Communications in Analysis and Geometry

Volume 23 (2015)

Number 4

Twistor geometry of Riemannian 4-manifolds by moving frames

Pages: 819 – 839

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n4.a4

Authors

Jixiang Fu (School of Mathematical Sciences, Fudan University, Shanghai, China)

Xianchao Zhou (Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, China)

Abstract

In this paper, we use the method of moving frames to characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z, g_t, \mathbb{J}_{\pm})$. Some special metric conditions (including the balanced metric condition, the first Gauduchon metric condition) on $(Z, g_t, \mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary connection on the vertical tangent bundle over the twistor space $Z$, we can recover J. Fine and D. Panov’s result on the condition of the first Chern form being symplectic and P. Gauduchon’s result on the condition of the first Chern form being a $(1,1)$-form respectively.

Keywords

twistor space, anti-self-dual, principal bundle, the first Chern form

2010 Mathematics Subject Classification

Primary 53B21, 53B35, 53C28. Secondary 53C24, 53C25, 53C56.

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Published 13 August 2015