On some compact Einstein almost Kähler manifolds

The curvature and the integrability of almost-Kähler manifolds: A survey

“…Indirectly, the Goldberg conjecture predicts that compatible Einstein metrics are very rare on compact symplectic manifolds. The conjecture is still far from being solved, but there are cases when it has been be confirmed: Sekigawa [81] proved that the conjecture is true if the scalar curvature is non-negative (see Theorem 2 below) and there are further positive partial results in dimension four under other additional curvature assumptions [4,12,13,75]. Moreover, a number of subtle topological restrictions to the existence of Einstein metrics on compact 4-manifolds are now known [55,61,62,63], and these can be thought as further support for the conjecture.…”

Some critical almost Kähler structures

“…In [3], Goldberg conjectured that a compact almost Kähler Einstein manifold is integrable. This conjecture is true in the case where the scalar curvature is non-negative ( [8]). However, it is still open in the remaining case.…”

Curvature identities on almost Hermitian manifolds and applications

“…In [11], Goldberg also proposed a conjecture that any almost Kähler Einstein manifold must be Kähler. The conjecture was solved with a further assumption that the scalar curvature is nonnegative by Sekigawa in [22]. However the full conjecture is still open.…”