. . . "\u626D\u7387\u5F35\u91CF"@zh . "\u0422\u0435\u043D\u0437\u043E\u0440\u043E\u043C \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0432 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0432\u0435\u043A\u0442\u043E\u0440\u043E\u0437\u043D\u0430\u0447\u043D\u0438\u0439 \u0442\u0435\u043D\u0437\u043E\u0440, \u0449\u043E \u043A\u043E\u0436\u043D\u0456\u0439 \u043F\u0430\u0440\u0456 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u0445 \u043F\u043E\u043B\u0456\u0432 \u043A\u043B\u0430\u0441\u0443 , \u0437\u0430\u0434\u0430\u043D\u0438\u0445 \u043D\u0430 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u0433\u043B\u0430\u0434\u043A\u043E\u043C\u0443 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456 \u0437 \u0432\u0432\u0435\u0434\u0435\u043D\u043E\u044E \u0430\u0444\u0456\u043D\u043D\u043E\u044E \u0437\u0432'\u044F\u0437\u043D\u0456\u0441\u0442\u044E \u043F\u0440\u0438\u0441\u0432\u043E\u044E\u0454 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0435 \u043F\u043E\u043B\u0435 \u043A\u043B\u0430\u0441\u0443 . \u0420\u0430\u0437\u043E\u043C \u0456\u0437 \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C \u043A\u0440\u0438\u0432\u0438\u043D\u0438 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0454 \u043E\u0434\u043D\u0438\u043C \u0437 \u0433\u043E\u043B\u043E\u0432\u043D\u0438\u0445 \u0456\u043D\u0432\u0430\u0440\u0456\u0430\u043D\u0442\u0456\u0432 \u0430\u0444\u0456\u043D\u043D\u043E\u0457 \u0437\u0432'\u044F\u0437\u043D\u043E\u0441\u0442\u0456. \u0417\u043E\u043A\u0440\u0435\u043C\u0430 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0432\u0456\u0434\u0456\u0433\u0440\u0430\u0454 \u0434\u0443\u0436\u0435 \u0432\u0430\u0436\u043B\u0438\u0432\u0443 \u0440\u043E\u043B\u044C \u0443 \u0432\u0438\u0432\u0447\u0435\u043D\u043D\u0456 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0433\u0435\u043E\u0434\u0435\u0437\u0438\u0447\u043D\u0438\u0445 \u043B\u0456\u043D\u0456\u0439 \u043D\u0430 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0430\u0445."@uk . . . . . "\u041A\u0440\u0443\u0447\u0435\u043D\u0438\u0435 \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0438"@ru . "Tensor skr\u0119cenia \u2013 obiekt opisuj\u0105cy przekr\u0119cenie ramki poruszaj\u0105cej si\u0119 wzd\u0142u\u017C krzywej."@pl . . . . . . . "Torsionstensor"@de . . . . . "\u041A\u0440\u0443\u0301\u0447\u0435\u043D\u0438\u0435 \u0430\u0444\u0444\u0438\u0301\u043D\u043D\u043E\u0439 \u0441\u0432\u044F\u0301\u0437\u043D\u043E\u0441\u0442\u0438 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043A \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0435\u0439 \u0432 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438. \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u043F\u043E\u043D\u044F\u0442\u0438\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u044B, \u0438\u043C\u0435\u044E\u0449\u0435\u0433\u043E \u0441\u043C\u044B\u0441\u043B \u0434\u043B\u044F \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0438 \u0432 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u043B\u044C\u043D\u043E\u043C \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438 \u0438\u043B\u0438 \u0434\u0430\u0436\u0435 \u0432 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438, \u043A\u0440\u0443\u0447\u0435\u043D\u0438\u0435 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u043E \u043B\u0438\u0448\u044C \u0434\u043B\u044F \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0435\u0439 \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438 (\u0438\u043B\u0438, \u0447\u0443\u0442\u044C \u0431\u043E\u043B\u0435\u0435 \u043E\u0431\u0449\u043E, \u0432 \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F\u0445, \u0441\u043D\u0430\u0431\u0436\u0451\u043D\u043D\u044B\u0445 \u043E\u0442\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0435\u043C \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0435 \u2014 \u0441\u043A\u0430\u0436\u0435\u043C, \u043A\u043E\u043D\u0442\u0430\u043A\u0442\u043D\u043E\u043C \u043F\u043E\u0434\u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438). \u0415\u0441\u043B\u0438 \u2014 \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u044C \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438, \u0435\u0451 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u0438\u044F \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043A\u0430\u043A . \u041D\u0435\u043F\u043E\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u043C \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u0435\u043C \u043F\u0440\u043E\u0432\u0435\u0440\u044F\u0435\u0442\u0441\u044F, \u0447\u0442\u043E \u0434\u0430\u043D\u043D\u044B\u0439 \u043E\u043F\u0435\u0440\u0430\u0442\u043E\u0440 \u043B\u0438\u043D\u0435\u0435\u043D \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0435\u043B\u044C\u043D\u043E \u0443\u043C\u043D\u043E\u0436\u0435\u043D\u0438\u044F \u043D\u0430 \u0444\u0443\u043D\u043A\u0446\u0438\u0438, \u0438, \u0441\u043B\u0435\u0434\u043E\u0432\u0430\u0442\u0435\u043B\u044C\u043D\u043E, \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0442\u0435\u043B\u044C\u043D\u043E \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442 \u0442\u0435\u043D\u0437\u043E\u0440 \u0432\u0438\u0434\u0430 . \u0418\u043D\u044B\u043C\u0438 \u0441\u043B\u043E\u0432\u0430\u043C\u0438, \u043F\u0430\u0440\u0435 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u044B\u0445 \u0432\u0435\u043A\u0442\u043E\u0440\u043E\u0432 \u0432 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043A\u0440\u0443\u0447\u0435\u043D\u0438\u0435 \u043A\u043E\u0441\u043E\u0441\u0438\u043C\u043C\u0435\u0442\u0438\u0447\u0435\u0441\u043A\u0438\u043C \u043E\u0431\u0440\u0430\u0437\u043E\u043C \u0441\u043E\u043F\u043E\u0441\u0442\u0430\u0432\u043B\u044F\u0435\u0442 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u044B\u0439 \u0432\u0435\u043A\u0442\u043E\u0440."