A Normal Form for Generic 2-Dimensional Almost-Riemannian Structures at a tangency point

Ugo Boscain (1) , Grégoire Charlot (2) , Roberta Ghezzi (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 IF - Institut Fourier
3 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket span a one-dimensional space and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding a normal form at a generic tangency point. The problem happens to be equivalent to finding a smooth canonical parameterized curve passing through the point and transversal to the distribution. It is known that the cut locus from the point is not a good candidate since it is not smooth. Therefore, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a crest of the Gaussian curvature. Such crest is uniquely determined and has a natural parametrization.

Domaines: Optimisation et contrôle [math.OC]; Géométrie métrique [math.MG]

Ugo Boscain, Grégoire Charlot, Roberta Ghezzi. A Normal Form for Generic 2-Dimensional Almost-Riemannian Structures at a tangency point. 2010. ⟨hal-00512380v1⟩