Some integrability conditions for almost Kähler manifolds

@inproceedings{Kirchberg2004SomeIC,
  title={Some integrability conditions for almost K{\"a}hler manifolds},
  author={Klaus-Dieter Kirchberg},
  year={2004}
}
Abstract Among other results, a compact almost Kahler manifold is proved to be Kahler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact almost Kahler manifolds with harmonic Weyl tensor and non-parallel semi-positive Ricci tensor. Stronger results are obtained in dimension 4.