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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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