The principal investigator will study problems concerning manifolds with positive Ricci curvature, harmonic maps between spheres, and geometry and topology of manifolds with boundary. Obstructions to the existence of metrics on Riemannian manifolds with prescribed curvature conditions will be investigated. Dirac bundles and Dirac operators on manifolds with boundary of dimension 4 and 5 will be analyzed. The mathematical object known as a torus has the shape of an inner tube and is positively curved on the outer parts, while negatively curved around the hole. As a fact, its total curvature is zero. A long time ago mathematicians learned that this meant, with a different concept of distance, the torus can appear flat. The principal investigator will continue his research into analogous problems concerning such prescribed curvature on higher-dimensional manifolds.
