"Hyperbolic manifolds and the Bochner technique"
Abstract:The Bochner technique is one of the most important methods in Riemannian geometry. In various guises, it involves simply applying Stokes theorem to harmonic sections of geometrically defined bundles over a Riemannian manifold, and then carefully keeping track of the signs. It has a long interesting history proving rigidity theorems for locally symmetric spaces. In joint work with Steven Kerckhoff, we use this method to prove a new rigidity theorem for a large class of infinite volume hyperbolic manifolds. I will discuss the history of these ideas, and current work.