On Einstein four-manifolds
type: Item
instance of: scholarly article
title: On Einstein four-manifolds
zbMATH Open document ID: 1078.53034
author: Ézio de Araujo Costa
published in: Journal of Geometry and Physics
publication date: 2005-06-23T00:00:00Z
review text: Topological obstructions to the existence of Einstein metrics on a given compact and oriented 4-manifold were found by \textit{N. Hitchin} [J. Differ. Geom. 9, 435--441 (1974; Zbl 0281.53039)] and by \textit{M. J. Gursky} and \textit{C. LeBrun} [Ann. Global Anal. Geom. 17, No. 4, 315--328 (1999; Zbl 0967.53029)]. In the paper under review the author gives several improvements of these results. One of the results states that under the assumption \(\text{inf\,}K\geq 2\rho/3\) and \(\rho< 0\) the inequality \(\chi\geq 15|\tau|/8\) holds where \(K\) denotes the sectional curvature, \(\rho\) is the Ricci curvature and \(\tau\) the signature. In a related result, a characterization of the standard metrics on the 4-sphere, \(\mathbb{R} \mathbb{P}^4\) and \(\mathbb{C} \mathbb{P}^2\) is given. Here the assumption is \(\sup K\leq 2\rho/3\) and \(\rho > 0\). Further results are obtained concerning rigidity of Einstein metrics under similar assumptions on the sectional curvature.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 2182242
zbMATH Keywords: curvature operator
zbMATH Keywords: Ricci curvature
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: j.geomphys.2003.10.013
OpenAlex ID: W2007135083
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DOI: 10.1016/J.GEOMPHYS.2003.10.013
description: scientific article; zbMATH DE number 2182242