Graduate Student Lecture

Robert Bryant (Duke University)

Title: Curvature-Homogeneous Riemannian Manifolds

Abstract: A Riemannian manifold (M, g) is said to be curvature-homogeneous if, for any two points x and y in M, there is an isometry of tangent spaces TxM and TyM that identifies the curvature operators at the two points. In other words, the Riemannian metric g is “homogeneous up to second order”. When M is a surface, this is the same as having constant Gauss curvature, and, in this case, the surface is actually locally homogeneous. In higher dimensions, this is no longer true, but exactly how ‘flexible’ curvature homogeneous spaces are is not well-understood. In this talk, I will survey some classical and recent results about curvature-homogeneous metrics and describe some of the tools that can be used to study this problem.