Jost maps, ball-homogeneous and harmonic manifolds

Given a real number epsilon > 0, small enough, an associated Jost map J(epsilon) between two Riemannian manifolds is defined. Then we prove that connected Riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball (i.e. for which the identity is a J(epsilon) map) are ball-homogeneous. In the analytic case we characterize such manifolds in terms of the Euclidean Laplacian and we show that they have constant scalar curvature. Under some restriction on the Ricci curvature we prove that Riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic. (C) 2002 Elsevier Science (USA). All rights reserved.