Fundamental Groups of Positively Curved Manifolds

【作者】 李想；

【导师】 戎小春； 于祖焕；

【作者基本信息】 首都师范大学 ， 基础数学， 2007， 硕士

【摘要】 本文以06年秋季学期戎小春教授于首都师范大学组织的“正曲率，对称群和拓扑”的讨论班上报告的一些内容为基础，简要回顾了其中一研究方向的基本定义与定理，并侧重整理了关于正曲率流形基本群的一些的结果。同时，作者对讨论班上所提出的问题进行了进一步的探究。本文主要证明的结论是：定理A(Myers定理的逆问题)：“任意一个有限群可作为某一个正Ricci曲率闭流形的基本群”。戎小春教授在他未公开的讲义中提出了关于这个定理证明梗概，在本文中作者通过群扩张的方法，对该定理进行了完整的证明；定理B：“设M是一个具有正截面曲率的闭流形，环群T~k(k≥1)等距地作用于M上，φ是M上的一个等距映射，且与T~k可交换，则φ保持一个T~k的圆轨道。”该定理是戎小春教授关于等距环群作用于正曲率流形研究中的一个重要结果。此结果建立了交换李群作用与正曲率流形基本群的关系。在文章【12】中，戎小春教授证明了流形维数是奇数的情形，并指出了维数是偶数的情形定理亦成立。本文中将给出该定理的一个统一证明。定理C，D和E也是讲义【11】和文章【12】中关于正曲率流形基本群的重要结果。作为研究报告，我们也将给出这三个定理的详细证明。更多还原

【Abstract】 This thesis is based on parts of the report in the seminar of "positive curvature, symmetry and topology" in the fall semester of 2006 at Capital Normal University. We first briefly review basic notions and theorems related to this thesis. Then we will collect some results on "Fundamental group of positively curved manifolds" and we will investigate some problems arose.The main results in this thesis are: Theorems A (the inverse problem of the Myers’s theorem): "Any finite group can be the fundamental group of some closed manifold of positive Ricci curvature." Professor Xiaochun Rong has sketched a proof in his unpublished lecture notes. Here we use the method of group’s extension to give a detailed proof; Theorems B: "Let M be a closed manifold of positive sectional curvature on which a torus T~k(k≥1) acts isometrically. If φ is an isometry on M commuting with the T~k-action, then φ preserves some T~k-orbit which is a circle." This theorem is a basic result on "isometric T~k-action on positively curved manifolds", which establish as a bridge between an isometric torus group action on a manifold with positive sectional curvature and its fundamental group. Professor Rong gave a proof for odd dimension case in [12] and asserted that the theorem still holds in even dimensions. (though even-dimensional case may not be needed as long as fundamental groups are concerned) In this thesis, we will give a unify proof of the theorem in both odd and even dimensional cases. Theorem C, D and E are important applications of Theorems B in [11]and [12]. For the completeness, we will also give self-contained proof, in details.更多还原