Two theorems on harmonic manifolds

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 (c eλ r r^m) , with c, λ complex constants.