Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 801 - 822, Volume 20, Issue 4, April 2008      doi:10.3934/dcds.2008.20.801

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Andrei Agrachev - SISSA, via Beirut 2-4, 34014 Trieste, Italy (email)
Ugo Boscain - SISSA, via Beirut 2-4 34014 Trieste, Italy (email)
Mario Sigalotti - Institut de Mathématiques Élie Cartan de Nancy, UMR 7502 INRIA/Universités de Nancy/CNRS, POB 239, 54506 Vandoeuvre-lès-Nancy, France (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 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, Grushin plane, rank-varying distributions, Gauss-Bonnet formula, conjugate points, optimal control.
Mathematics Subject Classification:  Primary: 49j15; Secondary: 53c17.

Received: December 2006;      Revised: August 2007;      Published: January 2008.