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Inmathematics—more specifically, indifferential geometry—themusical isomorphism (orcanonical isomorphism) is anisomorphism between thetangent bundle${\displaystyle \mathrm{T}M}$ and thecotangent bundle${\displaystyle {\mathrm{T}}^{\ast }M}$ of aRiemannian orpseudo-Riemannian manifold induced by itsmetric tensor. There are similar isomorphisms onsymplectic manifolds. These isomorphisms are global versions of the canonical isomorphism between aninner product space and itsdual. The termmusical refers to the use of themusical notation symbols${\displaystyle \mathrm{♭}}$ (flat) and${\displaystyle \mathrm{♯}}$ (sharp).^[1][2]

In the notation ofRicci calculus andmathematical physics, the idea is expressed as theraising and lowering of indices. Raising and lowering indices are a form ofindex manipulation in tensor expressions.

In certain specialized applications, such as onPoisson manifolds, the relationship may fail to be an isomorphism atsingular points, and so, for these cases, is technically only a homomorphism.

Inlinear algebra, afinite-dimensional vector space is isomorphic to itsdual space (the space oflinear functionals mapping the vector space to its base field), but not canonically isomorphic to it. This is to say that given a fixed basis for the vector space, there is a natural way to go back and forth between vectors and linear functionals: vectors are represented in the basis bycolumn vectors, and linear functionals are represented in the basis byrow vectors, and one can go back and forth bytransposing. However, without a fixed basis, there is no way to go back and forth between vectors and linear functionals. This is what is meant by that there is no canonical isomorphism.

On the other hand, a finite-dimensional vector space${\displaystyle V}$ endowed with a non-degeneratebilinear form${\displaystyle ⟨\cdot ,\cdot ⟩}$ is canonically isomorphic to its dual. The canonical isomorphism${\displaystyle V\to V^{\ast }}$ is given by

${\displaystyle v\mapsto ⟨v,\cdot ⟩}$.

The non-degeneracy of${\displaystyle ⟨\cdot ,\cdot ⟩}$ means exactly that the above map is an isomorphism. An example is where${\displaystyle V={\mathbb{R}}^n}$ and${\displaystyle ⟨\cdot ,\cdot ⟩}$ is thedot product.

In a basis${\displaystyle e_i}$, the canonical isomorphism above is represented as follows. Let${\displaystyle g_{ij}=⟨e_i,e_j⟩}$ be the components of the non-degenerate bilinear form and let${\displaystyle g^{ij}}$ be the components of the inverse matrix to${\displaystyle g_{ij}}$. Let${\displaystyle e^i}$ be the dual basis of${\displaystyle e_i}$. A vector${\displaystyle v}$ is written in the basis as${\displaystyle v=v^i{e}_i}$ usingEinstein summation notation, i.e.,${\displaystyle v}$ has components${\displaystyle v^i}$ in the basis. The canonical isomorphism applied to${\displaystyle v}$ gives an element of the dual, which is called a covector. The covector has components${\displaystyle v_i}$ in the dual basis given by contracting with${\displaystyle g}$:

${\displaystyle v_i=g_{ij}{v}^j.}$

This is what is meant by lowering the index. Conversely, contracting a covector${\displaystyle \alpha ={\alpha }_i{e}^i}$ with the inverse of${\displaystyle g}$ gives a vector with components

${\displaystyle {\alpha }^i=g^{ij}{\alpha }_j.}$

in the basis${\displaystyle e_i}$. This process is called raising the index.

Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in${\displaystyle g_{ij}}$ and${\displaystyle g^{ij}}$ being inverses:

${\displaystyle g^{ij}{g}_{jk}=g_{kj}{g}^{ji}={{\delta }^i}_k={{\delta }_k}^i}$

where${\displaystyle {\delta }_j^i}$ is theKronecker delta oridentity matrix.

