  We propose a novel second-order ODE as the continuous-time limit of a
Riemannian accelerated gradient-based method on a manifold with curvature
bounded from below. This ODE can be seen as a generalization of the ODE derived
for Euclidean spaces, and can also serve as an analysis tool. We study the
convergence behavior of this ODE for different classes of functions, such as
geodesically convex, strongly-convex and weakly-quasi-convex. We demonstrate
how such an ODE can be discretized using a semi-implicit and Nesterov-inspired
numerical integrator, that empirically yields stable algorithms which are
faithful to the continuous-time analysis and exhibit accelerated convergence.
