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In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

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  • Curvature of Riemannian manifolds
  • Кривизна римановых многообразий
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  • Кривизна римановых многообразий численно характеризует отличие римановой метрики многообразия от евклидовой в данной точке. В случае поверхности кривизна в точке полностью описывается гауссовой кривизной. В размерностях 3 и выше кривизна не может быть полностью охарактеризована одним числом в заданной точке, вместо этого она определяется как тензор.
  • In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
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  • In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry. For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the differential geometry of surfaces. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
  • Кривизна римановых многообразий численно характеризует отличие римановой метрики многообразия от евклидовой в данной точке. В случае поверхности кривизна в точке полностью описывается гауссовой кривизной. В размерностях 3 и выше кривизна не может быть полностью охарактеризована одним числом в заданной точке, вместо этого она определяется как тензор.
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