A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds

Abstract

We consider a generalization of Riemannian geometry that naturallyarises in the framework of control theory. Let $X$ and $Y$ be twosmooth vector fields on a two-dimensional manifold $M$. If $X$ and$Y$ are everywhere linearly independent, then they define aclassical Riemannian metric on $M$ (the metric for which they areorthonormal) and they give to $M$ the structure of metric space. If$X$ and $Y$ become linearly dependent somewhere on $M$, then thecorresponding Riemannian metric has singularities, but under genericconditions the metric structure is still well defined. Metricstructures that can be defined locally in this way are calledalmost-Riemannian structures. They are special cases of \ar s,which are naturally defined in terms of submodules of the space ofsmooth vector fields on $M$. Almost-Riemannian structures showinteresting phenomena, in particular those which concern therelation between curvature, presence of conjugate points, andtopology of the manifold. The main result of the paper is ageneralization to almost-Riemannian structures of the Gauss-Bonnetformula.