Stability problems in a theorem of F. Schur |
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Authors: | I G Nikolaev |
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Institution: | (1) Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall MC-382, 1409, W. Green St., 61801 IL, Urbana, USA |
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Abstract: | 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) withL
1-small integral anisotropy haveL
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 theW
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. |
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Keywords: | 53C20 53C21 53C45 |
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