Harmonic manifolds with some specific volume densities

Details Harmonic manifolds with some specific volume densities Harmonic manifolds with some specific volume densities

1996-03-24

arxiv.org/abs/dg-ga/9603013v1

Mathematics

Differential Geometry

10 pages, latex (e-mail: kram@..., aranjan@ganit.math.iitb.ernet.in)

Scientific paper

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 K\"{a}hler 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 K\"{a}hler 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.

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