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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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