  This paper provides some partial regularity results for geodesics (i.e.,
isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler
manifolds. Our strategy is to study infinitesimal and asymptotic properties of
geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show
that tangents of Carnot geodesics are geodesics in some groups of lower
nilpotency step. Namely, every blowup curve of every geodesic in every Carnot
group is still a geodesic in the group modulo its last layer. Then as a
consequence we get that in every sub-Riemannian manifold any $s$ times iterated
tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian
manifold in question. With a similar approach, we also show that blowdown
curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups
of lower rank. This latter result is also extended to rough geodesics.

Metric Geometry;Differential Geometry;Optimization and Control