  In order to minimize a differentiable geodesically convex function, we study
a second-order dynamical system on Riemannian manifolds with an asymptotically
vanishing damping term of the form $\alpha/t$. For positive values of $\alpha$,
convergence rates for the objective values and convergence of trajectory is
derived. We emphasize the crucial role of the curvature of the manifold for the
distinction of the modes of convergence. There is a clear correspondence to the
results that are known in the Euclidean case. When $\alpha$ is larger than a
certain constant that depends on the curvature of the manifold, we improve the
convergence rate of objective values compared to the previously known rate and
prove the convergence of the trajectory of the dynamical system to an element
of the set of minimizers. For $\alpha$ smaller than this curvature-dependent
constant, the best known sub-optimal rates for the objective values and the
trajectory are transferred to the Riemannian setting. We present computational
experiments that corroborate our theoretical results.
