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October  2008, 20(4): 801-822. doi: 10.3934/dcds.2008.20.801

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

Andrei Agrachev 1, , Ugo Boscain 2, and  Mario Sigalotti 3,

1. 

SISSA, via Beirut 2-4, 34014 Trieste, Italy
2. 

SISSA, via Beirut 2-4 34014 Trieste
3. 

Institut de Mathématiques Élie Cartan de Nancy, UMR 7502 INRIA/Universités de Nancy/CNRS, POB 239, 54506 Vandoeuvre-lès-Nancy, France

Received  December 2006 Revised  August 2007 Published  January 2008

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 two-dimensional 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 almost-Riemannian structures. They are special cases of \ar s, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian 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 almost-Riemannian structures of the Gauss-Bonnet formula.
Keywords: Generalized Riemannian geometry, conjugate points, optimal control., Grushin plane, rank-varying distributions, Gauss-Bonnet formula.
Mathematics Subject Classification: Primary: 49j15; Secondary: 53c1.
Citation: Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801
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