Problem
Problem 6.52 Let $(M, \Omega)$ be an almost symplectic manifold. An almost complex structure $J$ on the manifold $M$ is called compatible with the almost symplectic structure $\Omega$, if $g(X, Y)=\Omega(J X, Y)$ is an Hermitian metric on $M$, i.e.
(a) $\Omega(J X, X)>0$ for any non-zero tangent vector $X \in T_{p} M, p \in M$.
(b) $\Omega(J X, Y)+\Omega(X, J Y)=0$ for any tangent vectors $X, Y \in T_{p} M, p \in M$.
Prove that on any almost symplectic manifold $(M, \Omega)$ there exists an almost complex structure $J$ which is compatible with $\Omega$.
The relevant theory is developed, for instance, in Gromov [14] and Aebisher et al. [1] .