Inserted: 7 jul 2018
Last Updated: 7 jul 2018
We study infinitesimal and asymptotic properties of geodesics (i.e., isometric images of intervals) in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of 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. With the same approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result can be extended to rough geodesics.
Keywords: Carnot groups, sub-Riemannian geometry, Geodesics, tangent cones, sub-Finsler geometry, regularity of length minimizers, asymptotic cones