On almost Blaschke manifolds. I
type: Item
instance of: scholarly article
title: On almost Blaschke manifolds. I
zbMATH Open document ID: 0549.53040
DOI: 10.1007/BF01159164
author: Jin-Ichi Itoh
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173610
review text: Let \(i(M)\) be the injectivity radius of a Riemannian manifold \((M,g)\) and \(d(M)\) be its diameter. The purpose of this paper is to determine the topology of a Riemannian manifold \((M,g)\) when \(i(M)\) is close to \(d(M)\) in lower \((2,3)\) dimensional cases. In the 2-dimensional case under the assumption that the Gaussian curvature of \((M,g) \geq -\delta^ 2\) (resp. \(\geq 0)\) and \(\sinh (\delta i(M))>(\sqrt{3}/2)\cdot\sinh (\delta d(M))\) (resp. \(i(M)>(\sqrt{3}/2)d(M))\), it follows that \(M\) is diffeomorphic to \(S^ 2\) or \(RP^ 2\). In the 3-dimensional case under the assumption that the sectional curvature of (M,g) \(\geq\delta^ 2\) and \(\sin (\delta i(M))>(\sqrt{3}/2)\cdot\sin (\delta d(M)),\) it follows that M is diffeomorphic to \(S^ 3\) or \(RP^ 3\). These results are best possible. Under the hypothesis that i(M) is sufficiently close to d(M) the same result follows without any positive curvature assumption. To prove these results the author uses the structure of the cut locus.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3876127
zbMATH Keywords: pinching conditions
zbMATH Keywords: injectivity radius
zbMATH Keywords: diameter
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: sectional curvature
MaRDI profile type: Publication
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OpenAlex ID: W1993571491
description: scientific article; zbMATH DE number 3876127