Manifolds with Positive Sectional Curvature Almost Everywhere

几乎到处都有正截面曲率的流形

• 批准号: 0104086
• 负责人: Wolfgang Ziller
• 金额: $ 10.38万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104086&HistoricalAwards=false
• 关键词: Manifolds; Positive; Sectional; Curvature; Almost

Abstract for DMS - 0104086In his major project the investigator plans to continue hiswork on Riemannian manifolds with positive sectional curvatureon open dense sets of points. A particular emphasis shall beput on the question how to distinguish this class from the twowell studied classes of manifolds that have positive respectivelynonnegative curvature everywhere. In further projects the investigator plans to obtain rigidity results for metric foliations of Euclidean spheres and formanifolds all of whose geodesics are closed.The surfaces of eggs and footballs or the idealized surface ofthe earth are examples of 2-dimensional Riemannian manifolds withpositive sectional curvature. For more than a century geometersunderstand that surfaces with positive curvature must look simple.They can not develop any holes, so they look for examplefundamentally different from the surface of a doughnut. In contrastthere are two dimensional manifolds with nonnegative sectionalcurvature, which essentially look like the surface of a doughnut.This elementary observation reveals a fundamental differencebetween positive and nonnegative curvature. The main aim is tounderstand this fundamental difference and its implications inhigher dimensions as well.

DMS 摘要 - 0104086 在他的主要项目中，研究者计划继续研究开密集点集上具有正截面曲率的黎曼流形。应特别强调如何将此类流形与两个经过充分研究的流形类区分开来，这两个流形类处处具有正曲率和非负曲率。在进一步的项目中，研究人员计划获得欧几里得球体和所有测地线均闭合的流形的度量叶状结构的刚性结果。鸡蛋和足球的表面或地球的理想化表面是具有正截面曲率的二维黎曼流形的示例。一个多世纪以来，几何学家们都知道，具有正曲率的表面看起来一定很简单。它们不能形成任何孔，因此它们看起来与甜甜圈的表面有根本的不同。相比之下，存在具有非负截面曲率的二维流形，其本质上看起来像甜甜圈的表面。这一基本观察揭示了正曲率和非负曲率之间的根本区别。主要目的是理解这种根本差异及其在更高维度上的含义。