|Sfb 288 Differential Geometry and Quantum Physics|
Abstract for Sfb Preprint No. 584
Some integrability conditions for almost Kähler manifolds.
Among other results, a compact almost K"ahler manifold is proved to be K"ahler 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 K"ahler manifolds with harmonic Weyl tensor and non-parallel semi-positive Ricci tensor. Stronger results are obtained in dimension 4.
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