@ru . . . . . "\u041A\u0440\u0443\u0301\u0447\u0435\u043D\u0438\u0435 \u0430\u0444\u0444\u0438\u0301\u043D\u043D\u043E\u0439 \u0441\u0432\u044F\u0301\u0437\u043D\u043E\u0441\u0442\u0438 \u2014 \u043E\u0434\u043D\u0430 \u0438\u0437 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043A\u0438\u0445 \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043A \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0435\u0439 \u0432 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0438\u0438. \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u043F\u043E\u043D\u044F\u0442\u0438\u044F \u043A\u0440\u0438\u0432\u0438\u0437\u043D\u044B, \u0438\u043C\u0435\u044E\u0449\u0435\u0433\u043E \u0441\u043C\u044B\u0441\u043B \u0434\u043B\u044F \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0438 \u0432 \u043F\u0440\u043E\u0438\u0437\u0432\u043E\u043B\u044C\u043D\u043E\u043C \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438 \u0438\u043B\u0438 \u0434\u0430\u0436\u0435 \u0432 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u043E \u0442\u0440\u0438\u0432\u0438\u0430\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438, \u043A\u0440\u0443\u0447\u0435\u043D\u0438\u0435 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u0435\u043D\u043E \u043B\u0438\u0448\u044C \u0434\u043B\u044F \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u0435\u0439 \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438 (\u0438\u043B\u0438, \u0447\u0443\u0442\u044C \u0431\u043E\u043B\u0435\u0435 \u043E\u0431\u0449\u043E, \u0432 \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u044F\u0445, \u0441\u043D\u0430\u0431\u0436\u0451\u043D\u043D\u044B\u0445 \u043E\u0442\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0435\u043C \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u0435 \u2014 \u0441\u043A\u0430\u0436\u0435\u043C, \u043A\u043E\u043D\u0442\u0430\u043A\u0442\u043D\u043E\u043C \u043F\u043E\u0434\u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438). \u0415\u0441\u043B\u0438 \u2014 \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u044C \u0432 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u0440\u0430\u0441\u0441\u043B\u043E\u0435\u043D\u0438\u0438, \u0435\u0451 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u0438\u044F \u043E\u043F\u0440\u0435\u0434\u0435\u043B\u044F\u0435\u0442\u0441\u044F \u043A\u0430\u043A ."@ru . . . . . . . . . . . . . . . "Torsion tensor"@en . "In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet\u2013Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet\u2013Serret frame about the tangent vector). In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames \"roll\" along a curve \"without twisting\". More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued 2-form on the manifold. If \u2207 is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields X and Y, by where [X,Y] is the Lie bracket of vector fields. Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein\u2013Cartan theory."