Quantum geodesic flows and curvature

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.