Blowups and blowdowns of geodesics in Carnot groups

TL;DR

This paper investigates the local and asymptotic behavior of geodesics in Carnot groups and sub-Riemannian manifolds, revealing that their tangents and blowdowns simplify to geodesics or lines in lower-dimensional structures.

Contribution

It introduces a novel analysis of infinitesimal and asymptotic properties of geodesics in Carnot groups with sub-Finsler metrics, showing their tangents are geodesics in lower-step groups and blowdowns lie in lower-rank subgroups.

Findings

Tangents of Carnot geodesics are geodesics in lower-step groups.

Iterated tangents of geodesics are lines in sub-Riemannian manifolds.

Blowdown curves are contained in lower-rank subgroups.

Abstract

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.