The multidimensional problem of the correctness of Schur's theorem

This paper continues an earlier one (Mat. Sb. (N.S.), 116(158) (1981), 527–538). A function $\varepsilon(x)$ measuring the extent to which a Riemannian space is nonisotropic at the point $x$ is studied. Using $\varepsilon(x)$, definitions of the notion of correctness of Schur's theorem are given in the multidimensional case. The relations between these definitions are clarified, and sufficient conditions for the correctness of Schur's theorem are given. It is shown that by a small deformation of the given metric it is possible to obtain one in which Schur's theorem is not correct. The methods developed in the paper are applied to study some geometric properties of geodesically parallel surfaces.
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Bibliography: 11 titles.