@en . "\u0422\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F"@uk . . . "En g\u00E9om\u00E9trie diff\u00E9rentielle, la torsion constitue, avec la courbure, une mesure de la fa\u00E7on dont une \u00E9volue le long des courbes, et le tenseur de torsion en donne l'expression g\u00E9n\u00E9rale dans le cadre des vari\u00E9t\u00E9s, c'est-\u00E0-dire des \u00AB espaces courbes \u00BB de toutes dimensions. Le tenseur de torsion, qui est en r\u00E9alit\u00E9 un champ tensoriel, en est une version \u00E9tendue au cadre des vari\u00E9t\u00E9s dif\u00E9rentielles munies d'une connexion D. Il est d\u00E9fini par la formule"@fr . . . . "Der Torsionstensor ist ein mathematisches Objekt aus dem Bereich der Differentialgeometrie. Eingef\u00FChrt wurde dieses Tensorfeld von \u00C9lie Cartan in seinen Studien zur Geometrie und Gravitation."@de . . . . . . . . . "En geometria diferencial, la idea de torsi\u00F3 \u00E9s una manera de caracteritzar un gir o cargol d'un marc m\u00F2bil al voltant d'una corba. Per exemple, torsi\u00F3 d'una corba, que apareix a les f\u00F3rmules de Frenet\u2013Serret, quantifica el moviment de la corba al voltant del seu vector tangent a mesura que la corba avan\u00E7a. En la geometria de superf\u00EDcies, la torsi\u00F3 geod\u00E8sica descriu com una superf\u00EDcie gira al voltant d'una corba sobre la superf\u00EDcie."@ca . . . . . "En geometria diferencial, la idea de torsi\u00F3 \u00E9s una manera de caracteritzar un gir o cargol d'un marc m\u00F2bil al voltant d'una corba. Per exemple, torsi\u00F3 d'una corba, que apareix a les f\u00F3rmules de Frenet\u2013Serret, quantifica el moviment de la corba al voltant del seu vector tangent a mesura que la corba avan\u00E7a. En la geometria de superf\u00EDcies, la torsi\u00F3 geod\u00E8sica descriu com una superf\u00EDcie gira al voltant d'una corba sobre la superf\u00EDcie."@ca . . . . . "Der Torsionstensor ist ein mathematisches Objekt aus dem Bereich der Differentialgeometrie. Eingef\u00FChrt wurde dieses Tensorfeld von \u00C9lie Cartan in seinen Studien zur Geometrie und Gravitation."@de . . . . . "In geometria differenziale, la torsione \u00E8 un tensore che misura il grado di torsione degli spazi tangenti lungo una geodetica in una variet\u00E0 differenziabile dotata di connessione (e quindi di un trasporto parallelo che permette di spostare gli spazi tangenti lungo la curva). La nozione \u00E8 quindi ispirata a quella di torsione di una curva nello spazio usata nella geometria differenziale delle curve. In una variet\u00E0 riemanniana la torsione \u00E8 sempre nulla. Infatti la connessione di Levi-Civita usata in geometria riemanniana \u00E8 precisamente l'unica connessione senza torsione che preserva la metrica."@it . . . . "Tensor skr\u0119cenia"@pl . . "( \uACF5\uAC04 \uACE1\uC120\uC758 \uBE44\uD2C0\uB9BC\uC5D0 \uB300\uD574\uC11C\uB294 \uACE1\uC120 \uBE44\uD2C0\uB9BC \uBB38\uC11C\uB97C \uCC38\uACE0\uD558\uC2ED\uC2DC\uC624.) \uBBF8\uBD84\uAE30\uD558\uD559\uC5D0\uC11C \uBE44\uD2C0\uB9BC \uD150\uC11C(\uC601\uC5B4: torsion tensor)\uB294 \uC8FC\uB2E4\uBC1C\uC758 \uCF54\uC958 \uC811\uC18D\uC774 \uB808\uBE44\uCE58\uBE44\uD0C0 \uC811\uC18D\uC5D0\uC11C \uC5BC\uB9C8\uB098 \uBC97\uC5B4\uB098\uB294\uC9C0\uB97C \uCE21\uC815\uD558\uB294, (1,2)\uCC28 \uD150\uC11C\uC7A5\uC774\uB2E4."