Integrable geodesic flows and Masloy's quantization condition

可积测地线流和 Masloy 量子化条件

• 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

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 球体上构建了一系列黎曼尼南度量，其测地线流具有纤维度 k 的第一积分，对于每个大于 2 的 k。它们是第一个例子，除了 k=3,4 的情况外，由于 Bolsinov和福缅科。此外，所构造的流形具有每个测地线都是闭的性质。因此，它们是 Guillemin 以抽象方式证明其存在的流形的具体例子。我们还研究了一般类型的 Kahler-Liouville 流形的结构，即不一定是 (A) 类型。因此，我们证明了每个紧致的卡勒-刘维尔流形都具有束结构，使得纤维是一个卡勒-刘维尔流形，其测地流可积，并且基是（局部）一维卡勒流形的乘积。此外，我们还获得了另一类卡勒-刘维尔流形，称为 (B) 型，其测地流可积。这一类在(A)型流形的纤维束结构研究中已经出现，但我们现在得到了它的内在定义。此外，我们还研究了Hermite-Liouville流形的局部结构，并基本阐明了它们。此外，我们在复射影空间上构造了 Hermite-Liouville 流形的结构。构造方式类似于 Kahler-Liouvlle 流形，即实数 Liouville 流形的复化。然而，在埃尔米特情况下，多个刘维尔流形产生一个埃尔米特-刘维尔流形。因此，我们通过这种方式得到了相当多的可积测地流的例子。