  In 2004, Manning showed that the topological entropy of the geodesic flow for
a surface of negative curvature decreases as the metric evolves under the
normalised Ricci flow. It is an interesting open problem, also due to Manning,
to determine to what extent such behaviour persists for higher dimensional
manifolds. In this short note, we describe the problem and give a strong
criterion under which monotonicity of the topological entropy can be
established for a short time. In particular, the criterion applies to metrics
of negative sectional curvature which are in the same conformal class as a
metric of constant negative sectional curvature.
