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Harmonic radial vector fields on harmonic spaces

We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian $Δ^0$ on functions and the Laplacian $Δ^1$ on 1-forms. We examine the nature of the singularity as the geodesic distance $r$ tends to zero of radial eigen-functions and 1-forms. Via duality, our results give rise to corresponding results for radial vector fields. Many of our results extend to the context of spaces which are harmonic with respect to a single point.