On compact Hermitian manifolds with flat Gauduchon connections

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 $\nabla^s = (1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$the$s$-Gauduchonconnectionof$M$,where$\nabla^c$and$\nabla^b$ 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 \cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connection) are classified respectively by Boothby \cite{Boothby} in the 1950s or by Q. Wang and the authors recently \cite{WYZ}. In this article, we observe that if either $s\ge 4+2\sqrt{3}\approx 7.46$ or $s\le 4-2\sqrt{3}\approx 0.54$ and $s\neq 0$,then$g$isK\ddot{a}hler\mathrm{.}Wealsoshowthat,when$n=2$,$g$ is always K"ahler unless $s=2$. Note that non-K"ahler compact Bismut flat surfaces are exactly those isosceles Hopf surfaces by \cite{WYZ}.