Four-manifolds with positive curvature operator
type: Item
instance of: scholarly article
title: Four-manifolds with positive curvature operator
zbMATH Open document ID: 0628.53042
published in: Journal of Differential Geometry
publication date: 1986-01-01T00:00:00Z
review text: The main result of this paper states that a compact Riemannian \(M^ 4\) with nonnegative curvature operator is diffeomorphic to a quotient of \(S^ 4\) or \(\mathbb CP^ 2\) or \(S^ 3\times \mathbb R^ 1\) or \(S^ 2\times \mathbb R^ 2\) or \(\mathbb R^ 4\) by a group of isometries (in the standard metric). A corresponding result in dimension 3, with curvature operator replaced by Ricci curvature, is also obtained. In the proof of these results, the parabolic equation \(\partial g/\partial t=2/nrg-2 Ric\) is considered, where \(Ric\) is the Ricci-tensor and \(r\) the mean of the Ricci curvature. Under the assumption of the theorems, it is shown that the solution either converges, as \(t\to \infty\), to a metric of constant sectional curvature or the geometry can be already identified.
reviewed by: Werner Ballmann
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 57R99
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 4022085
zbMATH Keywords: parabolic Einstein equation
zbMATH Keywords: compact Riemannian
zbMATH Keywords: nonnegative curvature operator
zbMATH Keywords: Ricci curvature
author: Richard S. Hamilton
MaRDI profile type: Publication
Wikidata QID: Q115185218
full work available at URL: https://doi.org/10.4310/jdg/1214440433
OpenAlex ID: W1587723526
DOI: 10.4310/JDG/1214440433
description: scientific article; zbMATH DE number 4022085