One says that the manifold is a
HARMONIC SPACE if the volume density function Θ
is such that
Θ(r,θ;P)=Θ(r) is a radial function independent of the
angle and the base point.
There are a number of equivalent geometrical
characterizations that will be discussed,
in particular the manifold
is a harmonic space if and only if small geodesic spheres have
constant mean curvature if and only if every harmonic function
satisfies the mean value property.
We will present a brief survey of
what is known about harmonic spaces and discuss some recent
results
that are joint work with J. H. Park.