  We study geodesics flows on curved quantum Riemannian geometries using a
recent formulation in terms of bimodule connections and completely positive
maps. We complete this formalism with a canonical $*$ operation on
noncommutative vector fields. We show on a classical manifold how the Ricci
tensor arises naturally in our approach as a term in the convective derivative
of the divergence of the geodesic velocity field, and use this to propose a
similar object in the noncommutative case. Examples include quantum geodesic
flows on the algebra of 2 x 2 matrices, fuzzy spheres and the $q$-sphere.