@ko . . . "\u5728\u5FAE\u5206\u51E0\u4F55\u4E2D\uFF0C\u626D\u7387\u6216\u7A31\u6320\u7387\u6B64\u4E00\u6982\u5FF5\u662F\u523B\u753B\u6CBF\u7740\u66F2\u7EBF\u79FB\u52A8\u7684\u6807\u67B6\u7684\u626D\u66F2\u6216\u7684\u65B9\u6CD5\u3002\u4F8B\u5982\u66F2\u7EBF\u7684\u6320\u7387\uFF0C\u51FA\u73B0\u5728\u5F17\u83B1\u7EB3\u516C\u5F0F\u4E2D\uFF0C\u91CF\u5316\u4E86\u4E00\u6761\u66F2\u7EBF\u53D8\u5316\u65F6\u5173\u4E8E\u5B83\u7684\u5207\u5411\u91CF\u7684\u626D\u66F2\u7A0B\u5EA6\uFF08\u66F4\u786E\u5207\u7684\u8BF4\u5F17\u83B1\u7EB3\u6807\u67B6\u5173\u4E8E\u5207\u5411\u91CF\u7684\u65CB\u8F6C\uFF09\u3002\u5728\u66F2\u9762\u7684\u51E0\u4F55\u4E2D\uFF0C\u201C\u6D4B\u5730\u6320\u7387\u201D\u63CF\u8FF0\u4E86\u66F2\u9762\u5173\u4E8E\u66F2\u9762\u4E0A\u4E00\u6761\u66F2\u7EBF\u7684\u626D\u66F2\u3002\u76F8\u4F34\u7684\u66F2\u7387\u6982\u5FF5\u5EA6\u91CF\u4E86\u6CBF\u7740\u66F2\u7EBF\u7684\u6D3B\u52A8\u6807\u67B6\u201C\u6CA1\u6709\u626D\u66F2\u7684\u8F6C\u52A8\u201D\u3002 \u66F4\u4E00\u822C\u5730\uFF0C\u5728\u88C5\u5907\u4E00\u4E2A\u4EFF\u5C04\u8054\u7EDC\uFF08\u5373\u5207\u4E1B\u7684\u4E00\u4E2A\u8054\u7EDC\uFF09\u7684\u5FAE\u5206\u6D41\u5F62\u4E0A\uFF0C\u6320\u7387\u4E0E\u66F2\u7387\u6784\u6210\u4E86\u8054\u7EDC\u7684\u4E24\u4E2A\u57FA\u672C\u4E0D\u53D8\u91CF\u3002\u5728\u8FD9\u79CD\u610F\u4E49\u4E0B\uFF0C\u6320\u7387\u7ED9\u51FA\u4E86\u5207\u7A7A\u95F4\u5173\u4E8E\u4E00\u6761\u66F2\u7EBF\u5E73\u884C\u79FB\u52A8\u600E\u6837\u626D\u66F2\u7684\u5185\u8574\u523B\u753B\uFF1B\u800C\u66F2\u7387\u63CF\u8FF0\u4E86\u5207\u7A7A\u95F4\u6CBF\u7740\u66F2\u7EBF\u600E\u6837\u65CB\u8F6C\u3002\u6320\u7387\u53EF\u5177\u4F53\u7684\u63CF\u8FF0\u4E3A\u4E00\u4E2A\u5F20\u91CF\uFF0C\u6216\u4E00\u4E2A\u5411\u91CF\u503C2-\u5F62\u5F0F\u3002\u5982\u679C \u2207 \u662F\u5FAE\u5206\u6D41\u5F62\u4E0A\u4E00\u4E2A\u8054\u7EDC\uFF0C\u90A3\u4E48\u6320\u7387\u5F20\u91CF\u7528\u5411\u91CF\u573A X \u4E0E Y \u8868\u793A\u5B9A\u4E49\u4E3A\uFF1A \u8FD9\u91CC [X,Y] \u662F\u5411\u91CF\u573A\u7684\u674E\u62EC\u53F7\u3002 \u6320\u7387\u5728\u6D4B\u5730\u7EBF\u51E0\u4F55\u7684\u7814\u7A76\u7279\u522B\u91CD\u8981\u3002\u7ED9\u5B9A\u4E00\u4E2A\u53C2\u6570\u5316\u6D4B\u5730\u7EBF\u7CFB\u7EDF\uFF0C\u6211\u4EEC\u4E00\u5B9A\u6307\u5B9A\u4E00\u65CF\u4EFF\u5C04\u8054\u7EDC\u5177\u6709\u8FD9\u4E9B\u6D4B\u5730\u7EBF\uFF0C\u4F46\u662F\u5177\u6709\u4E0D\u540C\u7684\u6320\u7387\u3002\u5177\u6709\u60DF\u4E00\u201C\u5438\u6536\u6320\u7387\u201D\u7684\u8054\u7EDC\uFF0C\u5C06\u5217\u7EF4-\u5947\u7EF4\u5854\u8054\u7EDC\u63A8\u5E7F\u5230\u5176\u4ED6\uFF0C\u4E5F\u8BB8\u6CA1\u6709\u5EA6\u91CF\u7684\u60C5\u5F62\uFF08\u6BD4\u5982\u82AC\u65AF\u52D2\u51E0\u4F55\uFF09\u3002\u5438\u6536\u6320\u7387\u5728G-\u7ED3\u6784\u4E0E\u7684\u7814\u7A76\u4E2D\u4E5F\u8D77\u7740\u91CD\u8981\u7684\u4F5C\u7528\u3002\u6320\u7387\u901A\u8FC7\u5173\u8054\u7684\u5728\u7814\u7A76\u6D4B\u5730\u7EBF\u975E\u53C2\u6570\u65CF\u4E5F\u5F88\u6709\u7528\u3002\u5728\u76F8\u5BF9\u8BBA\u4E2D\uFF0C\u8FD9\u79CD\u60F3\u6CD5\u4EE5\u7231\u56E0\u65AF\u5766-\u5609\u5F53\u7406\u8BBA\u7684\u5F62\u5F0F\u63D0\u4F9B\u4E86\u5DE5\u5177\u3002"@zh . "In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet\u2013Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves (or rather the rotation of the Frenet\u2013Serret frame about the tangent vector). In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames \"roll\" along a curve \"without twisting\"."