Harmonic Manifolds and the Volume of Tubes about Curves

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.Comment: 17 pages, no figures. This version is different from the journal versio

Harmonic Manifolds and Tubes

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.Comment: 14 pages, grant numbers are update

A situation alaysis of end-of-life care in Hungary and opportunities for enhancement

9

HARDNESS FUNCTIONS TO PREDICT WELDABILITY OF LOW CARBON STEEL (Data to Computer Data Bank)

The determination of the expectable hardness of a welded joint can be carried out in the knowledge of the cooling time from 850 °C to 500 °C with function HV-Δt8 / 5 characteristic of the given material. On the basis of the known hardness criterion it can be determined whether the technology causing the given cooling time can be used or not. Direct connection is tried to be found between hardness and cooling time starting from relations in literature and chemical composition of base metals. In general we can state these relations reflect the character of hardness change but are not reliable to calculate the actual hardness values. Relations relying on separate, definite measurement results ensure the reliable prediction of expectable hardness of heat affected zone at characteristic types of steel. Parameters at our disposal in the form of computer data bank, on the other hand, make the fast establishment of the necessary relations possible