Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) with L ₁-small integral anisotropy have L _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in the W _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.