Complete classification of compact four-manifolds with positive isotropic curvature
type: Item
instance of: scholarly article
title: Complete classification of compact four-manifolds with positive isotropic curvature
zbMATH Open document ID: 1257.53053
author: Siu-Hung Tang
author: Xi-Ping Zhu
author: Bing-Long Chen
published in: Journal of Differential Geometry
publication date: 2012-09-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.1999
full work available at URL: https://projecteuclid.org/euclid.jdg/1343133700
reviewed by: Giovanni Calvaruso
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 6081388
zbMATH Keywords: positive isotropic curvature
MaRDI profile type: Publication
OpenAlex ID: W2962931256
arXiv ID: 0810.1999
Wikidata QID: Q115170175
DOI: 10.4310/JDG/1343133700
review text: Let \((M,<,>)\) be an \(n\)-dimensional Riemannian manifold. The curvature operator of \(M\) at a point \(p\) is the self-adjoint linear endomorphism of \(\Lambda ^2 T_p M\) defined by NEWLINE\[NEWLINE<\mathcal{R} (X \wedge Y),U\wedge V>=<R(X,Y)U,V>,NEWLINE\]NEWLINE where \(R\) is the curvature tensor. The Riemannian metric \(<,>\) extends to a Hermitian inner product \(<<,>>\) on \(T_p M \otimes \mathbb C\), and \(\mathcal{R}\) to a complex linear map on \(\Lambda ^2 T_p M \otimes \mathbb C\), denoted again by \(\mathcal{R}\). For any two-plane \(\sigma \subset T_p M \otimes \mathbb C\), the complex sectional curvature is defined by NEWLINE\[NEWLINEK_{\mathbb C} (\sigma)=<<\mathcal{R}(Z\wedge W),Z\wedge W>>,NEWLINE\]NEWLINE where \(\{Z,W\}\) is a basis of \(\sigma\), unitary with respect to \(<<,>>\). If \(K_{\mathbb C} (\sigma) >0\) for any totally isotropic plane at any point, then \(M\) is said to have positive isotropic curvature.NEWLINENEWLINEPositive isotropic curvature was introduced by \textit{M. J. Micallef} and \textit{J. D. Moore} [Ann. Math. (2) 127, No. 1, 199--227 (1988; Zbl 0661.53027)], who proved that a compact simply connected \(n(\geq 4)-\)dimen\-sional manifold with positive isotropic curvature is homeomorphic to a sphere. Schoen conjectured that an \(n(\geq 4)\)-dimensional compact manifold with positive isotropic curvature has a finite cover diffeomorphic to either \(\mathbb S^n\), \(\mathbb S^{n-1}\times \mathbb S^1\), or to a connected sum of them.NEWLINENEWLINEIn the present paper, this conjecture is proved in dimension four. More precisely, the following complete classification result is proved: a compact \(4\)-dimensional manifold admits a metric with positive isotropic curvature if and only if it is diffeomorphic to either \(\mathbb S^4\), \(\mathbb R \mathbb P ^4\), \(\mathbb S^3 \times \mathbb R /G\), or a connected sum of them, where \(G\) is a cocompact fixed-point free discrete subgroup of the isometry group of the standard metric o…
description: scientific article; zbMATH DE number 6081388