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

Bernard Bonnard (1) , Grégoire Charlot (2) , Roberta Ghezzi (3) , Gabriel Janin (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IF - Institut Fourier
3 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

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.

Domaines: Optimisation et contrôle [math.OC]

Bernard Bonnard, Grégoire Charlot, Roberta Ghezzi, Gabriel Janin. The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry. 2010. ⟨hal-00517193⟩