A new version of differentiable sphere theorem
type: Item
instance of: scholarly article
title: A new version of differentiable sphere theorem
zbMATH Open document ID: 0688.53016
DOI: 10.1007/BF01388850
author: Yukio Otsu
author: Katsuhiro Shiohama
author: Takao Yamaguchi
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143726
review text: The classical differentiable sphere theorem states that a simply connected complete Riemannian manifold satisfying some strong curvature pinching is diffeomorphic to the standard sphere \(S^ n\). In this paper the upper curvature bound is replaced by a lower bound on the volume. More precisely the authors show that there exists a positive number \(\epsilon =\epsilon (n)\) such that every connected complete Riemannian manifold M of dimension n with \(K_ M\geq 1\) and \(Vol(M)\geq Vol(S^ n)- \epsilon\) is diffeomorphic to \(S^ n\). They also present an analogue for the real projective space involving an additional upper bound on the diameter.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4128005
zbMATH Keywords: sphere theorem
zbMATH Keywords: curvature pinching
zbMATH Keywords: bound on the volume
reviewed by: Christian Bär
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4128005