  In this paper, we study curve shortening flows on rotational surfaces in
$\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and
that some condition related to the Gauss curvature and the curvature of
embedded curve holds on them. Under these assumptions, we prove that the curve
remains a graph over the parallels of the rotational surface along the flow.
Also, we prove the comparison principle and the long-time existence of the
flow.
