权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds

Zoll 曲面的射影几何和 Finsler 流形上的切割轨迹

基本信息

批准号：
20K03595
负责人：
SABAU Vasile.Sor
金额：
$ 1.25万
依托单位：
Tokai University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2020
资助国家：
日本
起止时间：
2020-04-01 至 2024-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-20K03595/
关键词：
Finsler manifolds geodesics cut locus isometry group cohomology group Zoll manifolds Liouville manifolds closed geodesics cylinders of revolution Randers metrics conjugate locus theory of geodesics Riemannian manifolds Cut Locus

项目摘要

We have studied the cut locus of Randers metrics in a more general case than the solutions of Zermelo's navigation problem with Killing vector fields. Indeed, the structure of cut locus on a Randers manifold can be determined without any curvature or Killing-related property. This shows that there are very large classes of Finsler metrics whose cut locus structure can be determined.The construction is done in 2 steps. First step is to construct Finsler metrics as solutions of Zermelo's navigation problem solution for the Riemannian metric h and a Killing field V, followed by a beta-change by means of a closed one-form. The construction naturally extends to the case of the Zermelo's navigation for (F,V), where F is an a-priori given Finsler metric of Randers type and V an F-Killing field. The study of Finsler Killing fields is a complex topic in modern Finsler gometry. The dimension of the isometry group of the Finsler metric F and the cohomology group of the manifold are related.Moreover, the construction given here was further generalized to the case of a sequence of Riemannian metrics and a sequence of Killing fields that leads to sequences of new Finsler metrics with computable geodesics, curvatures and cut loci. This is a completely new trend in modern Finsler geometry that needs further attention.

我们在比使用 Killing 矢量场解决 Zermelo 导航问题更一般的情况下研究了 Randers 度量的切割轨迹。事实上，兰德斯流形上的切割轨迹的结构可以在没有任何曲率或杀伤相关属性的情况下确定。这表明存在非常大的 Finsler 度量类别，其切割轨迹结构可以确定。构建分两步完成。第一步是构建 Finsler 度量作为黎曼度量 h 和 Killing field V 的 Zermelo 导航问题解决方案的解决方案，然后通过封闭的单一形式进行 beta 更改。该结构自然地扩展到 Zermelo 的 (F,V) 导航情况，其中 F 是先验给定的 Randers 类型的 Finsler 度量，V 是 F-Killing 场。芬斯勒杀戮场的研究是现代芬斯勒几何学中的一个复杂课题。芬斯勒度量 F 的等距群的维数和流形的上同调群是相关的。此外，这里给出的构造被进一步推广到黎曼度量序列和 Killing 场序列的情况，从而产生具有可计算测地线、曲率和切割轨迹的新芬斯勒度量序列。这是现代芬斯勒几何学的一个全新趋势，需要进一步关注。