Two-Dimensional Almost-Riemannian Structures with Tangency Points

Andrei Agrachev (1) , Ugo Boscain (2) , Grégoire Charlot (3) , Roberta Ghezzi (1) , Mario Sigalotti (4, 5)

1 SISSA / ISAS - Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 IF - Institut Fourier
4 IECN - Institut Élie Cartan de Nancy
5 CORIDA - Robust control of infinite dimensional systems and applications

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. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

Domains: Differential Geometry [math.DG]; Optimization and Control [math.OC]

Andrei Agrachev, Ugo Boscain, Grégoire Charlot, Roberta Ghezzi, Mario Sigalotti. Two-Dimensional Almost-Riemannian Structures with Tangency Points. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2010, 27 (3), pp.793-307. ⟨10.1016/j.anihpc.2009.11.011⟩. ⟨hal-00410127⟩