Title: On compact Hermitian manifolds with flat Gauduchon connections,
Abstract: 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 $s$-Gauduchon connection of $M$, 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
$s\geq 4+2\sqrt{3} \approx 7.46$ or $s\leq 4-2\sqrt{3}\approx 0.54$ and $s\neq
0$, then $g$ is Kähler. We also show that, when $n=2$, $g$ is always Kähler
unless $s=2$. Note that non-Kähler compact Bismut flat surfaces are exactly
those isosceles Hopf surfaces by \cite{WYZ}.