Problems related to integrable geodesic flows

与可积测地流相关的问题

基本信息

批准号：
18540087
负责人：
KIYOHARA Kazuyoshi
金额：
$ 2.14万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

项目摘要

It is well known that the geodesic flow of ellipsoid is completely integrable. We studied in this research much finer properties of it. One of the results we obtained is the determination of the cut loci for any points; they are closed balls of codimension one for general points and those of codimension two for special points. The other result is the clarification of the structure of the first conjugate loci for general points. In particular, we showed that the set of singularities of conjugate locus of a general point consists of three connected component and each component (an open and dense part) is a cuspidal edge. This is a higher dimensional version of the so-called Jacobi's last geometric statement: "The conjugate locus of any non-umbilic point on two-dimensional ellipsoid has exactly four cusps". The main ingradient in the proofs of those results is the detailed investigation of Jacobi fields and their zeros. Moreover, we showed that the above results equally hold for some Liouville manifolds.Also, we investigated local structures of Hermite-Liouville manifolds and clarified them completely, even when they do not have the action of infinitesimal automor-phisms as for the case of Kahler-Liouville manifolds. Moreover, we illustrated a way of local construction of Hermite-Liouville manifolds in the case of having infinitesimal automorphisms, which almost corresponds to a global construction of Hermite-Liouville manifolds on complex projective spaces. In this construction, one can easily check which one is Kahler-Liouville and which one is not.

众所周知，椭球体的测地线流是完全可积的。我们在这项研究中研究了它的更精细的特性。我们获得的结果之一是确定任意点的切割轨迹；一般点为余维一闭球，特殊点为余维二闭球。另一个结果是阐明了一般点的第一共轭位点的结构。特别地，我们证明了一般点的共轭轨迹奇点集由三个连通分量组成，每个分量（开放和密集部分）是尖边。这是所谓的雅可比最后几何陈述的高维版本：“二维椭球上任何非脐点的共轭轨迹恰好有四个尖点”。这些结果证明的主要内容是对雅可比域及其零点的详细研究。此外，我们证明了上述结果对于一些刘维尔流形同样成立。此外，我们研究了埃尔米特-刘维尔流形的局部结构并完全阐明了它们，即使它们不具有无穷小自同构作用，如卡勒的情况-刘维尔流形。此外，我们还说明了在具有无穷小自同构的情况下局部构造 Hermite-Liouville 流形的方法，这几乎对应于复射影空间上 Hermite-Liouville 流形的全局构造。在这种结构中，人们可以很容易地检查哪一个是卡勒-刘维尔，哪一个不是。