_{X}Z) - \omega(\nabla_{Y}Z, X) - \\ &\qquad \omega(Z, \nabla_{Y}Z) - \omega(\nabla_{Z}X, Y) - \omega(X, \nabla_{Z}Y)\\ & = X\omega(Y, Z) + Y\Omega(Z, X) + Z \omega(X, Y)\\ &\quad - \omega(\nabla_{X}Y- \nabla_{Y}X, Z) - \omega(\nabla_{Y}Z - \nabla_{Z}Y, X) - \omega(\nabla_{Z}X - \nabla_{X}Z, Y)\\ & = X\omega(Y, Z) + Y\Omega(Z, X) + Z \omega(X, Y) \\ &\quad - \omega([X, Y], Z) - \omega([Y, Z], X) - \omega([Z, X], Y)\\ & = d\omega(X, Y, Z). \end{split} \end{align}