Isometries, rigidity, and universal covers

Details Isometries, rigidity, and universal covers Isometries, rigidity, and universal covers

2005-06-27

arxiv.org/abs/math/0506544v2

Mathematics

Differential Geometry

26 pages. to appear in Annals of Math

Scientific paper

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.

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