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Superintegrable geodesic flows versus Zoll metrics
Journal of Geometry and Physics ( IF 1.5 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.geomphys.2020.103873
Galliano Valent

Abstract Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly ( 1 , 2 ) . More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. ( 1 , 3 ) , up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S 2 are Zoll. We will generalize these results to the case ( 1 , n ) for any n ≥ 2 . Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S 2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S 2 , as it is already the case for the ( 1 , 2 ) model constructed by Koenigs.

中文翻译:

超可积测地线流与 Zoll 度量

摘要 Koenigs 构造了一系列二维超可积 (SI) 模型,在动量 (1, 2) 中具有一个线性积分和两个二次积分。最近 Matveev 和 Shevchishin 已经表明,这种构造确实可以推广到具有一个线性积分和两个三次积分的模型,即 (1, 3),直到非线性常微分方程的解。我们对该方程的显式解允许构建这些 SI 系统,并证明在 S 2 上全局定义的系统是 Zoll。我们将这些结果推广到任何 n ≥ 2 的情况 (1, n)。我们的方法再次具有建设性,并显示了当 n 为奇数时,在 S 2 上全局定义的度量确实是 Zoll 的存在(在参数的适当限制下),
更新日期:2021-01-01
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