Title: Harmonic manifolds and the geometry of tubes about curves (Harmonikus
 sokaságok és a görbék körüli csövek geometriája)

Abstract: Harmonic manifolds form an important class of Riemannian
manifolds. In this talk we show 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, and 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, assuming in the case
of total scalar curvature that the space has dimension at least 4. 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. At the end of the talk we show that
in a 3-dimensional manifold, the total scalar curvature of a tubular surface
about a regular simple arc depends only on the length of the arc and the
radius of the tube if and only if the space is D'Atri.