Almost Hermitian 4-manifolds of pointwise constant anti-holomorphic sectional curvature (Q1407635)

Let \((M,J,g)\) be an almost Hermitian manifold. A 2-plane \(\alpha\) in \(T_p M\) is called anti-holomorphic (or totally real) if it is orthogonal to \(J \alpha\). We say that \(M\) is of pointwise constant anti-holomorphic sectional curvature if at each point \(p\), the Riemannian sectional curvature is independent on the choice of the anti-holomorphic section \(\alpha\) in \(T_p M\). If \(M\) is 4-dimensional, the main result here (which generalizes one in [\textit{V. Apostolov, G. Ganchev} and \textit{S. Ivanov}, Proc. Am. Math. Soc. 125, No. 12, 3705--3714 (1997; Zbl 0898.53025)]), states that \((M,J,g)\) satisfies the above property iff it is self-dual with \(J\)-invariant Ricci tensor and \(K_{1212}=0\), where \(K\) is the complexification of the Riemannian curvature tensor. Some sufficient conditions for such a manifold to be Kähler are given. A theorem of Schur type for anti-holomorphic sectional curvature in the case of conformally flat quasi-Kähler 4-manifolds is obtained.