Function-theoretic characterization of Einstein spaces and harmonic spaces

Abstract

Introduction. Consider the following general problem: Let P be a function-theoretic property which holds at the points of a (local or global) Euclidean space E. and which can be stated, formally, for points of a Riemannian space Rn. Find a statement Q which is an intrinsic (geometric) property of R. such that P implies Q and Q implies P. Conversely, given Q, one may look for P. In this paper we solve particular problems of the above type. We take for Q the statements that R. (which is always assumed to have a positive definite metric) is an Einstein space, an harmonic space (to be defined in ?3) with a particular fundamental solution, and a space with constant curvature. P then stands for various statements about the mean value of solutions of certain equations. Denoting by M(u, x?, R) the mean value of u on the geodesic sphere with center x? and radius R we obtain the following results: An Einstein space is characterized (in ?1) by