  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.
