Mathematical Sciences: Manifolds With Non-Negative Curvature

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The principal investigator will conduct research into Riemannian manifolds with positive Ricci curvature and harmonic maps between spheres. He will investigate as to whether products of two closed 1-connected manifolds must have positive Ricci curvature when each has such curvature. And he will investigate the problem of whether closed 1-connected 5-manifolds carry metrics with positive Ricci curvature. The study of differential geometry has focused on problems concerning curvature for over a century. Today there are several competing notions of curvature including sectional, Gauss, and Ricci. Recently, in collaboration, the principal investigator constructed a complete Riemannian manifold with positive Ricci curvature which did not have non-negative sectional curvature. In this project he will study related questions for hypersurfaces of dimension four and five.