Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products

1996/07/01 by Jeff Cheeger, Tobias Colding, Tobias H. Colding · 615 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bounded function #Combinatorics #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Pure mathematics #Ricci curvature #Rigidity (electromagnetism) #Tangent #Tangent cone #Upper and lower bounds

paper · doi:10.2307/2118589

published in Annals of Mathematics 144(1), 189 (Princeton University)

openalex publication_date 1996/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/27

Abstract

The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the implies metric cone theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain stability conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete

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