A Riemannian manifold is called harmonic, if for any point it admits a nonconstant harmonic function depending only on the distance to . A.Lichnerowicz conjectured that any harmonic manifold is two-point homogeneous. This conjecture is proved in dimension and also for some classes of manifolds, but disproved in general, with the first counterexample of dimension . We prove the Lichnerowicz Conjecture in dimension : a five-dimensional harmonic manifold has constant sectional curvature. We also obtain a functional equation for the volume density function of a harmonic manifold and show that is an exponential polynomial, a finite linear combination of the terms of the form , with complex constants.
Cite this article
Yuri Nikolayevsky, Two theorems on harmonic manifolds. Comment. Math. Helv. 80 (2005), no. 1, pp. 29–50DOI 10.4171/CMH/2