Harmonic radial vector fields on harmonic spaces

TL;DR

This paper characterizes harmonic spaces by analyzing radial eigen-spaces of Laplacians on functions and 1-forms, examining singularities at zero distance, and extending results to harmonic spaces relative to a point.

Contribution

It provides a new characterization of harmonic spaces through the dimensions of radial eigen-spaces and explores the nature of singularities and duality for radial vector fields.

Findings

Characterization of harmonic spaces via radial eigen-spaces.

Analysis of singularity behavior of radial eigen-functions and 1-forms.

Extension of results to spaces harmonic with respect to a single point.

Abstract

We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian $\Delta^0$ on functions and the Laplacian $\Delta^1$ 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.