Curvature properties of the Chern connection of twistor spaces. (English) Zbl 1160.53013

Summary: The twistor space \(\mathcal Z\) 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 this paper we compute the first Chern form of the almost Hermitian manifold \((\mathcal Z,h_t,J_n)\), \(n=1,2\) and find the geometric conditions on \(M\) under which the curvature of its Chern connection \(D^n\) is of type \((1,1)\). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to \(D^n\) and show that the Nijenhuis tensor of \(J_2\) is \(D^2\)-parallel provided the base manifold \(M\) is Einstein and self-dual.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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