1996/03/24 by K. Ramachandran, Koushik Ramachandran, Ramachandran, K. +2
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #dg-ga #math.DG
paper · pdf · doi:10.48550/arxiv.dg-ga/9603013
10 pages, latex (e-mail: kram@..., [email protected])
arxiv created 1996/03/24 · arxiv updated 2009/11/30
We show that noncompact simply connected harmonic manifolds with volume density Θp(r) =\sinh n-1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θp(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.