Curvature characterizations of twistor spaces over four-dimensional Riemannian manifolds (Q1872578)

The author studies the complex contact structure of a twistor space over a self-dual Einstein 4-manifold with nonzero scalar curvature. The author gives a characterization of such twistor spaces as those satisfying a curvature identity and he describes how this result fits in with other areas of research in complex contact geometry. The Hermitian geometry associated to the complex contact strucure is used by the author to prove the following main result. Theorem 6.1. Let \((Z,J)\) be a three-dimensional manifold. Then \(Z\) is biholomorphic to the twistor space of a self-dual, Einstein four-manifold with nonzero scalar curvature, if and only if \(Z\) has a Hermitian metric \(h\) and a \(J\)-invariant splitting \(TZ={\mathcal V}\oplus {\mathcal H}\) such that \(\dim_\mathbb{R} {\mathcal H}=4\), \({\mathcal V}\) is totally geodesic (and hence integrable), \({\mathcal H}\) is holomorphic with \(h\) bundlelike through \({\mathcal V}\), and, for every unit \(U\in{\mathcal V}\), the Riemannian curvature \(R\) of \(Z\) satisfies \[ \begin{multlined} R_{XY}U= \lambda u(Y){\mathcal H}{\mathcal X}+\mu v(Y)J'X-\lambda u(X){\mathcal H}Y- \mu v(X)J'Y+\\ \bigl[\nu_1 h(J'X,Y)+ \nu_2h(J''X,Y) \bigr] V,\end{multlined} \] where \(V=-JU\), \(u(E)=h(U,E)\), \(v(E)=h(V,E)\) and \(J'=J \circ {\mathcal H}\) and \(J''=J\circ {\mathcal V}\) for some constants, \(\lambda,\mu, \nu_1\), and \(\nu_2\), such that \(\lambda+ \mu\neq 0\) and \(\nu_2>0\). The author discusses the relation of this theorem with another theorem of his own [Kodai Math. J. 23, 12-26 (2000; Zbl 1028.53049)], and with the concept of normal complex contact manifold due to \textit{S. Ishihara} and \textit{M. Konishi} in [Kodai Math. J. 3, 385-396 (1980; Zbl 0455.53032)] and \textit{B. Korkmaz} [Rocky Mt. J. Math. 30, 1343-1380 (2000; Zbl 0990.53080)].