Two-dimensional almost-Riemannian structures with tangency points

  • A.A. Agrachev

    SISSA, via Beirut 2-4, 34014 Trieste, Italy
  • U. Boscain

    Centre de Mathématiques Appliquées, École Polytechnique Route de Saclay, 91128 Palaiseau Cedex, France
  • G. Charlot

    Institut Fourier, UMR 5582, CNRS/Université Grenoble 1, 100 rue des Maths, BP 74, 38402 St Martin d'Hères, France
  • R. Ghezzi

    SISSA, via Beirut 2-4, 34014 Trieste, Italy
  • M. Sigalotti

    INRIA Nancy – Grand Est, Équipe-projet CORIDA, France, Institut Élie Cartan, UMR CNRS/INRIA/Nancy Université, BP 239, 54506 Vandœuvre-lès-Nancy, France

Abstract

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.

Cite this article

A.A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 3, pp. 793–807

DOI 10.1016/J.ANIHPC.2009.11.011