A rigidity phenomenon on Riemannian manifolds with reverse volume pinching
type: Item
instance of: scholarly article
title: A rigidity phenomenon on Riemannian manifolds with reverse volume pinching
zbMATH Open document ID: 1152.53028
author: Peihe Wang
author: Yuliang Wen
published in: Annals of Global Analysis and Geometry
publication date: 2008-07-30T00:00:00Z
review text: The well-known topological sphere theorem states that a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval \((1,4]\) is homeomorphic to the \(n\)-sphere. The proof of this celebrated theorem undergone several steps and was computed in 1961 by Klingenberg. The differentiable sphere theorem which claims that the manifold is actually diffeomorphic to the \(n\)-sphere with the standard differentiable structure under the same hypothesis was recently proved by [\textit{S. Brendle} and \textit{R. Schoen}, J. Am. Math. Soc. 22, 287--307 (2009; doi:10.1090/S0894-0347-08-00613-9)]. The main result of the paper under review is that a compact simply connected \(2n\)-dimensional Riemannian manifold without boundary is diffeomorphic to the sphere \(S^{2n}\) if its curvature and volume are bounded by: \(0<K_M \leq 1\), \(V(M)\leq (1+\eta)V(S^{2n})\), where \(\eta\) is a constant dependent only on \(n\). The topological version of this was proved in [\textit{D. Coghlan} and \textit{Y. Itokawa}, Proc. Am. Math. Soc. 111, No. 3, 815--819, (1991; Zbl 0719.53019)] where the same curvature bound was used whilst the volume bound was \(V(M)< \frac{3}{2}V(S^{2n})\).
reviewed by: Ye-Lin Ou
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 5306111
zbMATH Keywords: volume comparison theorem
zbMATH Keywords: Hausdorff convergence
zbMATH Keywords: differentiable sphere theorem
zbMATH Keywords: harmonic coordinate
zbMATH Keywords: harmonic radius
MaRDI profile type: Publication
Wikidata QID: Q115384679
full work available at URL: s10455-007-9094-4
OpenAlex ID: W2008694222
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DOI: 10.1007/S10455-007-9094-4
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description: scientific article; zbMATH DE number 5306111