Local models and integrability of certain almost Kähler 4-manifolds (Q1849483)

In this well-motivated and interesting paper the authors treat non-Kähler almost Kähler four-dimensional manifolds \((M,g,J)\) such that the Riemann curvature tensor satisfies the identity \(R(X,Y,Z,W)= R(JX,JY,J Z,JW)\). This last condition is equivalent to the fact that the Ricci tensor \(\rho\) is invariant with respect to \(J\), i.e., \(\rho(X,Y)= \rho(JX,JY)\), and the Kähler form \(\Omega\) is an eigenform of the Weyl tensor viewed as a symmetric traceless endomorphism acting on the space of two-forms. We only mention the following two main results of the paper: (i) a local classification, up to local isometry, of the spaces mentioned above; (ii) a proof, by using this classification, of the theorem stating that a compact four-dimensional almost Kähler manifold such that its curvature tensor satisfies the above identity is necessarily Kählerian.