Manifolds with Positive Sectional Curvature Almost Everywhere

摘要DMS -0104086在他的主要项目中，研究人员计划继续他的工作，黎曼流形上的正截面曲率的开放密集集的点。一个特别的重点应beput上的问题，如何区分这类从twowell研究类的流形，有积极的各自nonnegative曲率无处不在。在进一步的项目中，研究人员计划获得欧氏球面和形式流形的度量叶理的刚性结果，这些球面和形式流形的测地线都是闭合的，鸡蛋和足球的表面或理想化的地球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，几何学家都认为正曲率的曲面看起来一定很简单，它们不会形成任何孔洞，因此它们看起来与甜甜圈的表面有着本质的不同。与此相反，有非负截面曲率的二维流形，它们基本上看起来像甜甜圈的表面。这个基本的观察揭示了正曲率和非负曲率之间的根本区别。主要的目的是理解这个基本的区别和它在更高维度的含义。

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.