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

Bonnard, Bernard; Caillau, Jean‐Baptiste; Janin, Gabriel

doi:10.1051/cocv/2012020

Cited by 6 publications

(6 citation statements)

References 22 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(compare with the form in [3] obtained in Hamiltonian formalism). One can easily see that H ′ (ν) < 0 < H ′′ (ν) (4.4) and hence the h-cut locus of a point q on the equator is a subarc of the equator (Lemma 3.3 in [4]), and the h-cut locus of a point q, distinct from equator of (S 2 , h λ ) is a subarc of the opposite parallel (Lemma 3.4 in [4]).…”

Section: Remark 43mentioning

confidence: 86%

“…For instance, numerical simulations show that (H + F ) ′′ (ν) ≤ 0, for λ ≤ 1.5, while for λ > 1.5 the function (H + F ) ′′ (ν) can take both, positive and negative values, where ν ∈ (0, √ λ + 1), see Figure 6. Example 4.4 Another example is obtained from the Riemannian 2-sphere of revolution (S 2 , h λ ) given in [3], where h λ is given by (4.1) and m λ (r) = sin r 1 − λ sin 2 r , r ∈ [0, π], λ ∈ (0, 1).…”

Section: Remark 43mentioning

confidence: 99%

“…It is clear that for λ ∈ (0, 1), G ′ λ vanishes at the pair of poles and the equator and the Gaussian curvature, G λ is monotone for λ ∈ (0, 2 3 ) with a local extremum of λ = 2 3 (see [3] for details).…”

Section: Examplesmentioning

confidence: 99%

See 2 more Smart Citations

The cut locus of a Randers rotational 2-sphere of revolution

Hama¹,

Kasemsuwan²,

Sabau³

2018

Publ. Math. Debrecen

View full text Add to dashboard Cite

In the present paper we study structure of the cut locus of a Randers rotational 2-sphere of revolution (M, F = α + β). We show that in the case when Gaussian curvature of the Randers surface is monotone along a meridian the cut locus of a point q ∈ M is a point on a subarc of the opposite half bending meridian or of the antipodal parallel (Theorem 1.1). More generally, when the Gaussian curvature is not monotone along the meridian, but the cut locus of a point q on the equator is a subarc of the same equator, then the cut locus of any point q ∈ M different from poles is a subarc of the antipodal parallel (Theorem 1.2). Some examples are also given at the last section. * Mathematics Subject Classification (2010) : 53C60, 53C22. † Keywords: Randers metrics, 2-sphere of revolution, cut locus, Gaussian curvature.2 The 2-sphere of revolution 2.1 The Riemannian 2-sphere of revolution A compact Riemannian manifold (M, h) homeomorphic to a 2-sphere is called a 2sphere of revolution if M admits a point p, called pole, such that for any two points q 1 , q 2 on M with d h (p, q 1 ) = d h (p, q 2 ), there exists an h-isometry f on M satisfying f (q 1 ) = q 2 , and f (p) = p, where d h (·, ·) denoted the h-Riemannian distance function on M.Let (r, θ) denote geodesic polar coordinates around a pole p of (M, h). The Riemannian metric can be expressed as h = dr 2 + m 2 (r)dθ 2 on M \ {p, q}, where q denotes the unique

show abstract

Section: Remark 43mentioning

confidence: 86%

Section: Remark 43mentioning

confidence: 99%

See 1 more Smart Citation

The cut locus of a Randers rotational 2-sphere of revolution

Hama¹,

Kasemsuwan²,

Sabau³

2018

Publ. Math. Debrecen

View full text Add to dashboard Cite

show abstract

“…4.1. Mechanical representation in the Riemannian averaged Kepler case [7,8] In this section, we give a brief description of the mecanical representation in the Riemannian case, following the example of the averaged Kepler metric. In the Riemannian setting, roughly speaking, the time and energy minimization problems are equivalent.…”

Section: Mechanical System and Generalized Morse-reeb Classificationmentioning

confidence: 99%

Zermelo navigation problems on surfaces of revolution and geometric optimal control

Bonnard¹,

Cots²,

Wembe³

2023

ESAIM: COCV

View full text Add to dashboard Cite

In this article, the historical study from Carathéodory-Zermelo about computing the quickest nautical path is generalized to Zermelo navigation problems on surfaces of revolution, in the frame of geometric optimal control. Using the Maximum Principle, we present two methods dedicated to analyzing the geodesic flow and to compute the conjugate and cut loci. We apply these calculations to investigate case studies related to applications in hydrodynamics, space mechanics and geometry.

show abstract

“…which in the case k = 1 corresponds to the standard Grushin metric on S 2 introduced in [9], whose conjugate and cut loci were described in [6,7].…”

Section: Introductionmentioning

confidence: 99%

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

et al. 2014

Self Cite

View full text Add to dashboard Cite

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S 2 associated to the dynamics of spin particles with Ising coupling is analysed using both geometric techniques and numerical simulations.. * Work supported by ANR Geometric Control Methods (project No. NT09 504490).

show abstract

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

Cited by 6 publications

References 22 publications

The cut locus of a Randers rotational 2-sphere of revolution

The cut locus of a Randers rotational 2-sphere of revolution

Zermelo navigation problems on surfaces of revolution and geometric optimal control

Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

Contact Info

Product

Resources

About