## On polyharmonic Riemannian manifolds.(English)Zbl 0628.53057

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
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