Riemannian and Sub-Riemannian geodesic flow

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 describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion $\pi:M \to B$, we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in $M$ coincide.