In his major project the investigator plans to continue his work on Riemannian manifolds with positive sectional curvature on open dense sets of points. A particular emphasis shall be put on the question how to distinguish this class from the two well studied classes of manifolds that have positive respectively nonnegative curvature everywhere. In further projects the investigator plans to obtain rigidity results for metric foliations of Euclidean spheres and for manifolds all of whose geodesics are closed.
The surfaces of eggs and footballs or the idealized surface of the earth are examples of 2-dimensional Riemannian manifolds with positive sectional curvature. For more than a century geometers understand that surfaces with positive curvature must look simple. They can not develop any holes, so they look for example fundamentally different from the surface of a doughnut. In contrast there are two dimensional manifolds with nonnegative sectional curvature, which essentially look like the surface of a doughnut. This elementary observation reveals a fundamental difference between positive and nonnegative curvature. The main aim is to understand this fundamental difference and its implications in higher dimensions as well.