  Let $(M, g(t))$, $t\in[0,T)$ be a closed Riemannian $n$-manifold whose
Riemannian metric $g(t)$ evolves by the geometric flow $ \frac{\partial
}{\partial t} g_{ij}=-2S_{ij} $, where $S_{ij}(t)$ is a symmetric two-tensor on
$(M,g(t))$. We discuss differential Harnack estimates for positive solution to
the porous medium equation with potential, $\frac{\partial u}{\partial
t}=\Delta u^{p}+S u$, where $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$, on
time-dependent Riemannian metric evolving by the above geometric flow.
