  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.
