Stability problems in a theorem of F. Schur
type: Item
instance of: scholarly article
title: Stability problems in a theorem of F. Schur
zbMATH Open document ID: 0837.53031
DOI: 10.1007/BF02566005
published in: Commentarii Mathematici Helvetici
publication date: 1996-05-24T00:00:00Z
full work available at URL: https://eudml.org/doc/140366
review text: Schur's theorem states that an isotropic Riemannian manifold of dimension \(\geq 3\) has constant curvature. For a Riemannian manifold \((M, g)\) let the curvature anisotropy at \(p\in M\) be the maximum deviation of the sectional curvature at \(p\) from the normalized scalar curvature at \(p\). Cheeger's class \({\mathfrak R}(n, d, V, \Lambda)\) consists of all \(n\)- dimensional manifolds with diameter \(\leq d\), volume \(\geq V\) and sectional curvature \(|K|\leq \Lambda\). The main result of this paper states that if a manifold in \({\mathfrak R}(n, d, V, \Lambda)\) has \(L_1\)-small curvature anisotropy then the oscillation of the scalar curvature over \(M\) is \(L_p\)-small. Furthermore, there exists a metric of constant sectional curvature on \(M\) which is \(W^2_p\)-near to \(g\). The author first proves that a geodesically complete metric space of Urysohn-Menger dimension \(> 2\) for which the Wald curvature exists at each point is isometric to a space form. Then he uses Cheeger's finiteness and Gromov's compactness theorems for \({\mathfrak R}(n, d, V, \Lambda)\) to derive the stability results.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 790579
zbMATH Keywords: isotropic manifold
zbMATH Keywords: Schur's theorem
zbMATH Keywords: constant curvature
zbMATH Keywords: curvature anisotropy
zbMATH Keywords: Wald curvature
author: Igor G. Nikolaev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 790579