Cut loci and conjugate loci on Liouville surfaces

Itoh, Jin-ichi; Kiyohara, Kazuyoshi

doi:10.1007/s00229-011-0433-1

Cited by 14 publications

(10 citation statements)

References 18 publications

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“…For the case n = dim M = 2, we need (4.1) for 1 ≤ k ≤ 2, as described in our earlier paper [11]. A typical example satisfying the condition (4.1) is the ellipsoid, in which case A(λ) = √ λ.…”

Section: A Monotonicity Condition For A(λ)mentioning

confidence: 99%

“…We also assume that n = dim M ≥ 3 in the following theorem, since it is necessary in some part of our proof, and since the two-dimensional case is treated in another paper [11].…”

Section: Cut Locus (1)mentioning

confidence: 99%

“…Furthermore, we have seen in [11] that there are many surfaces possessing such simple cut loci. Surfaces we considered in [11] are so-called Liouville surfaces, i.e., surfaces whose geodesic flows possess first integrals which are fiberwise quadratic …”

mentioning

confidence: 99%

“…Also, we gave the complete proof of Jacobi's last geometric statement ([15], [16], see also [28], which contains historical remarks). Furthermore, we have seen in [11] that there are many surfaces possessing such simple cut loci. Surfaces we considered in [11] are so-called Liouville surfaces, i.e., surfaces whose geodesic flows possess first integrals which are fiberwise quadratic…”

mentioning

confidence: 99%

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The Cut Loci on Ellipsoids and Certain Liouville Manifolds

Itoh¹,

Kiyohara

2010

Asian Journal of Mathematics

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Abstract. We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.Key words. Cut locus, ellipsoid, Liouville manifold, integrable geodesic flow.AMS subject classifications. Primary 53C22, Secondary 53A051. Introduction. On a complete riemannian manifold, any geodesic γ(t) starting at a point γ(0) = p has the property that any segment {γ(t) | 0 ≤ t ≤ T } is minimal, i.e., the length of the segment is equal to the distance between the points p and γ(T ), if T > 0 is small. If the supremum t 0 of the set of such T is finite, then the point γ(t 0 ) is called the cut point of p along the geodesic γ(t) (t ≥ 0). The cut locus of the point p is then defined as the set of all cut points of p along the geodesics starting at p. For the general properties of cut loci, we refer to [19], [26].The study of cut locus was started at 1905 by H. Poincaré [22] in the case of convex surfaces, and there are several classical results, for example, [21], [35], [36]. From its definition, the cut locus of a point p on a compact manifold M is homotopically equivalent to M − {p}, but it can be very complicated, see [5], [9]. The structure of cut locus was studied in connection with the singularity theory, see It is well known that the cut locus of any point on the sphere of constant curvature consists of a single point, and it is also known that this property characterizes the sphere of constant curvature (an affirmatively solved case of the Blaschke conjecture, see [1] ([29] is an experimental work). Especially in higher dimensional case there are not many results without symmetric spaces and some singular spaces [14], even if using computational approximations.In the earlier paper [10], we proved that the cut locus of a non-umbilic point on a tri-axial ellipsoid is a segment of the curvature line containing the antipodal point, inspired by an experimental work [12]. Also, we gave the complete proof of Jacobi's last geometric statement ([15], [16], see also [28], which contains historical remarks). Furthermore, we have seen in [11] that there are many surfaces possessing such simple cut loci. Surfaces we considered in [11] are so-called Liouville surfaces, i.e., surfaces whose geodesic flows possess first integrals which are fiberwise quadratic

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Section: A Monotonicity Condition For A(λ)mentioning

confidence: 99%

“…We also assume that n = dim M ≥ 3 in the following theorem, since it is necessary in some part of our proof, and since the two-dimensional case is treated in another paper [11].…”

Section: Cut Locus (1)mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The Cut Loci on Ellipsoids and Certain Liouville Manifolds

Itoh¹,

Kiyohara

2010

Asian Journal of Mathematics

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“…The conjugate locus is a classical topic in Differential Geometry (see Jacobi [16], Poincaré [22], Blaschke [3] and Myers [21]) which is undergoing renewed interest due to the recent proof of the famous Last Geometric Statement of Jacobi by Itoh and Kiyohara [13] (Figure 5, see also [14], [15], [24]) and the recent development of interesting applications (for example [27], [28], [4], [5]). Recent work by the author [30] showed how, as the base point is moved in the surface, the conjugate locus may spontaneously create or annihilate pairs of cusps; in particular when two new cusps are created there is also a new "loop" of a particular type, and this suggests a relationship between the number of cusps and the number of loops.…”

Section: Introductionmentioning

confidence: 99%

The conjugate locus on convex surfaces

Waters

2018

Geom Dedicata

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The conjugate locus of a point on a surface is the envelope of geodesics emanating radially from that point. In this paper we show that the conjugate locus of generic points on convex surfaces satisfy a simple relationship between the rotation index and the number of cusps. As a consequence we prove the 'vierspitzensatz': the conjugate locus of a generic point on a convex surface must have at least four cusps. Along the way we prove certain results about evolutes in the plane and we extend the discussion to the existence of 'smooth loops' and geodesic curvature.geodesic and conjugate locus and Jacobi field and rotation index and evolute and convex arXiv:1806.00278v1 [math.DG]

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Two Applications of Geometric Optimal Control to the Dynamics of Spin Particles

Bonnard

Chyba

2014

The Impact of Applications on Mathematics

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Cut loci and conjugate loci on Liouville surfaces

Cited by 14 publications

References 18 publications

The Cut Loci on Ellipsoids and Certain Liouville Manifolds

The Cut Loci on Ellipsoids and Certain Liouville Manifolds

The conjugate locus on convex surfaces

Two Applications of Geometric Optimal Control to the Dynamics of Spin Particles

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