  Let $(M,g)$ be a compact Riemann surfaces, and $\pi:P\to M$ be a principal
$\mathbb S^1$-bundle over $M$ endowed with a connection $A$. Fixing an inner
product on the Lie algebra of $\mathbb{S}^1$, the connection $A$ and metric $g$
define a Riemannian metric $g_A$ on $P$. In this article, we show that the
Ricci flow equation of metric $g_A$ is equivalent to a system of differential
equations. We will give an explicit solution of normalized Ricci flow equation
of metric $g_A$ in the case where the base manifold is of constant curvature
and the the initial connection $A$ is Yang-Mills. Finally, we will describe
some asymptotic behaviors of these flows.
