On Compact Hermitian Manifolds with Flat Gauduchon Connections

Given a Hermitian manifold (Mn, 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.s2).c+s2.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+23...7.46 or s.4.23...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.