On polyharmonic Riemannian manifolds. (English) Zbl 0628.53057

Schimming, R.; Kowolik, J.

Z. Anal. Anwend. 6, 331-339 (1987).

A pseudo-Riemannian manifold (M,g) is called a k-harmonic space if the differential equation \(\Delta^ kF=0(\Delta^ k\) being the iterated Laplacian) has a local solution around each point p which is a radial function. (This means that F(x) depends only on the Synge’s two-point function \(\sigma\) (p,x), which becomes half of the square of the distance in the proper Riemannian case.) For \(k=1\), the famous Lichnerowicz conjecture says that all Riemannian harmonic spaces are locally two-point homogeneous. The k-harmonic spaces for \(k>1\) (in a more special setting) have been studied only recently by R. Caddeo, J. Eichhorn, A. Gray, P. Matzeu, R. Schimming and L. Vanhecke. The present authors give a rich variety of results for the general pseudo-Riemannian case. The only restriction is that the functions F(\(\sigma)\) under consideration behave asymptotically like some power of \(\sigma\) for \(\sigma\to 0\).

Reviewer: O.Kowalski

MSC:

53C50

Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

pseudo-Riemannian manifold; polyharmonic function; k-harmonic space; iterated Laplacian; Synge’s two-point function