On Compact Hermitian Manifolds with Flat Gauduchon Connections

• Published in: Acta mathematica Sinica. English series Vol. 34; no. 8; pp. 1259 - 1268
• Main Authors: Yang, Bo, Zheng, Fang Yang
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.08.2018; Springer Nature B.V
• Subjects: Flat surfaces; Joints; Manifolds; Mathematics; Mathematics and Statistics; Topological manifolds; Kähler manifolds; Hermitian connections; 53C55; Hermitian manifolds
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-018-7409-y

Given a Hermitian manifold ( M n , g ), the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call ∇ s = ( 1 − s 2 ) ∇ c + s 2 ∇ b the s -Gauduchon connection of M , where ∇ c and ∇ b are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat s -Gauduchon connection. This is related to a question asked by Yau. The cases with s = 0 (a flat Chern connection) or s = 2 (a flat Bismut connection) are classified respectively by Boothby in the 1950s or by the authors in a recent joint work with Q. Wang. In this article, we observe that if either s ≥ 4 + 2 3 ≈ 7.46 or s ≤ 4 − 2 3 ≈ 0.54 and s ≠ 0, then g is Kähler. We also show that, when n = 2, g is always Kähler unless s = 2. Therefore non-Kähler compact Gauduchon flat surfaces are exactly isosceles Hopf surfaces.