@prefix dbo: . @prefix dbr: . dbr:Riemannian_geometry dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Normal dbo:wikiPageWikiLink dbr:Normal_coordinates ; dbo:wikiPageDisambiguates dbr:Normal_coordinates . dbo:wikiPageWikiLink dbr:Normal_coordinates . @prefix foaf: . @prefix wikipedia-en: . wikipedia-en:Normal_coordinates foaf:primaryTopic dbr:Normal_coordinates . dbr:Synchronous_frame dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Levi-Civita_parallelogramoid dbo:wikiPageWikiLink dbr:Normal_coordinates . dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Scalar_curvature dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Local_reference_frame dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Christoffel_symbols dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Gerhard_Huisken dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Ricci_curvature dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Geodesic_normal_coordinates dbo:wikiPageWikiLink dbr:Normal_coordinates ; dbo:wikiPageRedirects dbr:Normal_coordinates . @prefix rdf: . @prefix yago: . dbr:Normal_coordinates rdf:type yago:CoordinateSystem105728024 , yago:WikicatCoordinateSystemsInDifferentialGeometry , yago:Cognition100023271 , yago:PsychologicalFeature100023100 , yago:Arrangement105726596 , yago:Structure105726345 , yago:Abstraction100002137 , yago:WikicatCoordinateSystems . @prefix rdfs: . dbr:Normal_coordinates rdfs:label "Coordonn\u00E9es normales"@fr , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0435 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442\u044B"@ru , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442"@uk , "Riemannsche Normalkoordinaten"@de , "Normal coordinates"@en , "\u7B80\u6B63\u5750\u6807"@zh ; rdfs:comment "Riemannsche Normalkoordinaten (nach Bernhard Riemann; auch Normalkoordinaten oder Exponentialkoordinaten) bilden ein besonderes Koordinatensystem, welches in der Differentialgeometrie betrachtet wird. Hier wird der Tangentialraum an als lokale Karte der Mannigfaltigkeit in einer Umgebung von verwendet. Solche Koordinaten sind einfach zu handhaben und finden daher auch Anwendung in der allgemeinen Relativit\u00E4tstheorie."@de , "\u7B80\u6B63\u5750\u6807\u53C8\u53EB\u505A\u6B63\u5219\u5750\u6807\uFF0C\u662F\u7528\u6765\u63CF\u8FF0\u548C\u8BA1\u7B97\u5206\u5B50\u5185\u90E8\u8FD0\u52A8\u7684\u4E00\u4E2A\u5750\u6807\u4F53\u7CFB\u3002"@zh , "In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish."@en , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0435 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442\u044B \u2014 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0430\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0432 \u043E\u043A\u0440\u0435\u0441\u0442\u043D\u043E\u0441\u0442\u0438 \u0442\u043E\u0447\u043A\u0438 \u0440\u0438\u043C\u0430\u043D\u043E\u0432\u0430 \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F (\u0438\u043B\u0438, \u0431\u043E\u043B\u0435\u0435 \u043E\u0431\u0449\u043E, \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F \u0441 \u0430\u0444\u0444\u0438\u043D\u043D\u043E\u0439 \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u044C\u044E) \u043F\u043E\u043B\u0443\u0447\u0435\u043D\u043D\u0430\u044F \u0438\u0437 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u043D\u0430 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435 \u0432 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043F\u0440\u0438\u043C\u0435\u043D\u0435\u043D\u0438\u0435\u043C \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u043E\u0442\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u044F. \u0412 \u0431\u0430\u0437\u043E\u0432\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0441\u0438\u043C\u0432\u043E\u043B\u044B \u041A\u0440\u0438\u0441\u0442\u043E\u0444\u0444\u0435\u043B\u044F \u043E\u0431\u043D\u0443\u043B\u044F\u044E\u0442\u0441\u044F;\u044D\u0442\u043E \u0447\u0430\u0441\u0442\u043E \u0443\u043F\u0440\u043E\u0449\u0430\u0435\u0442 \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F."