Stability problems in a theorem of F. Schur
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.
scientific article; zbMATH DE number 790579