Harmonic manifolds and the volume of tubes about curves (Q2822142)

This article deals with the volume of tubes in Riemannian spaces. It is known that the volume of a tube of small radius around a curve in the Euclidean space depends only on the length of the curve and the radius; the same holds true for tubes in rank-1 symmetric spaces [\textit{A. Gray} and \textit{L. Vanhecke}, Proc. Lond. Math. Soc., III. Ser. 44, 215--243 (1982; Zbl 0491.53035); \textit{A. Gray}, Tubes. 2nd ed. Basel: Birkhäuser (2003; Zbl 1048.53040)]. It is suggested that similar results have to be valid for other classes of Riemannian spaces, and the main attention is paid to harmonic manifolds.NEWLINENEWLINEBy definition, a Riemannian manifold \(M\) is said to have the tube property, if there is a function \(V: [0,\infty) \to\mathbb{R}\) such that for any smooth injective regular curve \(\gamma: [a,b]\to M\) and any sufficiently small \(r\) the volume of the tube of radius \(r\) around \(\gamma\) in \(M\) is equal to \(V(r) l_\gamma\), where \(l_\gamma\) is the length of \(\gamma\).NEWLINENEWLINEIt is proved that harmonic manifolds have the tube property.NEWLINENEWLINETheorem 1. A Riemannian manifold has the tube property if and only if it is a d'Atri space and satisfies the tube property for geodesic curves.NEWLINENEWLINETheorem 2. Every connected harmonic manifold has the tube property.NEWLINENEWLINENext, particular harmonic manifolds, the Damek-Ricci spaces, are considered and an explicit expression for \(V(r)\) is derived. Similar results for an other class of harmonic manifolds, the two-point homogeneous spaces, were obtained earlier by Gray and Vanhecke [loc. cit.].NEWLINENEWLINEIt is conjectured that if a Riemannian manifold has the tube property then it is harmonic. The following results partially confirm this conjecture.NEWLINENEWLINETheorem 3. A manifold having the tube property is a \(2\)-Stein space.NEWLINENEWLINETheorem 4. If a symmetric space has the tube property, then it is harmonic.NEWLINENEWLINEIt is claimed that Theorem 4 holds true for locally symmetric spaces too.