Abstract:

In this talk I will report on joint work with Shmuel Weinberger, addressing the question: which metrics on a Riemannian manifold have the most symmetry? Our main theorem is essentially a classification (up to finite index subgroups) of the isometry groups of the universal cover of every closed Riemannian manifold. I will try to explain this theorem and some of its corollaries. These include: new characterizations of locally symmetric (and also of arithmetic) manifolds among all closed Riemannian manifolds; a classification of 1-connected, aspherical manifolds which cover both a compact and a (noncompact) finite volume manifold; and verification of the Hopf Conjecture for manifolds with nontrivial hidden symmetry.