• Harmonic manifolds with some specific volume densities

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      https://www.ias.ac.in/article/fulltext/pmsc/107/03/0251-0261

    • Keywords

       

      Harmonic manifolds; volume densities; Ricci curvature; second fundamental form

    • Abstract

       

      We show that a noncompact, complete, simply connected harmonic manifold (Md, g) with volume densityθm(r)=sinhd-1r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M2d, g) with volume densityθm(r)=sinh2d-1r coshr 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 flat. Finally a rigidity result for real hyperbolic space is presented.

    • Author Affiliations

       

      K Ramachandran1 A Ranjan1

      1. Department of Mathematics, Indian Institute of Technology, Powai, Mumbai - 400 076, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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