Manifolds with Positive Sectional Curvature Almost Everywhere

对于DMS-0104086，在他的主要项目中，研究者计划继续他在开稠密点集上具有正截面曲率的黎曼流形上的工作。特别要强调的问题是，如何将这类流形与处处具有正非负曲率的两个Well研究流形类区分开来。在进一步的项目中，研究者计划得到欧几里德球面和流形的度量叶面的刚性结果，它们的测地线都是闭合的。鸡蛋和足球的表面或地球的理想化表面都是具有正截面曲率的二维黎曼流形的例子。一个多世纪以来，几何学家一直认为，具有正曲率的曲面必须看起来简单。它们不会形成任何洞，所以他们寻找与甜甜圈表面根本不同的例子。相比之下，二维流形具有非负的截面曲率，本质上看起来像是甜甜圈的表面。这一初步观察揭示了正曲率和非负曲率之间的根本区别。主要目的是理解这一根本区别及其在更高维度上的影响。

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.