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Lower bounds on Ricci curvature and the almost rigidity of warped products. (English) Zbl 0865.53037

The authors prove quantitative versions of rigidity theorems for volume or diameter in the presence of a suitable lower bound on Ricci curvature. The model spaces here form a class of smooth manifolds with a warped product metric of a particular type. They show that if the volume or diameter are almost maximal, then the manifold is close to a warped product in the Gromov-Hausdorff sense. Among the applications are the splitting theorem for Gromov-Hausdorff limit spaces when the lower bound for Ricci curvature tends to 0, as well as Gromov’s conjecture that manifolds of almost positive Ricci curvature have almost nilpotent fundamental group.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C20 Global Riemannian geometry, including pinching
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