@en . . . . . . . "22263"^^ . . . . . "4342484"^^ . . . . . . . . . "\u6369\u308C\u30C6\u30F3\u30BD\u30EB\uFF08\u306D\u3058\u308C\u30C6\u30F3\u30BD\u30EB\u3001torsion\uFF09\u3068\u306F\u3001\u5FAE\u5206\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u6982\u5FF5\u306E1\u3064\u3067\u3001\u66F2\u7DDA\u306B\u95A2\u3059\u308B\uFF08moving frame\uFF09\u306E\u30C4\u30A4\u30B9\u30C8\u3084\u6369\u308C\u65B9\u3092\u7279\u5FB4\u3065\u3051\u308B\u65B9\u6CD5\u306E\u3053\u3068\u3092\u3044\u3046\u3002"@ja . . . . . "Tenseur de torsion"@fr . . "( \uACF5\uAC04 \uACE1\uC120\uC758 \uBE44\uD2C0\uB9BC\uC5D0 \uB300\uD574\uC11C\uB294 \uACE1\uC120 \uBE44\uD2C0\uB9BC \uBB38\uC11C\uB97C \uCC38\uACE0\uD558\uC2ED\uC2DC\uC624.) \uBBF8\uBD84\uAE30\uD558\uD559\uC5D0\uC11C \uBE44\uD2C0\uB9BC \uD150\uC11C(\uC601\uC5B4: torsion tensor)\uB294 \uC8FC\uB2E4\uBC1C\uC758 \uCF54\uC958 \uC811\uC18D\uC774 \uB808\uBE44\uCE58\uBE44\uD0C0 \uC811\uC18D\uC5D0\uC11C \uC5BC\uB9C8\uB098 \uBC97\uC5B4\uB098\uB294\uC9C0\uB97C \uCE21\uC815\uD558\uB294, (1,2)\uCC28 \uD150\uC11C\uC7A5\uC774\uB2E4."@ko . . . "\u5728\u5FAE\u5206\u51E0\u4F55\u4E2D\uFF0C\u626D\u7387\u6216\u7A31\u6320\u7387\u6B64\u4E00\u6982\u5FF5\u662F\u523B\u753B\u6CBF\u7740\u66F2\u7EBF\u79FB\u52A8\u7684\u6807\u67B6\u7684\u626D\u66F2\u6216\u7684\u65B9\u6CD5\u3002\u4F8B\u5982\u66F2\u7EBF\u7684\u6320\u7387\uFF0C\u51FA\u73B0\u5728\u5F17\u83B1\u7EB3\u516C\u5F0F\u4E2D\uFF0C\u91CF\u5316\u4E86\u4E00\u6761\u66F2\u7EBF\u53D8\u5316\u65F6\u5173\u4E8E\u5B83\u7684\u5207\u5411\u91CF\u7684\u626D\u66F2\u7A0B\u5EA6\uFF08\u66F4\u786E\u5207\u7684\u8BF4\u5F17\u83B1\u7EB3\u6807\u67B6\u5173\u4E8E\u5207\u5411\u91CF\u7684\u65CB\u8F6C\uFF09\u3002\u5728\u66F2\u9762\u7684\u51E0\u4F55\u4E2D\uFF0C\u201C\u6D4B\u5730\u6320\u7387\u201D\u63CF\u8FF0\u4E86\u66F2\u9762\u5173\u4E8E\u66F2\u9762\u4E0A\u4E00\u6761\u66F2\u7EBF\u7684\u626D\u66F2\u3002\u76F8\u4F34\u7684\u66F2\u7387\u6982\u5FF5\u5EA6\u91CF\u4E86\u6CBF\u7740\u66F2\u7EBF\u7684\u6D3B\u52A8\u6807\u67B6\u201C\u6CA1\u6709\u626D\u66F2\u7684\u8F6C\u52A8\u201D\u3002 \u66F4\u4E00\u822C\u5730\uFF0C\u5728\u88C5\u5907\u4E00\u4E2A\u4EFF\u5C04\u8054\u7EDC\uFF08\u5373\u5207\u4E1B\u7684\u4E00\u4E2A\u8054\u7EDC\uFF09\u7684\u5FAE\u5206\u6D41\u5F62\u4E0A\uFF0C\u6320\u7387\u4E0E\u66F2\u7387\u6784\u6210\u4E86\u8054\u7EDC\u7684\u4E24\u4E2A\u57FA\u672C\u4E0D\u53D8\u91CF\u3002\u5728\u8FD9\u79CD\u610F\u4E49\u4E0B\uFF0C\u6320\u7387\u7ED9\u51FA\u4E86\u5207\u7A7A\u95F4\u5173\u4E8E\u4E00\u6761\u66F2\u7EBF\u5E73\u884C\u79FB\u52A8\u600E\u6837\u626D\u66F2\u7684\u5185\u8574\u523B\u753B\uFF1B\u800C\u66F2\u7387\u63CF\u8FF0\u4E86\u5207\u7A7A\u95F4\u6CBF\u7740\u66F2\u7EBF\u600E\u6837\u65CB\u8F6C\u3002\u6320\u7387\u53EF\u5177\u4F53\u7684\u63CF\u8FF0\u4E3A\u4E00\u4E2A\u5F20\u91CF\uFF0C\u6216\u4E00\u4E2A\u5411\u91CF\u503C2-\u5F62\u5F0F\u3002\u5982\u679C \u2207 \u662F\u5FAE\u5206\u6D41\u5F62\u4E0A\u4E00\u4E2A\u8054\u7EDC\uFF0C\u90A3\u4E48\u6320\u7387\u5F20\u91CF\u7528\u5411\u91CF\u573A X \u4E0E Y \u8868\u793A\u5B9A\u4E49\u4E3A\uFF1A \u8FD9\u91CC [X,Y] \u662F\u5411\u91CF\u573A\u7684\u674E\u62EC\u53F7\u3002 \u6320\u7387\u5728\u6D4B\u5730\u7EBF\u51E0\u4F55\u7684\u7814\u7A76\u7279\u522B\u91CD\u8981\u3002\u7ED9\u5B9A\u4E00\u4E2A\u53C2\u6570\u5316\u6D4B\u5730\u7EBF\u7CFB\u7EDF\uFF0C\u6211\u4EEC\u4E00\u5B9A\u6307\u5B9A\u4E00\u65CF\u4EFF\u5C04\u8054\u7EDC\u5177\u6709\u8FD9\u4E9B\u6D4B\u5730\u7EBF\uFF0C\u4F46\u662F\u5177\u6709\u4E0D\u540C\u7684\u6320\u7387\u3002\u5177\u6709\u60DF\u4E00\u201C\u5438\u6536\u6320\u7387\u201D\u7684\u8054\u7EDC\uFF0C\u5C06\u5217\u7EF4-\u5947\u7EF4\u5854\u8054\u7EDC\u63A8\u5E7F\u5230\u5176\u4ED6\uFF0C\u4E5F\u8BB8\u6CA1\u6709\u5EA6\u91CF\u7684\u60C5\u5F62\uFF08\u6BD4\u5982\u82AC\u65AF\u52D2\u51E0\u4F55\uFF09\u3002\u5438\u6536\u6320\u7387\u5728G-\u7ED3\u6784\u4E0E\u7684\u7814\u7A76\u4E2D\u4E5F\u8D77\u7740\u91CD\u8981\u7684\u4F5C\u7528\u3002\u6320\u7387\u901A\u8FC7\u5173\u8054\u7684\u5728\u7814\u7A76\u6D4B\u5730\u7EBF\u975E\u53C2\u6570\u65CF\u4E5F\u5F88\u6709\u7528\u3002\u5728\u76F8\u5BF9\u8BBA\u4E2D\uFF0C\u8FD9\u79CD\u60F3\u6CD5\u4EE5\u7231\u56E0\u65AF\u5766-\u5609\u5F53\u7406\u8BBA\u7684\u5F62\u5F0F\u63D0\u4F9B\u4E86\u5DE5\u5177\u3002"@zh . "Tensor skr\u0119cenia \u2013 obiekt opisuj\u0105cy przekr\u0119cenie ramki poruszaj\u0105cej si\u0119 wzd\u0142u\u017C krzywej."@pl . . . . "En g\u00E9om\u00E9trie diff\u00E9rentielle, la torsion constitue, avec la courbure, une mesure de la fa\u00E7on dont une \u00E9volue le long des courbes, et le tenseur de torsion en donne l'expression g\u00E9n\u00E9rale dans le cadre des vari\u00E9t\u00E9s, c'est-\u00E0-dire des \u00AB espaces courbes \u00BB de toutes dimensions. La torsion se manifeste en g\u00E9om\u00E9trie diff\u00E9rentielle classique comme une valeur num\u00E9rique associ\u00E9e \u00E0 chaque point d'une courbe de l'espace euclidien. En termes imag\u00E9s, si la courbure quantifie le caract\u00E8re plus ou moins accentu\u00E9 des virages pris par une courbe en comparant celle-ci \u00E0 un cercle dit \u00AB osculateur \u00BB, la torsion marque la tendance \u00E0 sortir du plan de ce cercle, en vrillant soit dans le m\u00EAme sens qu'une vis, soit dans le sens inverse. Le tenseur de torsion, qui est en r\u00E9alit\u00E9 un champ tensoriel, en est une version \u00E9tendue au cadre des vari\u00E9t\u00E9s dif\u00E9rentielles munies d'une connexion D. Il est d\u00E9fini par la formule o\u00F9 [X,Y] est le crochet de Lie des champs de vecteurs X et Y. La connexion est dite sans torsion quand ce tenseur est constamment nul. C'est le cas par exemple de la connexion de Levi-Civita en g\u00E9om\u00E9trie riemannienne."