On compact conformally flat 4-manifolds
type: Item
instance of: scholarly article
title: On compact conformally flat 4-manifolds
zbMATH Open document ID: 0785.53029
author: Hiroyasu Izeki
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1994-02-07T00:00:00Z
review text: Let \(M\) be a compact connected manifold and \(C\) a conformal class of metrics on \(M\). The resolution of the Yamabe problem asserts that, for any \((M,C)\), there exists a metric \(g\) contained in \(C\) such that the volume of \((M,g)\) is equal to 1 and the scalar curvature of \(g\) is constant and equal to the Yamabe invariant \(\mu(M,C)\) of \((M,C)\). Let \(M\) be an oriented 4-manifold with positive Euler number and suppose that \((M,C)\) is conformally flat. Denote by \(C_ 0\) the conformal class containing the standard metric on \(S^ n\). Then the result of the present paper is stated as follows: (1) If \(\mu(M,C) > -\mu(S^ n,C_ 0)\), then \((M,C)\) is conformal to \((S^ n,C_ 0)\). (2) If \(\mu(M,C) = - \mu(S^ n,C_ 0)\), then \(C\) contains a negative constant curvature metric. A similar result for non-orientable 4-manifolds is also proved in the paper.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 495487
zbMATH Keywords: Yamabe problem
zbMATH Keywords: Yamabe invariant
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227345
OpenAlex ID: W1967995694
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Wikidata QID: Q125937780
DOI: 10.2748/TMJ/1178227345
description: scientific article; zbMATH DE number 495487