Topology and Geometry of Manifolds with Lower Curvature Bounds

点击翻译按钮获取中文摘要

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.