A GaussBonnetlike formula on twodimensional almostRiemannian manifolds
Andrei Agrachev  SISSA, via Beirut 24, 34014 Trieste, Italy (email) Abstract: We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a twodimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almostRiemannian structures. They are special cases of \ar s, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. AlmostRiemannian structures show interesting phenomena, in particular those which concern the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almostRiemannian structures of the GaussBonnet formula.
Keywords: Generalized Riemannian geometry, Grushin plane, rankvarying
distributions, GaussBonnet formula, conjugate points, optimal
control.
Received: December 2006; Revised: August 2007; Available Online: January 2008. 
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