The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds
The Structure of the Conjugate Locus of a General Point on Ellipsoids and Certain Liouville Manifolds
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椭球体和某些刘维尔流形上一般点的共轭轨迹的结构
DOI：
10.1007/s40598-020-00153-9
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
K. Kiyohara
中科院分区：
文献类型：
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作者：
J. Itoh;K. Kiyohara
众所周知，自雅可比以来，椭球的测地线流是“完全可积的”，这意味着测地线轨道以某种显式方式描述。然而，这并不直接表明测地线的任何全局行为变得容易看到。事实上，它发生在最近，证明声明“的共轭轨迹的一般点在二维椭球只有四个尖点”雅可比的Vorlesungen尤伯杯dynamik出现在文献中。本文研究了Liouville流形，这是一类包含椭球的黎曼流形。我们解决了测地线方程;调查的雅可比场的行为，特别是零点的位置;并澄清了一般点的共轭轨迹的结构。特别地，我们证明了在共轭轨迹中产生的奇点只是尖棱和拉格朗日奇点，这将是Jacobi陈述的高维对应物。
It is well known since Jacobi that the geodesic flow of the ellipsoid is “completely integrable”, which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement “The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps” in Jacobi’s Vorlesungen über dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges andLagrangian singularities, which would be the higher dimensional counterpart of Jacobi’s statement.