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，椭球的大地测量流是完全可整合的。我们在这项研究中研究了它的优质特性。我们获得的结果之一是确定任何点的切割基因座。它们是一般点的封闭的Codimension One的封闭球，而Condimension二的球是特殊点的封闭球。另一个结果是阐明了通用点的第一个共轭基因座的结构。特别是，我们表明，一个一般点的共轭基因座的奇异性集由三个连接的组件组成，每个组件（一个开放且密集的部分）是一个cuspidal边缘。这是所谓的雅各比的最后几何陈述的较高维度：“二维椭圆形上任何非bumbilic点的共轭基因座完全具有四个尖头”。这些结果证明的主要内科医生是对雅各比田及其零的详细研究。此外，我们证明了上述结果同样成立了一些liouville歧管。此外，我们研究了Hermite-liouville歧管的局部结构，并完全澄清了它们，即使它们没有无穷小型自动形态的作用，即Kahler-Liouville歧管的案例。此外，我们说明了在具有无穷小型自动形态的情况下，在当地建造Hermite-Liouville歧管的方法，这几乎与复杂的投影空间上的Hermite-Liouville歧管的全球结构相对应。在这种结构中，可以轻松检查哪个是Kahler-Liouville，哪个不是。