Riemannian and Sub-Riemannian Geodesic Flows

We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.