Harmonic radial vector fields on harmonic spaces (Q2050871)

\(M\) is a harmonic space if and only if \(\tilde{\theta}_P=\sqrt{g_{ij}}\) (the volume density function) is radial for every point \(P\) of \(M\). In this paper, the authors characterize harmonic spaces in terms of the dimensions of various spaces of radial eigenspaces of the Laplacian \(\Delta^0\) on functions and the Laplacian \(\Delta^1\) on 1-forms (Theorem 1.3). The paper has a first introductory section where the main results are stated (Theorems 1.3 and 1.5) and two other sections. In the second section the authors prove Theorem 1.3 and in the last section Theorem 1.5 is proved.