Riemannian Manifolds for which a Power of the Radius is $k$-harmonic

Abstract

Let $\sigma =\sigma (x,y)$ denote Synge’s function of a Riemannian manifold $(M,g)$ of any signature and consider the condition that some power of $\sigma$ or the logarithm of $\sigma$ is $k$-harmonic. Then in many, cases $(M,g)$ turns out to be flat. Certain classes of non-flat manifolds can be characterized just by a condition of the aforesaid typo.