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.