The musical isomorphisms are the global version of the canonical isomorphism${\displaystyle v\mapsto ⟨v,\cdot ⟩}$ and its inverse for thetangent bundle andcotangent bundle of a (pseudo-)Riemannian manifold${\displaystyle (M,g)}$. They are canonical isomorphisms ofvector bundles which are at any pointp the canonical isomorphism applied to thetangent space ofM atp endowed with the inner product${\displaystyle g_p}$.

Because everysmooth manifold can be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a smooth manifold is (non-canonically) isomorphic to its dual.

Let(M,g) be a (pseudo-)Riemannian manifold. At each pointp, the mapg_p is a non-degenerate bilinear form on the tangent spaceT_pM. Ifv is a vector inT_pM, itsflat is thecovector

${\displaystyle v^{\mathrm{♭}}=g_p(v,\cdot )}$

inT^∗
_pM. Since this is a smooth map that preserves the pointp, it defines a morphism ofsmooth vector bundles${\displaystyle \mathrm{♭}:\mathrm{T}M\to {\mathrm{T}}^{\ast }M}$. By non-degeneracy of the metric,${\displaystyle \mathrm{♭}}$ has an inverse${\displaystyle \mathrm{♯}}$ at each point, characterized by

${\displaystyle g_p({\alpha }^{\mathrm{♯}},v)=\alpha (v)}$

forα inT^∗
_pM andv inT_pM. The vector${\displaystyle {\alpha }^{\mathrm{♯}}}$ is called thesharp ofα. The sharp map is a smooth bundle map${\displaystyle \mathrm{♯}:{\mathrm{T}}^{\ast }M\to \mathrm{T}M}$.

Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for eachp inM, there are mutually inverse vector space isomorphisms betweenT_pM andT^∗
_pM.

The flat and sharp maps can be applied tovector fields andcovector fields by applying them to each point. Hence, ifX is a vector field andω is a covector field,

${\displaystyle X^{\mathrm{♭}}=g(X,\cdot )}$

and

${\displaystyle g({\omega }^{\mathrm{♯}},X)=\omega (X)}$.

Suppose{e_i} is amoving tangent frame (see alsosmooth frame) for the tangent bundleTM with, asdual frame (see alsodual basis), themoving coframe (amoving tangent frame for thecotangent bundle${\displaystyle {\mathrm{T}}^{\ast }M}$; see alsocoframe){eⁱ}. Then thepseudo-Riemannian metric, which is a 2-covarianttensor field, can be written locally in this coframe asg =g_ijeⁱ ⊗e^j usingEinstein summation notation.

Given a vector fieldX =Xⁱe_i and denotingg_ijXⁱ =X_j, its flat is

${\displaystyle X^{\mathrm{♭}}=g_{ij}{X}^i{\mathbf{e}}^j=X_j{\mathbf{e}}^j}$.

This is referred to as lowering an index, because the components ofX are written with an upper indexXⁱ, whereas the components of${\displaystyle X^{\mathrm{♭}}}$ are written with a lower indexX_j.

In the same way, given a covector fieldω =ω_ieⁱ and denotingg^ijω_i =ω^j, its sharp is

${\displaystyle {\omega }^{\mathrm{♯}}=g^{ij}{\omega }_i{\mathbf{e}}_j={\omega }^j{\mathbf{e}}_j}$,

whereg^ij are thecomponents of theinverse metric tensor (given by the entries of the inverse matrix tog_ij). Taking the sharp of a covector field is referred to asraising an index.

The musical isomorphisms may also be extended, for eachr,s,k, to an isomorphism between the bundle

${\displaystyle \underset{i=1}{\overset{s}{⨂}}\mathrm{T}M\otimes \underset{j=1}{\overset{r}{⨂}}{\mathrm{T}}^{\ast }M}$

of${\displaystyle (r,s)}$ tensors and the bundle of${\displaystyle (r-k,s+k)}$ tensors. Herek can be positive or negative, so long asr -k ≥ 0 ands +k ≥ 0.

Lowering an index of an${\displaystyle (r,s)}$ tensor gives a${\displaystyle (r-1,s+1)}$ tensor, while raising an index gives a${\displaystyle (r+1,s-1)}$. Which index is to be raised or lowered must be indicated.

