Curvature properties of the Chern connection of twistor spaces

2005/03/18 by J. Davidov, Davidov, J., G. Grantcharov +3 · 2 citations
Mathematics · #53C15 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C15

paper · pdf · doi:10.48550/arxiv.math/0503384

14 pages, to appear in Rocky Mountain J. Math

arxiv created 2005/03/18 · arxiv updated 2009/12/01

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 J₁ and J₂ 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 Dⁿ is of type (1,1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to Dⁿ and show that the Nijenhuis tensor of J₂ is D²-parallel provided the base manifold M is Einstein and self-dual.

Cited by

• Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture
• On Kodaira dimension and scalar curvature in almost Hermitian geometry

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