Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (Md, g) with volume densityθm(r)=sinhd-1r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M2d, g) with volume densityθm(r)=sinh2d-1r coshr 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 flat. Finally a rigidity result for real hyperbolic space is presented.