Finsler几何中的Gromov-Hausdorff理论  

On the Gromov-Hausdorff theory in Finsler geometry

作　　者：赵唯 张福娥 Wei Zhao;Fue Zhang

机构地区：[1]华东理工大学数学学院,上海200237 [2]石河子大学理学院,石河子832003

出　　处：《中国科学：数学》2024年第10期1751-1770,共20页Scientia Sinica：Mathematica

基　　金：上海市自然科学基金(批准号:21ZR1418300)资助项目。

摘　　要：本文详细介绍Finsler几何中的Gromov-Hausdorff理论.首先找到一类能刻画所有向前完备Finsler流形的度量空间,称为向前度量空间.其次发现,在这类空间上能很自然地建立/推广GromovHausdorff理论,特别地,Gromov预紧性定理依然成立.最后,在Finsler情形下得到一系列预紧性定理和稳定性结果.此外,本文提供了诸多有趣的例子,深刻揭示了Finsler几何与Riemann几何的本质不同.This paper is devoted to the introduction of the Gromov-Hausdorff theory in Finsler geometry.Firstly,we are able to find a special kind of irreversible metric spaces,called forward metric spaces,which can accurately describe forward complete Finsler manifolds.Secondly,we establish/generalize the Gromov-Hausdorff theory on such metric spaces.In particular,Gromov's precompactness theorem remains valid in this setting.Last but not least,we obtain a series of precompactness theorems and stable results in the framework of Finsler geometry.Besides,many interesting examples are presented in this paper,which deeply point out genuine differences between Finsler geometry and Riemannian geometry.

关 键 词：FINSLER流形 向前度量空间 Gromov-Hausdorff拓扑 预紧性定理 

分 类 号：O186.1[理学—数学]