The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry

TL;DR

This paper investigates the structure of the cut locus and conjugate locus at a tangency point in 2D almost-Riemannian geometry, linking it to the Martinet case and providing estimations of the exponential map.

Contribution

It introduces new estimations of the exponential map at tangency points and describes the accumulation pattern of the cut locus, connecting almost-Riemannian and sub-Riemannian geometries.

Findings

The cut locus accumulates at the tangency point as an asymmetric cusp.

The branches of the cusp are separated by the singular set.

The analysis links almost-Riemannian geometry with the Martinet case.

Abstract

We study the tangential case in 2-dimensional almost-Riemannian geometry. We analyse the connection with the Martinet case in sub-Riemannian geometry. We compute estimations of the exponential map which allow us to describe the conjugate locus and the cut locus at a tangency point. We prove that this last one generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.