Two theorems on harmonic manifolds

  • Yuri Nikolayevsky

    La Trobe University, Bundoora, Australia


A Riemannian manifold is called {\it harmonic}, if for any point xx it admits a nonconstant harmonic function depending only on the distance to xx. A.Lichnerowicz conjectured that any harmonic manifold is two-point homogeneous. This conjecture is proved in dimension n4n \le 4 and also for some classes of manifolds, but disproved in general, with the first counterexample of dimension 77. We prove the Lichnerowicz Conjecture in dimension 55: a five-dimensional harmonic manifold has constant sectional curvature. We also obtain a functional equation for the volume density function \T(r)\T(r) of a harmonic manifold and show that \T(r)\T(r) is an exponential polynomial, a finite linear combination of the terms of the form (ce\larrm)\Re (c e^{\la r} r^m), with c,\lac, \la complex constants.

Cite this article

Yuri Nikolayevsky, Two theorems on harmonic manifolds. Comment. Math. Helv. 80 (2005), no. 1, pp. 29–50

DOI 10.4171/CMH/2