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On the structure of spaces with Ricci curvature bounded below. I. (English) Zbl 0902.53034
The authors investigate the structure of spaces which are pointed Gromov-Hausdorff limits of sequences of complete, connected Riemannian manifolds whose Ricci curvatures have a definite lower bound. Sometimes they also assume a lower volume bound. The presented results, most of which were announced and proved in earlier papers of the authors [see J. Cheeger and T. H. Colding, Ann. Math., II. Ser. 144, 189-237 (1996; Zbl 0865.53037)], are applications of the “almost rigidity” theorems for manifolds of almost nonnegative Ricci curvature.
Applying metrics of doubly warped product type, the authors construct a number of examples of spaces which are Gromov-Hausdorff limits of sequences of pointed Riemannian manifolds of positive Ricci curvature.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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