  Considering Riemannian submersions, we find necessary and sufficient
conditions for when sub-Riemannian normal geodesics project to curves of
constant first geodesic curvature or constant first and vanishing second
geodesic curvatures. We describe a canonical extension of the sub-Riemannian
metric and study geometric properties of the obtained Riemannian manifold. This
work contains several examples illustrating the results.
