Two-dimensional almost-Riemannian structures with tangency points

Agrachev, Andrei; Boscain, Ugo; Charlot, G.; Ghezzi, Roberta; Sigalotti, Mario

doi:10.1109/cdc.2009.5400489

Cited by 22 publications

(45 citation statements)

References 12 publications

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“…Remark 5 Notice that the distinction between T ⊕ and T ⊖ was used also in [7,18], to obtain respectively a Gauss-Bonnet Theorem for 2-ARSs and a classification of 2-ARSs w.r.t. Lipschitz equivalence (T ⊕ corresponds to a contribution τ q = −1 and T ⊖ corresponds to a contribution τ q = 1, where τ q is defined in [7]).…”

Section: Crests and Valleys Of The Curvaturementioning

confidence: 99%

“…An open question is the convergence or the divergence of the integral of the geodesic curvature on the boundary of a tubular neighborhood of the singular set, close to a tangency point. This question arose in the proof of the Gauss-Bonnet theorem given in [7]. In that paper, thanks to numerical simulations, the authors conjecture the divergence of such integral.…”

Section: Introductionmentioning

confidence: 98%

“…More precisely, in [4,20] the authors studied the generic case without tangency points. In [7] this formula was generalized to the case in which tangency points are present. (For generalizations of Gauss-Bonnet formula in related contexts, see also [6,25,26].)…”

Section: Introductionmentioning

confidence: 99%

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Normal forms and invariants for 2-dimensional almost-Riemannian structures

Boscain

Charlot

Ghezzi

2013

Differential Geometry and its Applications

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Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are "complete" in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be nonsmooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization. *

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Section: Crests and Valleys Of The Curvaturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Normal forms and invariants for 2-dimensional almost-Riemannian structures

Boscain

Charlot

Ghezzi

2013

Differential Geometry and its Applications

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“…They appear naturally when studying limits of Riemmanian metrics or even in the study of certain class of hypoelliptic operators [14]. Such concept appeared first in the work [19] but has aroused interest more recently as shown in the works [2,3,7,8,9,10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

Almost-Riemannian Structures on nonnilpotent, solvable 3D Lie groups

Ayala¹,

Hernández²,

Silva³

2022

Preprint

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“…If for a generic point q of a sub-Riemannian manifold M we have rank D q = dim M , then we call such a structure almost-Riemannian. In this case the set of points q ∈ M where rank D q < dim M is called the singular set and we will denote it by Z. Almost-Riemannian structures were extensively studied in [1,3,[12][13][14]16,17,40]. Unlike sub-Riemannian manifolds they are equipped with an array of canonical Riemannian objects, such as a curvature or volume.…”

Section: Introductionmentioning

confidence: 99%

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

Beschastnyi¹

2021

Preprint

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The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

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Two-dimensional almost-Riemannian structures with tangency points

Cited by 22 publications

References 12 publications

Normal forms and invariants for 2-dimensional almost-Riemannian structures

Normal forms and invariants for 2-dimensional almost-Riemannian structures

Almost-Riemannian Structures on nonnilpotent, solvable 3D Lie groups

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

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