The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry

Bonnard, Bernard; Charlot, G.; Ghezzi, Roberta; Janin, Gabriel

doi:10.1007/s10883-011-9113-4

Cited by 44 publications

(34 citation statements)

References 11 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The singular set Z ⊂ N is the set of points where dim D q < n. Tagency points have deep consequences on the local structure of the almost-Riemannian metric structure, and have been studied, in the 2-dimensional case, in [3,7]. If Z is a smooth, embedded submanifold, for all q ∈ Z there exists a non-zero λ ∈ T * q N , defined up to multiplication by a constant, such that λ(T q Z) = 0.…”

Section: Almost-riemannian Geometrymentioning

confidence: 99%

Quantum confinement on non-complete Riemannian manifolds

2018

View full text Add to dashboard Cite

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure ω, possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V ∈ L 2 loc (M ). The main merit of this paper is the identification of an intrinsic quantity, the effective potential V eff , which allows to formulate simple criteria for quantum confinement. Let δ be the distance from the possibly non-compact metric boundary of M . A simplified version of the main result guarantees quantum completeness if V ≥ −cδ 2 far from the metric boundary andclose to the metric boundary.These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of M ; (ii) generalize the Kalf-Walter-Schmincke-Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace-Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace-Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [9].

show abstract

Section: Almost-riemannian Geometrymentioning

confidence: 99%

Quantum confinement on non-complete Riemannian manifolds

2018

View full text Add to dashboard Cite

show abstract

“…The concept of nilpotent approximation or model of order −1 allows to get estimates of the SR-balls with small radii and more precise approximations have to be used, especially to evaluate the conjugate locus. In particular, this notion was developed in [2,3] and [8,9] respectively in the contact and Martinet case. This kind of computations are recalled next, and are crucial in our analysis.…”

Section: Sr-classification In Dimension 3 and Strokes With Small Amplmentioning

confidence: 99%

“…The solution can be estimated with micro-local expansions, see [2,8] for the problem of computing conjugate points. Note that the weights are similar to analyze periodicity since conjugate times correspond to periods.…”

Section: The Contact Case (See [3] For Concepts and Details Of Computmentioning

confidence: 99%

Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case

et al. 2019

Self Cite

View full text Add to dashboard Cite

Abstract. In [22], based on copepod observations, Takagi proposed a model to interpret the swimming behaviour of these microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and compares them invoking the concept of efficiency. Our aim is to provide an interpretation of Takagi's results in the frame of optimal control theory and sub-Riemannian geometry. The Maximum principle is used to select two types of periodic control candidates as minimizers: sinusoidal up to time reparameterization and the sequential paddling, interpreted as an abnormal stroke in sub-Riemannian geometry. Geometric analysis combined with numerical simulations are decisive tools to compute the optimal solutions, refining Takagi computations. A family of simple strokes with small amplitudes emanating from a center is characterized as an invariant of SR-geometry and allows to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.Résumé. Dans [22],à partir de l'observation du mécanisme de nage d'une famille de microorganismes appelés copépodes, Takagi propose un modèle pour interpréter ces nages. Deux types de contrôles périodiques associés sont proposés : le premier correspondà des contrôles sinusoïdaux formant une courbe de nage lisse et simple et le second est un contrôle constant par morceaux produisant un triangle. Notre objectif est d'interpréter cela dans le cadre du contrôle optimal et de la géométrie sous-Riemannienne. Le principe du Maximum est utilisé pour sélectionner des nages géodésiques de deux types : des nages formant des courbes simples associéesà des contrôles sinusoïdauxà une reparamétrisation du temps près et le triangle est interprété comme une nage anormale. L'analyse géométrique combinée avec des simulations numériques permet de générer une famille de nages de petites amplitudes ce qui par continuation permet de calculer la nage la plus efficace. La notion d'efficacité est discutée en détails en relation avec le concept de normalité.

show abstract

“…In our study the Riemannian situation will be extended to the almost-Riemannian case where the mapping has a pole at the equator. It will be called the Grušin case if the pole is of order one and the tangential case if the pole is of order two (see [1,11] for the analysis of such metrics).…”

Section: Preliminariesmentioning

confidence: 99%

“…This second estimate is the invariant associated with the pole of order 2 at the equator, computed in the tangential case [11].…”

Section: Lemma 319mentioning

confidence: 99%

Conjugate-cut loci and injectivity domains on two-spheres of revolution

Bonnard¹,

Caillau²,

Janin³

2013

ESAIM: COCV

View full text Add to dashboard Cite

to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.Mathematics Subject Classification. 58B20, 49K15, 53C22.

show abstract

The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry

Cited by 44 publications

References 11 publications

Quantum confinement on non-complete Riemannian manifolds

Quantum confinement on non-complete Riemannian manifolds

Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case

Conjugate-cut loci and injectivity domains on two-spheres of revolution

Contact Info

Product

Resources

About