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This paper is devoted to the analysis of metric measure spaces satisfyinglocally the curvature-dimension condition CD(K,N) introduced by the secondauthor and also studied by Lott & Villani. We prove that the local version ofCD(K,N) is equivalent to a global condition CD*(K,N), slightly weaker than the(usual, global) curvature-dimension condition. This so-called reducedcurvature-dimension condition CD*(K,N) has the local-to-global property. Wealso prove the tensorization property for CD*(K,N). As an application weconclude that the fundamental group of a metric measure space (M,d,m) is finitewhenever it satisfies locally the curvature-dimension condition CD(K,N) withpositive K and finite N.

Localization and Tensorization Properties of the Curvature-Dimension Condition for Metric Measure Spaces