Four-dimensional almost Kähler Einstein manifolds (Q807993)

Let (M,J,g) be an almost Kähler manifold. The *-Ricci tensor is defined to be the trace of the map \(Z\to R(X,JZ)JY\), where R denotes the curvature tensor of M. M is said to be *-Einstein if the *-Ricci tensor is proportional to g. The authors prove that a 4-dimensional compact almost Kähler manifold which is Einstein and *-Einstein must be Kähler.