Integrable geodesic flows and Masloy's quantization condition

可积测地线流和 Masloy 量子化条件

基本信息

批准号：
13640054
负责人：
KIYOHARA Kazuyoshi
金额：
$ 2.5万
依托单位：
HOKKAIDO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640054/
关键词：
Integrable geodesic flow Integrable system Periodic geodesic flow Zoll Hamiltonian mechanics Symplectiic geometry Liouville manifolds Eillipsoid Zoll計量バミルトン力学

项目摘要

We constructed a continuous family of riemanninan metrics on 2-sphere whose geodesic flows possess first integrals of fiber-degree k, for every k greater than 2. They are the first examples, exect the cases where k=3,4, due to Bolsinov and Fomenko. Moreover, the constructed manifolds have the property that every geodesic is closed. Therefore they are conrete examples of the manifolds that Guillemin showed their existence in an abstract manner.We also investigated the structures of Kahler-Liouville manifolds of general type, I.e., not necessarlly of type (A). As a consequence, we showed that every compact, proper Kahler-Liouville manifold has a bundle structure such that the fiber is a Kahler-Liouville manifold whose geodesic flow is integrable, and the base is (locally) a product of one-dimensional Kahler manifolds. Also, we obtain another class, called of type (B), of Kahler-Liouville manifolds whose geodesic flows are integrable. This class had already appeared in the study of fiber bundle structure of type (A) manifolds, but we now obtained its intrinsic definition.Also, we investigated local structures of Hermite-Liouville manifolds and basically clarifled them. Moreover, we construct the structure of Hermite-Liouville manifolds on complex projective spaces. The way of construction is similar to that of a Kahler-Liouvlle manifold, I.e., a complexification of a real Liouville manifold. However, in the Hermite case, plural Liouville manifolds produce one Hermite-Liouville manifold. Therefore, we obtain quite many examples of integrable geodesic flows in this way.

我们在2个球体上构建了一个连续的riemanninan量子，其测量流具有纤维度k的第一积分，每一个大于2。它们是第一个示例。此外，构造的歧管具有关闭每个测量的属性。因此，它们是Guillemin以抽象的方式表现出其存在的歧管的例子。我们还研究了一般类型的Kahler-Liouville歧管的结构，即不必要（a）类型。结果，我们表明，每个紧凑的，适当的Kahler-Liouville歧管都有一个捆扎结构，因此纤维是Kahler-Liouville的歧管，其地理流量是可集成的，并且基部是（本地）一维Kahler歧管的产物。另外，我们获得了Kahler-Liouville歧管的另一个称为（b）类型的类别，其大地歧管是可以集成的。该类别已经出现在（a）类型类型的纤维束结构的研究中，但是我们现在获得了其内在的定义。此外，我们研究了Hermite-liouville歧管的局部结构，并基本上澄清了它们。此外，我们在复杂的投影空间上构建了Hermite-Liouville歧管的结构。构建方式类似于Kahler-liouvlle歧管的构造方式，即对真实的Liouville歧管的复杂化。然而，在赫米特案中，复数liouville流形产生了一个Hermite-liouville歧管。因此，我们以这种方式获得了许多可集成的大地测量流的示例。