Geodesic spheres and non radial eigenfunctions on Damek-Ricci spaces
type: Item
instance of: scholarly article
title: Geodesic spheres and non radial eigenfunctions on Damek-Ricci spaces
zbMATH Open document ID: 1273.53026
author: Roberto Camporesi
published in: Indagationes Mathematicae. New Series
publication date: 2013-04-16T00:00:00Z
review text: A Riemannian manifold is called \textit{harmonic} if all geodesic hyperspheres of sufficiently small radius have constant mean curvature. It was conjectured for some time (Lichnerowicz's conjecture) that harmonic manifolds must be locally symmetric. Z. I. Szabo proved the conjecture for compact harmonic manifolds with finite fundamental groups in 1990. However, in 1992, E. Damek and F. Ricci constructed noncompact counterexamples. Namely, there is a class of harmonic homogeneous simply-connected manifolds of negative curvature which are not symmetric. Their examples are of the form \(S=NA\), a semi-direct product of a connected simply-connected nilpotent Lie group \(N\) of Heisenberg type and the one-dimensional group \(A\cong\mathbb R^+\) acting on \(N\) by anisotropic dilations, equipped with a suitable left-invariant Riemannian metric. In this paper, the author calculates the non-radial invariant eigenfunctions of the Laplacian on \(S\) in terms of geodesic polar coordinates.
reviewed by: Claudio Gorodski
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 6154628
zbMATH Keywords: homogeneous harmonic spaces
zbMATH Keywords: Jacobi functions
MaRDI profile type: Publication
full work available at URL: j.indag.2012.11.006
OpenAlex ID: W2104804585
DOI: 10.1016/J.INDAG.2012.11.006
description: scientific article; zbMATH DE number 6154628