2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152122We 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 isometric to the euclidean space. Finally a rigidity result for real hyperbolic space is presented.10 pages, latex (e-mail: kram@..., aranjan@ganit.math.iitb.ernet.in)Differential Geometrytext