Manifolds with Positive Sectional Curvature Almost Everywhere

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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.