New extremal problems for the Riemannian recognition program via Alexandrov geometry

Abstract

In its most general form, the recognition problem in riemannian geometry asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold. Optimally one wants to recognize a manifold having made as few measurements as possible. Many results in riemannian geometry, including pinching theorems, can be viewed this way. Here we are only interested in measurements that assign real numbers to each (complete) riemannian manifold. Typical examples of such invariants are diameter, volume, curvature bounds, etc. When viewing one or several invariants, I = (II ' . .. ,II)' of this type as a map on a suitable class, L , of riemannian manifolds, the following problems pose themselves:

Locations

• Journal of the American Mathematical Society - View - PDF