Topology and Geometry of Manifolds with Lower Curvature Bounds

具有较低曲率界限的流形的拓扑和几何

• 批准号: 0203979
• 负责人: Igor Belegradek
• 金额: $ 8.64万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2003-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203979&HistoricalAwards=false
• 关键词: Topology; Geometry; Manifolds; Lower; Curvature

ABSTRACT DMS - 0203979.This project deals with three areas of Riemannian geometry, namely,nonnegative sectional curvature, negative sectional curvature,and almost nonnegative Ricci curvature. Specifically, we continuethe search of new examples of metrics of nonnegative sectionalcurvature and obstructions to their existence, especially in thesimply-connected case, we plan to analyze the structure of openpinched negatively curved manifolds with nilpotent fundamentalgroups, and to study the difference between nonnegative andalmost nonnegative Ricci curvatures.One of the main goals of modern geometry is to obtain globalqualitative information about a space by measuring its localquantitative properties. Say, it has been known since thenineteenth century that a space that locally looks like an eggmust globally look like an egg, not like a doughnut or a jungle gym.This project deals with similar matters for higher dimensionalspaces, some of which occur naturally in physics and engineering.

本课题研究黎曼几何的三个领域，即非负截面曲率、负截面曲率和几乎非负的Ricci曲率。具体地说，我们继续寻找非负截面曲率的度量的新例子以及它们存在的障碍，特别是在单连通的情况下，我们计划分析具有幂零基本群的开挤压负曲线流形的结构，并研究非负和几乎非负Ricci曲率之间的差异。现代几何的主要目标之一是通过测量空间的局部数量性质来获得关于空间的全局定性信息。比方说，从19世纪起，人们就知道，一个局部看起来像鸡蛋的空间，在全球范围内看起来一定像一个鸡蛋，而不是像甜甜圈或丛林体操。这个项目研究了更高维空间的类似问题，其中一些是在物理和工程中自然出现的。