Two-dimensional almost-Riemannian structures with tangency points

Two dimensional almost-Riemannian geometries are metric structures on surfaces defined locally by a Lie bracket generating pair of vector fields. We study the relation between the topology of an almost-Riemannian structure on a compact oriented surface and the total curvature. In particular, we analyse the case in which there exist tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper is a characterization of trivializable oriented almost-Riemannian structures on compact oriented surfaces in terms of the topological invariants of the structure. Moreover, we present a Gauss- Bonnet formula for almost-Riemannian structures with tangency points.