A note on almost Kähler manifolds (Q1969677)

The Goldberg conjecture asserts that the almost complex structure \(J\) of a compact Einstein almost Kähler manifold \((M,J,g)\) is integrable (and hence \(M\) a Kähler manifold). This has been proved by \textit{K. Sekigawa} [J. Math. Soc. Japan 39, No. 4, 677-684 (1987; Zbl 0637.53053)] for the case when the scalar curvature is non-negative. A different generalization of the spaces of constant sectional curvature than the Einstein manifolds, provides the conformally flat manifolds. In the present paper, a family of almost Kähler conformally flat manifolds \(M^{2n}\), \(n\geq 2\), which are not Kähler is constructed and their connection with existing examples is commented upon.