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Some integrability conditions for almost K\"ahler manifolds

Klaus-Dieter Kirchberg
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Abstract

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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Klaus-Dieter Kirchberg. 2003-04-07. Some integrability conditions for almost K\"ahler manifolds. https://doi.org/10.1016/s0393-0440(03)00085-8

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