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| {{unreferenced|date=December 2012}}
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| In [[mathematics]], the '''musical isomorphism''' (or '''canonical isomorphism''') is an [[isomorphism]] between the [[tangent bundle]] ''TM'' and the [[cotangent bundle]] ''T<sup>*</sup>M'' of a [[Riemannian manifold]] given by its [[Riemannian metric|metric]]. There are similar isomorphisms on [[symplectic manifold]]s.
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| It is also known as [[raising and lowering indices]].
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| ==Discussion==
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| Let <math> (M,g) </math> be a [[Riemannian manifold]]. Suppose <math> \{\partial_i\} </math> is a [[local frame]] for the tangent bundle <math> TM </math> with dual coframe <math> \{dx^i\} </math>. Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as <math> g=g_{ij}\,dx^i \otimes dx^j </math> (where we employ the [[Einstein summation convention]]). Given a vector field <math> X=X^i \partial_i </math> we define its flat by
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| :<math> X^\flat := g_{ij} X^i \, dx^j=X_j \, dx^j.</math>
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| This is referred to as 'lowering an index'. Using the traditional diamond bracket notation for inner product defined by ''g'', we obtain the somewhat more transparent relation
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| :<math>X^\flat (Y) = \langle X, Y \rangle</math>
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| for all vectors ''X'' and ''Y''.
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| Alternatively, given a covector field <math> \omega=\omega_i \, dx^i </math> we define its sharp by
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| :<math>\omega^\sharp :=g^{ij} \omega_i \partial_j = \omega^j \partial_j</math>
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| where <math> g^{ij} </math> are the elements of the inverse matrix to <math> g_{ij} </math>. Taking the sharp of a covector field is referred to as 'raising an index'. In inner product notation, this reads
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| :<math>\langle \omega^\sharp, Y\rangle = \omega(Y),</math>
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| for <math> \omega </math> an arbitrary covector and <math> Y </math> an arbitrary vector.
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| Through this construction we have two inverse isomorphisms <math> \flat:TM \to T^*M </math> and <math> \sharp:T^*M \to TM </math>. These are isomorphisms of [[vector bundles]] and hence we have, for each <math> p \in M </math>, inverse vector space isomorphisms between <math> T_pM </math> and <math> T^*_pM </math>.
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| The musical isomorphisms may also be extended to the bundles <math> \bigotimes ^k TM </math> and <math> \bigotimes ^k T^*M </math>. It must be stated which index is to be raised or lowered. For instance, consider the (2,0) tensor field <math> X=X_{ij} \, dx^i \otimes dx^j </math>. Raising the second index, we get the (1,1) tensor field <math>X^\sharp = g^{jk}X_{ij} \, dx^i \otimes \partial _k. </math>
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| ==Trace of a tensor through a metric==
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| Given a (2,0) tensor field <math> X=X_{ij} \, dx^i \otimes dx^j </math> we define the trace of <math> X </math> through the metric <math> g </math> by
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| :<math> \operatorname{tr}_g(X):=\operatorname{tr}(X^\sharp)=\operatorname{tr}(g^{jk}X_{ij}) = g^{ji}X_{ij} = g^{ij}X_{ij}. </math>
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| Observe that the definition of trace is independent of the choice of index we raise since the metric tensor is symmetric.
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| == See also ==
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| *[[Duality (mathematics)]]
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| *[[Raising and lowering indices]]
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| *[[Dual space#Bilinear products and dual spaces|Bilinear products and dual spaces]]
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| *[[Vector bundle]]
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| *[[Flat (music)]] and [[Sharp (music)]] about the signs {{music|flat}} and {{music|sharp}}
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| [[Category:Riemannian geometry]]
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| [[Category:Symplectic geometry]]
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