Mathematical Sciences: Manifolds With Non-Negative Curvature

首席研究员将对具有正里奇曲率的黎曼流形和球间调和映射进行研究。他将研究两个闭合的1连通流形的乘积是否必须有正的里奇曲率，当每个流形都有这样的曲率时。他将研究闭合1连通5流形是否携带正Ricci曲率的度量。一个多世纪以来，微分几何的研究一直集中在曲率问题上。今天有几个相互竞争的曲率概念，包括截面曲率、高斯曲率和里奇曲率。最近，在合作中，主要研究者构造了一个具有正Ricci曲率的完备黎曼流形，该流形不具有非负截面曲率。在这个项目中，他将研究四维和五维超曲面的相关问题。

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.