About: Musical isomorphism

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In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols (flat) and (sharp). In covariant and contravariant notation, it is also known as raising and lowering indices.

Attributes	Values
rdfs:label	Isomorfisme musical (ca) Isomorfismo musical (es) Isomorfismo musicale (it) Isomorphisme musical (fr) 음악 동형 (ko) Musical isomorphism (en) Izomorfizm muzyczny (pl) Isomorfismo musical (pt) 音乐同构 (zh)
rdfs:comment	A matemàtiques, el isomorfisme musical és un isomorfisme entre el fibrat tangent i el fibrat cotangent d'una varietat riemanniana, que ve induït per la seva mètrica. (ca) En matemáticas, el isomorfismo musical es un isomorfismo entre el fibrado tangente y el fibrado cotangente de una variedad riemanniana, que viene inducido por su métrica. (es) In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols (flat) and (sharp). In covariant and contravariant notation, it is also known as raising and lowering indices. (en) L'isomorfismo musicale è un isomorfismo tra uno spazio vettoriale reale e il suo spazio duale che è indotto da una forma bilineare simmetrica non degenere. Nell'ambito della geometria riemanniana, si tratta di un isomorfismo tra il fibrato tangente di una varietà riemanniana e il suo fibrato cotangente che è indotto dalla metrica (it) En mathématiques, plus précisément en géométrie différentielle, l'isomorphisme musical (ou isomorphisme canonique ) est un isomorphisme entre le fibré tangent et le fibré cotangent d'une variété pseudo-riemannienne induite par son tenseur métrique. Il existe des isomorphismes similaires sur les variétés symplectiques. Le terme musical fait référence à l'utilisation des symboles (plat) et (tranchant). En notation covariante et contravariante, il est également connu sous le nom d'indice d'élévation et d'abaissement. (fr) 미분기하학에서 음악 동형(音樂同型, 영어: musical isomorphism)은 매끄러운 다양체의 접다발과 공변접다발 사이의 동형 사상이다. 준 리만 다양체나 심플렉틱 다양체의 경우 표준적인 음악 동형이 존재한다. (ko) Izomorfizm muzyczny – izomorfizm między wiązką styczną a wiązką kostyczną rozmaitości riemannowskiej określony za pomocą jej metryki. Znany jest również jako podnoszenie i opuszczanie wskaźników. (pl) Em matemática, o isomorfismo musical (ou isomorfismo canônico) é um isomorfismo entre o fibrado tangente TM e o fibrado cotangente T∗M e uma variedade de Riemann dada por sua métrica. Existem isomorfismos similares em variedades simpléticas. O termo musical refere-se ao uso dos símbolos e . (pt) 在数学中，特別是黎曼幾何跟微分流形的理論裡，音乐同构（Musical isomorphism 或典范同构 canonical isomorphism）是指（伪）黎曼流形 M 的切丛 TM 与余切丛之间的同构，这个同构由黎曼度量给出。不過一般地，只要流形的切丛上有一个处处非退化的双线性形式（比如辛流形上的辛形式）便可定义这样的同构。在帶有內積（或更一般的，非退化的雙線性形式）的有限維向量空間，這些同構自然給出了和其對偶空間之間的同構，在這種情況一般稱這些映射為典範同構（canonical isomorphosm）。這些運算在流形上的張量場理論裡也称为指标的上升和下降。 (zh)
dcterms:subject	Symplectic geometry Differential geometry Riemannian manifolds Riemannian geometry
Wikipage page ID	3762980 (xsd:integer)
Wikipage revision ID	1118329642 (xsd:integer)
Link from a Wikipage to another Wikipage	Nondegenerate bilinear form Totally antisymmetric tensor Bilinear form Symplectic geometry Vector field Differential geometry Mathematics Tangent bundle Riemannian manifolds Hodge dual Linear algebra Symplectic manifold Dual basis Dual space Duality (mathematics) Exterior algebra Oxford University Press Frame bundle Isomorphism Einstein summation convention Moving frame Tensor field Pseudo-Riemannian manifold Inverse function Cotangent bundle Tensor Riemannian geometry Abuse of notation Coframe Symmetric bilinear form Differential geometry Manifold Bundle metric Inner product space Metric tensor Raising and lowering indices Sharp (music) Vector bundle Flat (music) Inverse matrix Pseudo Riemannian metric Vector bundles Finite-dimensional vector space Covariant and contravariant indices Covector
sameAs	Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism Musical isomorphism
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has abstract	A matemàtiques, el isomorfisme musical és un isomorfisme entre el fibrat tangent i el fibrat cotangent d'una varietat riemanniana, que ve induït per la seva mètrica. (ca) En matemáticas, el isomorfismo musical es un isomorfismo entre el fibrado tangente y el fibrado cotangente de una variedad riemanniana, que viene inducido por su métrica. (es) In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols (flat) and (sharp). In covariant and contravariant notation, it is also known as raising and lowering indices. (en)

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