On Polyharmonic Riemannian Manifolds

Abstract

A natural generalization of the harmonic manifolds is considered: a Riemannian manifold is called $k$-harmonic or polyharmonic if it admits a non-constant $k$-harmonic function depending only on the geodesic distance $r=r(x,y)$ or rather on Synge’s function $\sigma =\sigma (x,y)$, i.e. a solution $F$ of ${\Delta }^{k}F(\sigma )=0$. Certain theorems are generalized from harmonic to polyharmonic manifolds.