Complete Riemannian manifolds with pointwise pinched curvature
type: Item
instance of: scholarly article
title: Complete Riemannian manifolds with pointwise pinched curvature
zbMATH Open document ID: 0957.53011
published in: Inventiones Mathematicae
publication date: 2001-03-28T00:00:00Z
review text: This paper deals with the problem of finding pinching conditions for the compactness of a Riemannian manifold. Let \((M,g)\) be a Riemannian manifold, with \(\dim M>2\) and consider the orthogonal decomposition of the Riemannian curvature: \(R={\mathcal W}+U+V\) where \({\mathcal W}\) denotes the Weyl tensor, while \(U\) are \(V\) are the algebraic curvature tensor fields respectively determined by the scalar curvature \(\tau\) and the traceless Ricci tensor \(\rho_0\), that is: \[ U(X,Y,Z,W)=\tfrac\tau{n(n-1)} (g(X,Z)g(Y,W)-g(Y,Z)g (X,W)); \] \[ V(X,Y,Z,W)=\tfrac 1{n-1}(\rho_0 (X,Z)g(Y,W)-\rho_0(X,W) g(Y,Z)+ \rho_0(Y,W)g (X,Z)-\rho_0 (Y,Z) (Z,W)). \] In [J. Differ. Geom. 21, 47--62 (1985; Zbl 0606.53026)], \textit{G. Huisken} considered the following pointwise pinching condition: \(\|{\mathcal W}\|^2+ \|V\|^2 \leq\delta_n (1-\varepsilon)^2\|U\|^2\), where \(\varepsilon>0\) and \(\delta_n= {2\over(n-2) (n+1)}\) if \(n\geq 4\), \(n\neq 5\), \(\delta_5= {1\over 10}\). A detailed study of the Ricci flow on complete non compact manifolds with positive scalar curvature, combined with recent results due to R. S. Hamilton, G. Huisken and W. X. Shi, allows to state the following mean theorems. Theorem 1. Let \((M, g)\) be a complete, \(n\)-dimensional Riemannian manifold with positive and bounded scalar curvature. If \(n\geq 4\) and the pinching condition (1) holds, then \(M\) is compact. Theorem 2. Let \((M,g)\) be a 3-dimensional, complete, noncompact Riemannian manifold with bounded and non-negative sectional curvatures. If the Ricci tensor \(\rho\) satisfies \(\rho(X,Y) \geq\varepsilon \tau g(X,Y)\), for any vector fields \(X,Y\) and for some \(\varepsilon>0\), then \(M\) is flat. The paper is endowed with a wide bibliography.
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 1463435
zbMATH Keywords: Weyl tensor
zbMATH Keywords: pinching condition
zbMATH Keywords: Ricci flow
zbMATH Keywords: positive scalar curvature
author: Bing-Long Chen
author: Xi-Ping Zhu
MaRDI profile type: Publication
full work available at URL: s002220000061
OpenAlex ID: W2004171259
Wikidata QID: Q115388542
DOI: 10.1007/S002220000061
description: scientific article; zbMATH DE number 1463435