Harmonic Manifolds and the Volume of Tubes about Curves

TL;DR

This paper demonstrates that harmonic manifolds uniquely possess the tube volume property, extending classical results from Euclidean and symmetric spaces to a broader class, and characterizes the geometric conditions for this property.

Contribution

It generalizes the tube volume property to harmonic manifolds, computes volumes in Damek-Ricci spaces, and characterizes symmetric spaces with this property as harmonic.

Findings

Harmonic manifolds have the tube property.

Volumes in Damek-Ricci spaces are computed.

Symmetric spaces with the tube property are harmonic.

Abstract

H. Hotelling proved that in the n-dimensional Euclidean or spherical space, the volume of a tube of small radius about a curve depends only on the length of the curve and the radius. A. Gray and L. Vanhecke extended Hotelling's theorem to rank one symmetric spaces computing the volumes of the tubes explicitly in these spaces. In the present paper, we generalize these results by showing that every harmonic manifold has the above tube property. We compute the volume of tubes in the Damek-Ricci spaces. We show that if a Riemannian manifold has the tube property, then it is a 2-stein D'Atri space. We also prove that a symmetric space has the tube property if and only if it is harmonic. Our results answer some questions posed by L. Vanhecke, T. J. Willmore, and G. Thorbergsson.