Harmonic manifolds with some specific volume densities

TL;DR

This paper characterizes harmonic manifolds with specific volume densities, showing they are isometric to hyperbolic spaces or Euclidean space, and provides alternative proofs and rigidity results without relying on the Cheeger-Gromoll splitting theorem.

Contribution

It establishes new characterizations of harmonic manifolds with particular volume densities and offers alternative proofs for known rigidity results.

Findings

Harmonic manifolds with volume density sinh^{n-1} r are hyperbolic spaces.

Kähler harmonic manifolds with volume density sinh^{2n-1} r cosh r are complex hyperbolic spaces.

Ricci flat harmonic manifolds are isometric to Euclidean space.

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 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.