Harmonic manifolds with some specific volume densities

Abstract

We show that noncompact simply connected harmonic manifolds with volume density ${\Theta }_p(r)={\sinh }^{n-1}r$ is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density ${\Theta }_p(r)={\sinh }^{2n-1}r\cosh r$ 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.

arXiv comment10 pages, latex (e-mail: kram@..., [email protected])