On Compact Hermitian Manifolds with Flat Gauduchon Connections - Yang, Bo; Zheng, Fang Yang (Acta mathematica Sinica. English series, 2018)
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.