Two theorems on harmonic manifolds (Q1772466)

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 in dimension 7. The author proves the Lichnerowicz conjecture in dimension 5: a five-dimensional harmonic manifold has constant sectional curvature. He also obtains a functional equation for the volume density function \(\theta(r)\) of a harmonic manifold and shows that \(\theta(r)\) is an exponential polynomial, i.e., a finite linear combination of the terms of the form \(\Re(ce^{\lambda r,m})\), with \(c\), \(\lambda\) complex constants.