Geodesic spheres and non radial eigenfunctions on Damek-Ricci spaces (Q1946801)

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.