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    • A New Proof of the Theorem: Harmonic Manifolds with Minimal Horospheres are Flat

      Hemangi M Shah

      Volume 124 Issue 3 August 2014 pp 419-425

    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/124/03/0419-0425

    • Keywords

       

      Harmonic manifold; Busemann function; minimal horospheres.

    • Abstract

       

      In this note we reprove the known theorem: Harmonic manifolds with minimal horospheres are flat. It turns out that our proof is simpler and more direct than the original one. We also reprove the theorem: Ricci flat harmonic manifolds are flat, which is generally affirmed by appealing to Cheeger–Gromov splitting theorem. We also confirm that if a harmonic manifold 𝑀 has same volume density function as $\mathbb{R}^n$, then 𝑀 is flat.

    • Author Affiliations

       

      Hemangi M Shah1

      1. Harish Chandra Research Institute, Chhatnag Road, Jhusi 211 019, India

    • Dates

       
      Published
      August 2014
      Manuscript received
      28 May 2013
      Manuscript revised
      12 November 2013

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