Geometry and topology under positive/nonnegative sectional curvature

Abstract

This project is dedicated to the study of rigidity and obstructions for foliations and submersion on manifolds with positive or non-negative sectional curvature. The motivation of this project is the relation between topology and sectional curvature along with the common interaction between geometric constructions and symmetry: to mention, the main source of examples of non-negatively and positively curved spaces are homogeneous spaces or direct variations. In this project, such symmetries appear as Riemannian foliations. The purpose of the project is to address problems such as the following: - Classify totally geodesic folations in symmetrical spaces - Prove structural theorems for manifolds with non-negative curvature - Approach structural conjectures on manifolds of positive curvature - Construct examples of new manifolds with non-negative/positive curvature - Study metric variations through submersions and group actions (AU)