@ru , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u2014 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0432 \u043E\u043A\u043E\u043B\u0456 \u0442\u043E\u0447\u043A\u0438 \u0440\u0456\u043C\u0430\u043D\u043E\u0432\u043E\u0433\u043E \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 (\u0430\u0431\u043E, \u0431\u0456\u043B\u044C\u0448 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u043E, \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 \u0437 \u0430\u0444\u0456\u043D\u043D\u043E\u044E \u0437\u0432'\u044F\u0437\u043D\u0456\u0441\u0442\u044E), \u0449\u043E \u043E\u0434\u0435\u0440\u0436\u0443\u0454\u0442\u044C\u0441\u044F \u0456\u0437 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u043D\u0430 \u0434\u043E\u0442\u0438\u0447\u043D\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456 \u0432 \u0434\u0430\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u0437\u0430\u0441\u0442\u043E\u0441\u0443\u0432\u0430\u043D\u043D\u044F\u043C \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0456\u0439\u043D\u043E\u0433\u043E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F."@uk , "En g\u00E9om\u00E9trie diff\u00E9rentielle, les coordonn\u00E9es normales d'un point p dans une vari\u00E9t\u00E9 diff\u00E9rentielle munie d'une connexion affine sym\u00E9trique sont un syst\u00E8me de coordonn\u00E9es locales au voisinage de p obtenu par une application exponentielle \u00E0 l'espace tangent \u00E0 p. Dans un syst\u00E8me de coordonn\u00E9es normales, les symboles de Christoffel de la connexion disparaissent au point p. En coordonn\u00E9es normales, associ\u00E9es \u00E0 une connexion de Levi-Civita d'une vari\u00E9t\u00E9 riemannienne, on peut en outre faire en sorte que le tenseur m\u00E9trique soit le symbole de Kronecker au point p, et que les d\u00E9riv\u00E9es partielles premi\u00E8res de la m\u00E9trique \u00E0 p disparaissent."@fr . @prefix dcterms: . @prefix dbc: . dbr:Normal_coordinates dcterms:subject dbc:Riemannian_geometry , dbc:Coordinate_systems_in_differential_geometry ; dbo:abstract "Riemannsche Normalkoordinaten (nach Bernhard Riemann; auch Normalkoordinaten oder Exponentialkoordinaten) bilden ein besonderes Koordinatensystem, welches in der Differentialgeometrie betrachtet wird. Hier wird der Tangentialraum an als lokale Karte der Mannigfaltigkeit in einer Umgebung von verwendet. Solche Koordinaten sind einfach zu handhaben und finden daher auch Anwendung in der allgemeinen Relativit\u00E4tstheorie."@de , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u2014 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0430 \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0432 \u043E\u043A\u043E\u043B\u0456 \u0442\u043E\u0447\u043A\u0438 \u0440\u0456\u043C\u0430\u043D\u043E\u0432\u043E\u0433\u043E \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 (\u0430\u0431\u043E, \u0431\u0456\u043B\u044C\u0448 \u0437\u0430\u0433\u0430\u043B\u044C\u043D\u043E, \u043C\u043D\u043E\u0433\u043E\u0432\u0438\u0434\u0443 \u0437 \u0430\u0444\u0456\u043D\u043D\u043E\u044E \u0437\u0432'\u044F\u0437\u043D\u0456\u0441\u0442\u044E), \u0449\u043E \u043E\u0434\u0435\u0440\u0436\u0443\u0454\u0442\u044C\u0441\u044F \u0456\u0437 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u043D\u0430 \u0434\u043E\u0442\u0438\u0447\u043D\u043E\u043C\u0443 \u043F\u0440\u043E\u0441\u0442\u043E\u0440\u0456 \u0432 \u0434\u0430\u043D\u0456\u0439 \u0442\u043E\u0447\u0446\u0456 \u0437\u0430\u0441\u0442\u043E\u0441\u0443\u0432\u0430\u043D\u043D\u044F\u043C \u0435\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0456\u0439\u043D\u043E\u0433\u043E \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F."@uk , "\u041D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0435 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442\u044B \u2014 \u043B\u043E\u043A\u0430\u043B\u044C\u043D\u0430\u044F \u0441\u0438\u0441\u0442\u0435\u043C\u0430 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0432 \u043E\u043A\u0440\u0435\u0441\u0442\u043D\u043E\u0441\u0442\u0438 \u0442\u043E\u0447\u043A\u0438 \u0440\u0438\u043C\u0430\u043D\u043E\u0432\u0430 \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F (\u0438\u043B\u0438, \u0431\u043E\u043B\u0435\u0435 \u043E\u0431\u0449\u043E, \u043C\u043D\u043E\u0433\u043E\u043E\u0431\u0440\u0430\u0437\u0438\u044F \u0441 \u0430\u0444\u0444\u0438\u043D\u043D\u043E\u0439 \u0441\u0432\u044F\u0437\u043D\u043E\u0441\u0442\u044C\u044E) \u043F\u043E\u043B\u0443\u0447\u0435\u043D\u043D\u0430\u044F \u0438\u0437 \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u043D\u0430 \u043A\u0430\u0441\u0430\u0442\u0435\u043B\u044C\u043D\u043E\u043C \u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0435 \u0432 \u0434\u0430\u043D\u043D\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043F\u0440\u0438\u043C\u0435\u043D\u0435\u043D\u0438\u0435\u043C \u044D\u043A\u0441\u043F\u043E\u043D\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0433\u043E \u043E\u0442\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u044F. \u0412 \u0431\u0430\u0437\u043E\u0432\u043E\u0439 \u0442\u043E\u0447\u043A\u0435 \u043D\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u043A\u043E\u043E\u0440\u0434\u0438\u043D\u0430\u0442 \u0441\u0438\u043C\u0432\u043E\u043B\u044B \u041A\u0440\u0438\u0441\u0442\u043E\u0444\u0444\u0435\u043B\u044F \u043E\u0431\u043D\u0443\u043B\u044F\u044E\u0442\u0441\u044F;\u044D\u0442\u043E \u0447\u0430\u0441\u0442\u043E \u0443\u043F\u0440\u043E\u0449\u0430\u0435\u0442 \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F."