The Lichnerowicz conjecture on harmonic manifolds
type: Item
instance of: scholarly article
title: The Lichnerowicz conjecture on harmonic manifolds
zbMATH Open document ID: 0686.53042
published in: Journal of Differential Geometry
publication date: 1990-01-01T00:00:00Z
review text: There are many equivalent definitions for harmonic spaces. For example, a Riemannian manifold is said to be locally harmonic if and only if for each point \(m\in M\) the volume density function \(\theta_ m\) of the exponential map \(\exp_ m\) centered at m is constant on sufficiently small geodesic spheres or equivalently, if and only if the mean curvature of all sufficiently small geodesic spheres \(G_ m(r)\), with center m and radius r, is constant on \(G_ m(r)\). There is a corresponding notion of a globally harmonic space. All manifolds which are locally isometric to a two-point homogeneous space are trivially harmonic and it is a longstanding conjecture of A. Lichnerowicz that these spaces are the only examples. A positive answer was known only for dimensions \(\leq 4.\) This important paper contains the most striking result in this domain: Any simply connected compact harmonic space is a two-point homogeneous space. At the same time the author gives a new proof of the fact that any two-point homogeneous space is symmetric. The author uses extensively the compactness condition and, unfortunately, the method cannot be used to give an answer in the general case, not even for complete spaces. So, further resarch is needed to obtain a complete answer for the conjecture of Lichnerowicz.
reviewed by: Lieven Vanhecke
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 4124533
zbMATH Keywords: harmonic spaces
zbMATH Keywords: two-point homogeneous space
author: Zoltán Imre Szabó
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214444087
OpenAlex ID: W1823562442
Wikidata QID: Q115182435
DOI: 10.4310/JDG/1214444087
description: scientific article; zbMATH DE number 4124533