Harmonic manifolds with some specific volume densities
Abstract
Description
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 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)
10 pages, latex (e-mail: kram@..., aranjan@ganit.math.iitb.ernet.in)