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A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. (English) Zbl 1198.49041
Let \(M\) be a two-dimensional smooth manifold and consider a pair of smooth vector fields \((X,Y)\) on \(M\). If the pair is Lie bracket generating, that is, if \(\text{span}\{X(q),Y(q),[X,Y](q),[X,[X,Y]](q),\dots\}\) is full-dimensional at every \(q\in M\), then the control system \(\dot q=u X(q)+v Y(q)\), \(u^ 2+v^ 2\leq 1\), is completely controllable and the minimum-time function defines a continuous distance \(d\) on \(M\). If \(X\) and \(Y\) are everywhere linearly independent, such a distance is Riemannian and it corresponds to the metric for which \((X,Y)\) is an orthonormal moving frame. The idea is to study the geometry obtained starting from a pair of vector fields which may become collinear. The set \({\mathcal Z}\) of points of \(M\) at which \(X\) and \(Y\) are parallel is a one-dimensional embedded submanifold of \(M\). Metric structures that are defined locally by a pair of vector fields \((X, Y)\) with the above control system are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on \(M\). It is interesting to see that even in the case where the Gaussian curvature is everywhere negative, that is on \(M\setminus{\mathcal Z}\), geodesics may have conjugate points. For this reason it seems interesting to analyze the relations between the curvature, the presence of conjugate points, and the topology of the manifold. After providing a characterization of generic almost-Riemannian structures by means of local normal forms, in this paper the authors start this analysis by proving a generalization of the Gauss-Bonnet formula. Let \(M\) be compact, oriented, and endowed with an orientable almost-Riemannian structure with the Gaussian curvature \(K: M\setminus{\mathcal Z}\to\mathbb R\). In order to extend the Gauss-Bonnet formula, the authors present the meaning of \(\int_ M K \,dA\), the integral of \(K\) on \(M\) with respect to the Riemannian density \(dA\) induced by the Riemannian metric on \(M\setminus{\mathcal Z}\).

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
53C17 Sub-Riemannian geometry
49J15 Existence theories for optimal control problems involving ordinary differential equations
93B05 Controllability
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