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Riemannian manifolds for which a power of the radius is k-harmonic. (English) Zbl 0571.53012
Let (M,g) be a Riemannian manifold and y a point of M. Further let x be a point of a normal neighborhood N(y) of y and put $$d=d(x,y)$$. R. Caddeo, P. Matzeu, J. Eichborn and the reviewer studied Riemannian manifolds such that for all y and all $$x\in N(y)$$ $$\Delta^ kd^{2\ell}=0$$, where $$k\in {\mathbb{N}}_ 0$$ and $$\ell$$ is a real number. $$\Delta$$ denotes the Laplacian on (M,g).
In this paper a similar problem for pseudo-Riemannian manifolds is considered. Let $$\sigma$$ (x,y) denote Synge’s two-point function, that is, for $$x=\exp_ y\sum x^ ie_ i$$, $$\sigma (x,y)=g(y)_{\alpha \beta}x^{\alpha}x^{\beta}$$. Further, let $$N(y)^-=\{x\in N(y)|$$ $$\sigma$$ (x,y)$$\neq 0\}$$. The author studies in detail pseudo-Riemannian manifolds such that $$\Delta^ k| \sigma |^{\ell}=0$$ or $$\Delta^ k \log | \sigma | =0$$ for all $$y\in M$$ and all $$x\in N(y)^-$$. As could be expected, this theory is richer than in the Riemannian case.
Reviewer: L.Vanhecke

##### MSC:
 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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