@ru , "En g\u00E9om\u00E9trie diff\u00E9rentielle, les coordonn\u00E9es normales d'un point p dans une vari\u00E9t\u00E9 diff\u00E9rentielle munie d'une connexion affine sym\u00E9trique sont un syst\u00E8me de coordonn\u00E9es locales au voisinage de p obtenu par une application exponentielle \u00E0 l'espace tangent \u00E0 p. Dans un syst\u00E8me de coordonn\u00E9es normales, les symboles de Christoffel de la connexion disparaissent au point p. En coordonn\u00E9es normales, associ\u00E9es \u00E0 une connexion de Levi-Civita d'une vari\u00E9t\u00E9 riemannienne, on peut en outre faire en sorte que le tenseur m\u00E9trique soit le symbole de Kronecker au point p, et que les d\u00E9riv\u00E9es partielles premi\u00E8res de la m\u00E9trique \u00E0 p disparaissent. Un r\u00E9sultat classique de la g\u00E9om\u00E9trie diff\u00E9rentielle \u00E9tablit que les coordonn\u00E9es normales \u00E0 un point existent toujours sur une vari\u00E9t\u00E9 munie d'une connexion affine sym\u00E9trique. Dans ces coordonn\u00E9es, la d\u00E9riv\u00E9e covariante se r\u00E9duit \u00E0 une d\u00E9riv\u00E9e partielle (autour de p seulement), et les g\u00E9od\u00E9siques passant par p sont localement des fonctions lin\u00E9aires de t (param\u00E8tre affine). Cette id\u00E9e a \u00E9t\u00E9 mise en \u0153uvre par Albert Einstein dans la th\u00E9orie de la relativit\u00E9 g\u00E9n\u00E9rale : le principe d'\u00E9quivalence utilise les coordonn\u00E9es normales via les r\u00E9f\u00E9rentiels inertiels. Les coordonn\u00E9es normales existent toujours pour une connexion de Levi-Civita d'une vari\u00E9t\u00E9 riemannienne ou pseudo-riemannienne. \u00C0 l'inverse, il n'y a pas moyen de d\u00E9finir des coordonn\u00E9es normales pour un espace de Finsler."@fr , "\u7B80\u6B63\u5750\u6807\u53C8\u53EB\u505A\u6B63\u5219\u5750\u6807\uFF0C\u662F\u7528\u6765\u63CF\u8FF0\u548C\u8BA1\u7B97\u5206\u5B50\u5185\u90E8\u8FD0\u52A8\u7684\u4E00\u4E2A\u5750\u6807\u4F53\u7CFB\u3002"@zh , "In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish. A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable."@en ; dbo:wikiPageWikiLink dbr:Christoffel_symbols , dbc:Riemannian_geometry , dbr:Geodesic , dbr:Local_reference_frame , dbr:Kronecker_delta , , dbc:Coordinate_systems_in_differential_geometry , dbr:N_sphere , dbr:Orthonormal_basis , dbr:Affine_connection , , , dbr:Wiley_Interscience , dbr:Mathematische_Annalen , dbr:Foundations_of_Differential_Geometry , dbr:General_theory_of_relativity , dbr:Albert_Einstein , dbr:Differential_geometry , dbr:Spherical_coordinates , dbr:Block_diagonal , dbr:Diffeomorphism , dbr:Basis_of_a_vector_space , dbr:Pseudo-Riemannian , dbr:Differentiable_manifold , dbr:Riemannian_manifold , dbr:Tangent_space , , dbr:Partial_derivative , dbr:Finsler_manifold , dbr:Fermi_coordinates , dbr:Levi-Civita_connection , dbr:Metric_tensor , dbr:Torsion_tensor , dbr:Gradient , dbr:Local_coordinate_system , dbr:Equivalence_principle , dbr:Inertial_frame , . @prefix dbp: . @prefix dbt: . dbr:Normal_coordinates dbp:wikiPageUsesTemplate dbt:Citation , dbt:Short_description , dbt:Harv ; dbo:wikiPageRevisionID 1101353666 . @prefix xsd: . dbr:Normal_coordinates dbo:wikiPageLength "7720"^^xsd:nonNegativeInteger ; dbo:wikiPageID 7679762 . @prefix owl: . dbr:Normal_coordinates owl:sameAs , . @prefix wikidata: . dbr:Normal_coordinates owl:sameAs wikidata:Q2152232 , , . @prefix yago-res: . dbr:Normal_coordinates owl:sameAs yago-res:Normal_coordinates , , dbr:Normal_coordinates . @prefix dbpedia-de: . dbr:Normal_coordinates owl:sameAs dbpedia-de:Riemannsche_Normalkoordinaten , . @prefix prov: . dbr:Normal_coordinates prov:wasDerivedFrom ; foaf:isPrimaryTopicOf wikipedia-en:Normal_coordinates . dbo:wikiPageWikiLink dbr:Normal_coordinates . dbr:Normal_coordinate dbo:wikiPageWikiLink dbr:Normal_coordinates ; dbo:wikiPageRedirects dbr:Normal_coordinates . dbr:Normal_neighborhood dbo:wikiPageWikiLink dbr:Normal_coordinates ; dbo:wikiPageRedirects dbr:Normal_coordinates .
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