# Two theorems on harmonic manifolds

### Yuri Nikolayevsky

La Trobe University, Bundoora, Australia

## Abstract

A Riemannian manifold is called *harmonic*, if for any point $x$ it admits a nonconstant harmonic function depending only on the distance to $x$. A.Lichnerowicz conjectured that any harmonic manifold is two-point homogeneous. This conjecture is proved in dimension $n≤4$ and also for some classes of manifolds, but disproved in general, with the first counterexample of dimension $7$. We prove the Lichnerowicz Conjecture in dimension $5$: a five-dimensional harmonic manifold has constant sectional curvature. We also obtain a functional equation for the volume density function $θ(r)$ of a harmonic manifold and show that $θ(r)$ is an exponential polynomial, a finite linear combination of the terms of the form $Re(ce_{λr}r_{m})$, with $c,λ$ complex constants.

## Cite this article

Yuri Nikolayevsky, Two theorems on harmonic manifolds. Comment. Math. Helv. 80 (2005), no. 1, pp. 29–50

DOI 10.4171/CMH/2