Two theorems on harmonic manifolds

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 $n\le 4$ 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 $\theta (r)$ of a harmonic manifold and show that $\theta (r)$ is an exponential polynomial, a finite linear combination of the terms of the form $\mathrm{Re}(c{e}^{\lambda r}{r}^m)$, with $c,\lambda$ complex constants.