JournalscmhVol. 80 , No. 1DOI 10.4171/cmh/2

Two theorems on harmonic manifolds

  • Yuri Nikolayevsky

    La Trobe University, Bundoora, Australia
Two theorems on harmonic manifolds cover

Abstract

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.