Four-dimensional almost Kähler locally symmetric spaces (Q1924639)

The Goldberg conjecture states that a compact almost Kähler Einstein space must be Kählerian. Positive answers are known for several special cases but for the general case, a positive answer has not yet been published. In this paper, the authors prove that a four-dimensional compact almost Kähler manifold which is locally symmetric is a Kähler manifold. The proof is a case by case examination of the explicitly known list of all four-dimensional locally symmetric manifolds.