Curvature properties of the Chern connection of twistor spaces

Davidov, J.; Grantcharov, G.; Muskarov, O.

Curvature properties of the Chern connection of twistor spaces

Authors

Davidov, J.

Grantcharov, G.

Muskarov, O.

Description

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 this paper we compute the first Chern form of the almost Hermitian manifold (\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.
14 pages, to appear in Rocky Mountain J. Math