# Harmonic Manifolds and Tubes

## TL;DR

This paper characterizes harmonic manifolds by properties of tubular hypersurfaces around curves, linking volume and curvature measures to the manifold's harmonicity.

## Contribution

It proves that the volume and curvature properties of tubes around geodesic segments uniquely characterize harmonic manifolds.

## Key findings

- Volume of tubes depends only on length and radius in harmonic manifolds.
- Total mean curvature and scalar curvature of tubular hypersurfaces are characterized by harmonicity.
- Explicit formulas relate tubular hypersurface measures to the volume density function.

## Abstract

The authors showed in a preceding paper that in a connected locally harmonic manifold, the volume of a tube of small radius about a regularly parameterized simple arc depends only on the length of the arc and the radius. In this paper, we show that this property characterizes harmonic manifolds even if it is assumed only for tubes about geodesic segments. As a consequence, we obtain similar characterizations of harmonic manifolds in terms of the total mean curvature and the total scalar curvature of tubular hypersurfaces about curves. We find simple formulae expressing the volume, total mean curvature, and total scalar curvature of tubular hypersurfaces about a curve in a harmonic manifold as a function of the volume density function.