Localization and tensorization properties of the curvature-dimension condition for metric measure spaces

This paper is devoted to the analysis of metric measure spaces satisfying locally the curvature-dimension condition CD(K, N) introduced by the second author and also studied by Lott & Villani. We prove that the local version of CD(K, N) is equivalent to a global condition CD* (K, N), slightly weaker than the (usual, global) curvature-dimension condition. This so-called reduced curvature-dimension condition CD* (K, N) has the local-to-global property. We also prove the tensorization property for CD* (K, N). As an application we conclude that the fundamental group pi(1) (M, x(0)) of a metric measure space (M, d, m) is finite whenever it satisfies locally the curvature-dimension condition CD(K, N) with positive K and finite N. (C) 2010 Elsevier Inc. All rights reserved.