Isometries, rigidity, and universal covers

TL;DR

This paper classifies certain aspherical Riemannian manifolds based on their universal cover's isometry group, with applications to hyperbolic geometry, the Hopf Conjecture, and complex geometry, using advanced geometric and algebraic methods.

Contribution

It provides a comprehensive classification of aspherical Riemannian manifolds with indiscrete universal cover isometry groups, extending to non-aspherical cases and connecting to major conjectures.

Findings

Manifolds with hyperbolic fundamental groups are negatively curved, locally symmetric.

Classification of contractible manifolds covering finite volume manifolds.

New proofs and applications to the Hopf Conjecture and Kazhdan's Conjecture.

Abstract

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of all contractible Riemannian manifolds covering both compact and (noncompact, complete) finite volume manifold. There are also applications to the Hopf Conjecture, a new proof of Kazhdan's Conjecture (Frankel's Theorem) in complex geometry, etc. Ideas in the proof come from Lie theory, the homological theory of transformation groups, harmonic maps, and large-scale geometry. An extension to the non-aspherical case is also given.