ar X iv : d g - ga / 9 60 30 13 v 1 2 4 M ar 1 99 6 Harmonic manifolds with some specific volume densities

We show that noncompact simply connected harmonic manifolds with volume density Θ p (r) = sinh n−1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θ p (r) = sinh 2n−1 r cosh r is isometric to the complex hyperbolic space. A similar result is also proved for Quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is isomet-ric to the euclidean space. Finally a rigidity result for real hyperbolic space is presented.