Twistor spaces with Hermitian Ricci tensor

Davidov, Johann; Muškarov, Oleg

doi:10.1090/S0002-9939-1990-1017845-3

Twistor spaces with Hermitian Ricci tensor
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by Johann Davidov and Oleg Muškarov PDF

Proc. Amer. Math. Soc. 109 (1990), 1115-1120 Request permission

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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