The Euclidean Laplacian

Abstract

The following paper arose from joint work of Alfred Gray and the author. A formula due to Pizzetti (which expresses the mean-value of a real-valued function over a sphere of radius r in euclidean space as an infmite power series in r) is generalized to apply to the mean-value of functions over a geodesic sphere of radius r in a Riemannian manifold. This expansion is then used to obtain a characterization of Einstein and super-Einstein manifolds. In the course of this work, a new differential operator L (2k) is globally defined over the manifold for each integer k. The restriction of this operator to a point is the k-th iterate of the euclidean Laplacian at that point. The euclidean Laplacian plays a vital rôle in the development of the theory.