On almost complex structures which are not compatible with symplectic forms (Q533981)

The author proves the following result: Let \((M,g,J)\) be an almost Hermitian manifold admitting a Hermitian connection with skew-symmetric torsion. Let \(p\in M\) be such that \(N_p\neq 0\), where \(N(X,Y)=[JX, JY]-J[JX, Y]-J[X, JY]-[X, Y]\) is the Nijenhuis tensor. Then \(J\) does not satisfy the local symplectic property at \(p\). This means that there is no symplectic form \(\omega\) defined in some neighborhood of \(p\) which is compatible with \(J\).