Peter Gilkey, University of Oregon

Title: An introduction to the theory of harmonic Riemannian manifolds

Abstract: In geodesic coordinates centered a point P, one can write the Riemannian
volume element in the form dvol=Θ(r,θ;P)dr dθ.

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.