@fr . . "\uBE44\uD2C0\uB9BC \uD150\uC11C"@ko . . . "1117700944"^^ . . . . . "\u6369\u308C\u30C6\u30F3\u30BD\u30EB"@ja . "In geometria differenziale, la torsione \u00E8 un tensore che misura il grado di torsione degli spazi tangenti lungo una geodetica in una variet\u00E0 differenziabile dotata di connessione (e quindi di un trasporto parallelo che permette di spostare gli spazi tangenti lungo la curva). La nozione \u00E8 quindi ispirata a quella di torsione di una curva nello spazio usata nella geometria differenziale delle curve. In una variet\u00E0 riemanniana la torsione \u00E8 sempre nulla. Infatti la connessione di Levi-Civita usata in geometria riemanniana \u00E8 precisamente l'unica connessione senza torsione che preserva la metrica."@it . . . "\u0422\u0435\u043D\u0437\u043E\u0440\u043E\u043C \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0432 \u0434\u0438\u0444\u0435\u0440\u0435\u043D\u0446\u0456\u0430\u043B\u044C\u043D\u0456\u0439 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0432\u0435\u043A\u0442\u043E\u0440\u043E\u0437\u043D\u0430\u0447\u043D\u0438\u0439 \u0442\u0435\u043D\u0437\u043E\u0440, \u0449\u043E \u043A\u043E\u0436\u043D\u0456\u0439 \u043F\u0430\u0440\u0456 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0438\u0445 \u043F\u043E\u043B\u0456\u0432 \u043A\u043B\u0430\u0441\u0443 , \u0437\u0430\u0434\u0430\u043D\u0438\u0445 \u043D\u0430 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u0433\u043B\u0430\u0434\u043A\u043E\u043C\u0443 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0456 \u0437 \u0432\u0432\u0435\u0434\u0435\u043D\u043E\u044E \u0430\u0444\u0456\u043D\u043D\u043E\u044E \u0437\u0432'\u044F\u0437\u043D\u0456\u0441\u0442\u044E \u043F\u0440\u0438\u0441\u0432\u043E\u044E\u0454 \u0432\u0435\u043A\u0442\u043E\u0440\u043D\u0435 \u043F\u043E\u043B\u0435 \u043A\u043B\u0430\u0441\u0443 . \u0420\u0430\u0437\u043E\u043C \u0456\u0437 \u0442\u0435\u043D\u0437\u043E\u0440\u043E\u043C \u043A\u0440\u0438\u0432\u0438\u043D\u0438 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0454 \u043E\u0434\u043D\u0438\u043C \u0437 \u0433\u043E\u043B\u043E\u0432\u043D\u0438\u0445 \u0456\u043D\u0432\u0430\u0440\u0456\u0430\u043D\u0442\u0456\u0432 \u0430\u0444\u0456\u043D\u043D\u043E\u0457 \u0437\u0432'\u044F\u0437\u043D\u043E\u0441\u0442\u0456. \u0417\u043E\u043A\u0440\u0435\u043C\u0430 \u0442\u0435\u043D\u0437\u043E\u0440 \u043A\u0440\u0443\u0447\u0435\u043D\u043D\u044F \u0432\u0456\u0434\u0456\u0433\u0440\u0430\u0454 \u0434\u0443\u0436\u0435 \u0432\u0430\u0436\u043B\u0438\u0432\u0443 \u0440\u043E\u043B\u044C \u0443 \u0432\u0438\u0432\u0447\u0435\u043D\u043D\u0456 \u0433\u0435\u043E\u043C\u0435\u0442\u0440\u0456\u0457 \u0433\u0435\u043E\u0434\u0435\u0437\u0438\u0447\u043D\u0438\u0445 \u043B\u0456\u043D\u0456\u0439 \u043D\u0430 \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0430\u0445."@uk . "\u6369\u308C\u30C6\u30F3\u30BD\u30EB\uFF08\u306D\u3058\u308C\u30C6\u30F3\u30BD\u30EB\u3001torsion\uFF09\u3068\u306F\u3001\u5FAE\u5206\u5E7E\u4F55\u5B66\u306B\u304A\u3051\u308B\u6982\u5FF5\u306E1\u3064\u3067\u3001\u66F2\u7DDA\u306B\u95A2\u3059\u308B\uFF08moving frame\uFF09\u306E\u30C4\u30A4\u30B9\u30C8\u3084\u6369\u308C\u65B9\u3092\u7279\u5FB4\u3065\u3051\u308B\u65B9\u6CD5\u306E\u3053\u3068\u3092\u3044\u3046\u3002"@ja . . . . . . . . "Torsi\u00F3 d'una connexi\u00F3"@ca . . . . . . . . . . . . . "Torsione (geometria differenziale)"@it .