Noncompact harmonic manifolds

View/ Open
Date
2013-04-10MFO Scientific Program
Research in Pairs 2012Series
Oberwolfach Preprints;2013,08Author
Knieper, Gerhard
Peyerimhoff, Norbert
Metadata
Show full item recordOWP-2013-08
Abstract
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szab ́o [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR]provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on non-compact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.