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Noncompact harmonic manifolds

Author: Gerhard Knieper, Norbert Peyerimhoff
Url: http://arxiv.org/abs/1302.3841v1
Format: Ps, Pdf
Year: 2013
Category: Differential Geometry
Pages: 84
Clicks: 1044

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. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci 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 noncompact 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.

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