On nearly Schur-type theorems
Dr. Kwok-Kun Kwong
It is well-known that if the principal curvatures of a surface in the Euclidean space depends on the base point only (i.e. umbilic), then it has constant curvature. Another classical result of Schur states that if the Ricci curvature of a Riemannian manifold depends on the base point only, then its scalar curvature is constant. I will give a unified viewpoint for these types of results and give a quantitative version of these results for closed Riemannian manifolds.