Pinching below \(1\over 4\), injectivity radius, and conjugate radius
type: Item
instance of: scholarly article
title: Pinching below \(1\over 4\), injectivity radius, and conjugate radius
zbMATH Open document ID: 0813.53026
author: Wolfgang T. Meyer
author: Uwe Abresch
published in: Journal of Differential Geometry
publication date: 1995-02-26T00:00:00Z
review text: The injectivity radius of any simply connected, even dimensional Riemannian manifold \(M^ n\) with positive sectional curvature equals its conjugate radius. So far the corresponding result in odd-dimensions has only been known under the additional hypothesis that \(M^ n\) is weakly \({1\over 4}\)-pinched. Moreover, some famous examples due to M. Berger show that the statement is even false, unless \(M^ n\) is at least \({1\over 9}\)-pinched. It has been a long standing problem whether the pinching constant can be pushed below \({1\over 4}\) for odd dimensional manifolds or not. In this paper we prove that this is indeed possible. The pinching constant \(\delta\in [{1\over 9},{1\over 4})\) that is needed in our main theorem does not depend on the dimension. As an application we obtain a sphere theorem for simply connected, odd-dimensional, \(\delta_ n\)-pinched manifolds where the pinching constant \(\delta_ n\) is strictly less than \({1\over 4}\) and up to now still depends on the dimension.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 727344
zbMATH Keywords: injectivity radius
zbMATH Keywords: positive sectional curvature
zbMATH Keywords: conjugate radius
zbMATH Keywords: weakly \({1\over 4}\)-pinched
zbMATH Keywords: pinching
zbMATH Keywords: sphere theorem
Wikidata QID: Q115178864
reviewed by: Uwe Abresch
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214455781
OpenAlex ID: W1482449643
DOI: 10.4310/JDG/1214455781
description: scientific article; zbMATH DE number 727344