Topology and Geometry of Manifolds with Lower Curvature Bounds

摘要DMS - 0203979。本课题涉及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里奇曲率。具体地说，我们继续寻找非负截面曲率的度量及其存在障碍的新例子，特别是在单连通情况下，我们计划分析具有幂零基群的开缩负弯曲流形的结构，并研究非负和几乎非负Ricci曲率的区别。现代几何的主要目标之一是通过测量空间的局部定量特性来获得空间的整体定性信息。例如，自19世纪以来人们就知道，局部看起来像鸡蛋的空间，全局看起来一定像鸡蛋，而不是像甜甜圈或丛林游乐场。这个项目处理的是高维空间的类似问题，其中一些在物理学和工程学中自然存在。

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.