Kähler curvature identities for twistor spaces (Q1913498)

The twistor spaces \(Z\) of an oriented Riemannian 4-manifold admits a natural 1-parameter family of Riemannian metrics \(h_t\), \(t > 0\), 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 4-manifolds \(M\) whose twistor spaces \((Z, h_t, J_n)\), \(n = 1\) or 2, satisfy the Kähler curvature identities studied by \textit{A. Gray} [Tôhoku Math. J., II. Ser. 28, 601-612 (1976; Zbl 0351.53040)] and use this to supply examples giving a negative answer to a question raised by him. The proof is based on an explicit formula for the curvature of \((Z, h_t)\) in terms of the curvature of \(M\) [the first two authors, Acta Math. Hung. 58, No. 3/4, 319-332 (1991; Zbl 0747.53030); ibid. 65, No. 4, 421-422 (1994; Zbl 0807.53036)].