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

Harmonic manifold; Busemann function; minimal horospheres.

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.

Hemangi M Shah¹

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

Manuscript received

28 May 2013

Manuscript revised

12 November 2013