Isometries, rigidity and universal covers. (English) Zbl 1175.53055

Farb, Benson; Weinberger, Shmuel

Ann. Math. (2) 168, No. 3, 915-940 (2008).

The authors describe all closed, aspherical Riemannian manifolds \(M\) such that the isometry group \(\text{Isom}(\tilde M)\) of the universal cover \(\tilde M\) is not discrete (Theorem 1.2).
The description is given in terms of Riemannian orbifolds and orbibundles. Methods from Lie theory, harmonic maps, large-scale geometry, and homological theory of transformation groups are used in the proof.
The authors give applications of their main theorem and methods from the proof to some problems in geometry: characterizations of locally symmetric manifolds, compact complex manifolds, and some others.

Reviewer: Roman Urban (Wrocław)

MSC:

53C24	Rigidity results
53C35	Differential geometry of symmetric spaces

Keywords:

aspherical Riemannian manifolds; Riemannian orbifolds; Riemannian orbibundles