Four-manifolds with \(1/4\)-pinched flag curvatures
type: Item
instance of: scholarly article
title: Four-manifolds with \(1/4\)-pinched flag curvatures
zbMATH Open document ID: 1187.53066
author: J. Blot
published in: The Asian Journal of Mathematics
publication date: 2010-01-13T00:00:00Z
review text: The authors aim to characterize solutions of the \textit{Ricci-flow equation} on a connected compact Riemannian manifold \((M,g_0)\) with respect to the so-called \textit{pinching constant} \(\lambda=\text{min sec/max sec}\), i.e., by adding an upper as well as a lower bound on the sectional curvature (denoted {sec}). With respect to this scalar characterization of Riemannian manifolds, there is some interesting information on their global structure. Let us recall some of them. {\parindent6mm \begin{itemize}\item[(*)] (Brendle and Schoen) If \(M\) is a compact \(n\)-dimensional Riemannian manifold with \(\lambda\in({{1}\over{4}},1)\) (\textit{strict pinching}), \(M\) is diffeomorphic to a spherical space form. If \(\lambda\in[{{1}\over{4}},1]\) \(M\) is diffeomorphic to a spherical space form or isometric to a locally symmetric space. \item[(*)] (Berger-Klingenberg) If \(M\) is a compact simply connected manifold with \(\lambda\geq{{1}\over{4}}\) then \(M\) is either homeomorphic to \(S^n\) or isometric to \({\mathbb CP}^n\), \({\mathbb HP}^n\) or \(Ca{\mathbb P}^2\), with their standard Fubini metric. \item[(*)] (Synge) A manifold with positive curvature does not necessarily be simply connected. In fact if \(\dim M=n=2k\), one has \(\pi_1(M)=0\) if orientable and \(\pi_1(M)={\mathbb Z}_2\) if non-orientable. If \(\dim M=n=2k+1\) and positively curved \(M\) is orientable. \item[(*)] (Cheeger) Given a constant \(\epsilon>0\), there are only finitely many diffeomorphism types of compact simply connected \(2n\)-dimensional manifolds \(M\) with \(\lambda\geq\epsilon\). \item[(*)] (Fang-Rong-Petrunin-Tuschmann) Given a constant \(\epsilon>0\), there are only finitely many diffeomorphism types of compact \((2n+1)\)-dimensional manifolds \(M\) with \(\pi_1(M)=\pi_2(M)=0\) and \(\lambda\geq\epsilon\). \end{itemize}} In this paper, one considers four-dimensional manifolds with \textit{\(\lambda\)-pinched flag curvature}, (\(0\leq \lambda<1\)), i.e., \(R_u(v,v)\geq\lambda(x)R…
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 5657195
zbMATH Keywords: Ricci-flow equation
zbMATH Keywords: pinched flag curvatures
zbMATH Keywords: curvature tensor
zbMATH Keywords: Ricci-flow
zbMATH Keywords: Fubini-Study metric
zbMATH Keywords: Cauchy problem
zbMATH Keywords: \(2\)-jet-space
MaRDI profile type: Publication
DOI: 10.4310/AJM.2009.V13.N2.A5
description: scientific article; zbMATH DE number 5657195