Compact Liouville surfaces

Kiyohara, Kazuyoshi

doi:10.2969/jmsj/04330555

Cited by 49 publications

(45 citation statements)

References 2 publications

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“…For the projective plane and for the Klein bottle such examples can be easily constructed using Beltrami's examples on the sphere and Dini's examples on the torus. Corollary 3 almost follows from Kolokol'tsov [7] or Kiyohara [5]. More precisely, the integral I 0 is quadratic in velocities so that the existence of a projectively equivalent metric implies the existence of an integral quadratic in velocities.…”

Section: Corollary 2 Let G Be a Riemannian Metric On M N Suppose Tmentioning

confidence: 92%

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Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Matveev¹,

Topalov²

2000

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. Let g be a Riemannian metric with ergodic geodesic flow. Then if some metricḡ has the same geodesics (regarded as unparameterized curves) with g, then the metrics are homothetic. If two metrics on a closed surface of genus greater than one have the same geodesics, then they are homothetic.Let g,ḡ be two C 2 -smooth Riemannian metrics on a manifold M n of dimension n ≥ 2. They are projectively equivalent if they have the same geodesics regarded as unparameterized curves. For a given metric g, there always exist trivial examples of projectively equivalent metrics: for any positive constant C, the metric Cg is projectively equivalent to g.Consider the fiberwise-linear mapping G :and the fiberwise-linear mapping A : T M n → T M n given byConsider the characteristic polynomial det(A − λ Id) = c 0 λ n + c 1 λ n−1 + · · · + c n and the mappings S 0 , S 1 , . . . , S n−1 : T M n → T M n given byHere m lies in the set {1, 2, . . . , n} so that k = n − m lies in the set {0, 1, . . . , n − 1}. Consider the functions I 0 , I 1 , . . . , I n−1 : T * M n → R given by the general formulaIn invariant terms, if we identify T M n with T * M n via the metric g, the functions If the geodesic flow of a metric is ergodic, any smooth integral is a function of the Hamiltonian and therefore only trivial examples of projective equivalence are possible. Basically we need not ergodicity but transitivity (= the existence of an

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Section: Corollary 2 Let G Be a Riemannian Metric On M N Suppose Tmentioning

confidence: 92%

“…This fact was shown in [5] provided that the integral is not a linear combination of the Hamiltonian and the square of some linear in velocities integral; the latter condition does not allow us to apply directly the result of [5] to our situation.…”

Section: Corollary 2 Let G Be a Riemannian Metric On M N Suppose Tmentioning

confidence: 97%

Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Matveev¹,

Topalov²

2000

Electron. Res. Announc. Amer. Math. Soc.

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“…Global classification of Liouville metrics is discussed in [Ki,IKS,Kol,M 2 ]. The final classification was achieved in [M 1 ], see the review [BMF] for other contributions, results and references.…”

Section: Liouville Metrics: Some Global Questionsmentioning

confidence: 99%

Invariant characterization of Liouville metrics and polynomial integrals

Kruglikov

2008

Journal of Geometry and Physics

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A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals of geodesic flows.

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“…We present an example of a Liouville surface diffeomorphic to the torus which is of elementary type (see Proposition 3.8 of [32]). …”

Section: Tori: An Examplementioning

confidence: 99%

Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

Sinclair

Tanaka

2006

Math. Comp.

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Abstract. Numerical evidence is presented which strongly suggests that "Jacobi's last geometric statement"-that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment-holds for compact real analytic Liouville surfaces diffeomorphic to S 2 if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima, and two global maxima (as is the case for an ellipsoid). Our experiments suggest that this is a sufficient rather than a necessary condition. Furthermore, for compact real analytic Liouville surfaces diffeomorphic to S 2 upon which the Gaussian curvature can be negative but has exactly six critical points, these being two saddles, two global minima, and two global maxima, it appears that the cut locus is always a subarc of a line given by x 1 = const or x 2 = const, where (x 1 , x 2 ) are canonical coordinates with respect to which the metric has the form (f 1 (x 1 ) + f 2 (x 2 ))(dx 2 1 + dx 2 2 ). In the case of an ellipsoid, these curves are lines of curvature.The point of this paper is to present a conjecture, already contained in the abstract, as a contribution to pure mathematical research in global Riemannian geometry. The overwhelming bulk of the paper will however necessarily be devoted to a description of the computational methods used to motivate the conjecture, and, in particular, the checks which have been performed to verify that the software (which has been written specifically for this study) is indeed performing correctly. Such checks are vital in any experimental work. This paper is not intended to be a contribution to computational science as such, since the algorithms we have used, although perhaps combined in an unusual manner, are standard.The conjugate locus from a point p (which we will often refer to as the starting point) on a Riemannian manifold is the envelope of the geodesics emanating from p. In two dimensions, it can also be defined analogously to the nodes of a vibrating string as follows. A geodesic passing through p can be uniquely defined by the angle θ it makes at p with one fixed (reference) geodesic also passing through p. Varying this angle by a small amount (δθ) will cause the geodesic to deviate from its original path. The distance between points on the geodesic with angle θ (say g θ ) and the geodesic curve with angle θ + δθ (write g θ+δθ ) can be written as a real function ξ δθ of arclength s along g θ , where s = 0 corresponds to the point p. Clearly ξ δθ (0) = 0, since we are only considering geodesics which pass through p. The limit of ξ δθ (s)/δθ as δθ approaches zero may have further zeros (the nodes of the string, where, to

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Compact Liouville surfaces

Cited by 49 publications

References 2 publications

Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Invariant characterization of Liouville metrics and polynomial integrals

Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

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