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

By Andrei A. Agrachev, Ugo V. Boscain, Mario Sigalotti. 2006

123 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d73fca230499f4630cc5a8a50d976defe066cb8c

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Citations in the corpus

Normal forms and invariants for 2-dimensional almost-Riemannian structures (2010, methodology): “Concerning the local expression (F3) in Figure 1, in [4] the choice of the smooth parameterized curve was arbitrary and not canonical.”
Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces (2010, methodology): “It is not difficult to prove that property (H0) is generic among 2-ARSs (see [2]).”
The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry (2010, background): “For 2-dimensional ARS, it was proven in [2] that generically the singular set Z is an embedded submanifold of dimension 1 and only 3 types of points exist: the ordinary points where the metric is Riemannian, the Grushin points where the distribution ∆ generated by the vector fields has dimension 1 a”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, methodology): “Gauss–Bonnet type theorems have previously been obtained by Diniz and Veloso [23] for noncharacteristic surfaces in H, and by Agrachev, Boscain and Sigalotti [1] for almost-Riemannian structures.”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, methodology): “Indeed from the results of [7] immediately”