Two theorems on harmonic manifolds

Abstract

A Riemannian manifold is called {\it 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 $\T(r)$ of a harmonic manifold and show that $\T(r)$ is an exponential polynomial, a finite linear combination of the terms of the form $\Re (c e^{\la r} r^m)$, with $c, \la$ complex constants.