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Twistor spaces with Hermitian Ricci tensor

Authors: Johann Davidov and Oleg Muškarov
Journal: Proc. Amer. Math. Soc. 109 (1990), 1115-1120
MSC: Primary 53C55; Secondary 53C25
MathSciNet review: 1017845
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Abstract: The twistor space $ Z$ of an oriented Riemannian $ 4$-manifold $ M$ admits a natural $ 1$-parameter family of Riemannian metrics $ {h_t}$ compatible with the almost-complex structures $ {J_1}$ and $ {J_2}$ introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In the present note we describe the (real-analytic) manifolds $ M$ for which the Ricci tensor of $ \left( {Z,{h_t}} \right)$ is $ {J_n}$-Hermitian, $ n = 1\;{\text{or}}\;2$. This is used to supply examples giving a negative answer to the Blair and Ianus question of whether a compact almost-Kähler manifold with Hermitian Ricci tensor is Kählerian.

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