On almost Hermitian 4-manifolds with \(J\)-invariant Ricci tensor (Q1125244)

The author states the following theorem: ``Theorem 3.1: Every conformally flat, almost Hermitian 4-manifold with \(J\)-invariant Ricci tensor is either a space of constant curvature or a Kähler manifold. In the second case locally the manifold is flat or a product space of two 2-dimensional Kählerian manifolds of constant curvature \(K\) and \(-K\), respectively.'' This theorem is not completely correct as the following example shows: in \textit{V. Apostolov} and \textit{P. Gauduchon} [Int. J. Math. 8, 421-439 (1997; Zbl 0891.53054)], the authors provide a method (Corollary 3) to obtain non-Kähler and non-Einstein Hermitian metrics with \(J\)-invariant Ricci tensor on compact 4-manifolds; such method applied to \(\mathbb{C} P^1\times \Sigma_g\), \(g\geq 2\), gives examples which contradict the statement of the theorem above, in the paper under review.