For instance, consider the(0, 2) tensorX =X_ijeⁱ ⊗e^j. Raising the second index, we get the(1, 1) tensor

${\displaystyle X^{\mathrm{♯}}=g^{jk}{X}_{ij}{\mathrm{e}}^i\otimes {\mathrm{e}}_k.}$

In other words, the components${\displaystyle X_i^k}$ of${\displaystyle X^{\mathrm{♯}}}$ are given by

${\displaystyle X_i^k=g^{jk}{X}_{ij}.}$

Similar formulas are available for tensors of other orders. For example, for a${\displaystyle (0,n)}$ tensorX, all indices are raised by:^[3]

${\displaystyle X^{j_1{j}_2\cdots j_n}=g^{j_1{i}_1}{g}^{j_2{i}_2}\cdots g^{j_n{i}_n}{X}_{i_1{i}_2\cdots i_n}.}$

For a${\displaystyle (n,0)}$ tensorX, all indices are lowered by:

${\displaystyle X_{j_1{j}_2\cdots j_n}=g_{j_1{i}_1}{g}_{j_2{i}_2}\cdots g_{j_n{i}_n}{X}^{i_1{i}_2\cdots i_n}.}$

For a mixed tensor of order${\displaystyle (n,m)}$, all lower indices are raised and all upper indices are lowered by

${\displaystyle {X_{p_1{p}_2\cdots p_n}}^{q_1{q}_2\cdots q_m}=g_{p_1{i}_1}{g}_{p_2{i}_2}\cdots g_{p_n{i}_n}{g}^{q_1{j}_1}{g}^{q_2{j}_2}\cdots g^{q_m{j}_m}{X^{i_1{i}_2\cdots i_n}}_{j_1{j}_2\cdots j_m}.}$

Well-formulated expressions are constrained by the rules of Einstein summation notation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we can immediately see that an expression such as${\displaystyle g_{ij}{v}^i{u}^j}$is well formulated while${\displaystyle g_{ij}{v}_i{u}_j}$ is not.

In the context ofexterior algebra, an extension of the musical operators may be defined on⋀V and its dual⋀V^*, and are again mutual inverses:^[4]

${\displaystyle \mathrm{♭}:\underset{i=1}{\overset{k}{\bigwedge }}V\to \underset{i=1}{\overset{k}{\bigwedge }}{V}^{\ast },}$

${\displaystyle \mathrm{♯}:\underset{i=1}{\overset{k}{\bigwedge }}{V}^{\ast }\to \underset{i=1}{\overset{k}{\bigwedge }}V,}$

defined by

${\displaystyle (X\wedge \dots \wedge Z{)}^{\mathrm{♭}}=X^{\mathrm{♭}}\wedge \dots \wedge Z^{\mathrm{♭}},}$

${\displaystyle (\alpha \wedge \dots \wedge \gamma {)}^{\mathrm{♯}}={\alpha }^{\mathrm{♯}}\wedge \dots \wedge {\gamma }^{\mathrm{♯}}.}$

In this extension, in which♭ mapsk-vectors tok-covectors and♯ mapsk-covectors tok-vectors, all the indices of atotally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:$${\displaystyle Y^{\mathrm{♯}}=(Y_{i_1\dots i_j}{\mathbf{e}}^{i_1}\otimes \cdots \otimes {\mathbf{e}}^{i_j}{)}^{\mathrm{♯}}=g^{i_1{r}_1}\dots g^{i_j{r}_s}{Y}_{i_1\dots i_k}{\mathbf{e}}_{r_1}\otimes \cdots \otimes {\mathbf{e}}_{r_s}.}$$

This works not just fork-vectors in the context of linear algebra but also fork-forms in the context of a (pseudo-)Riemannian manifold:

${\displaystyle \mathrm{♭}:\underset{i=1}{\overset{k}{\bigwedge }}\mathrm{T}M\to \underset{i=1}{\overset{k}{\bigwedge }}{\mathrm{T}}^{\ast }M,}$

${\displaystyle \mathrm{♯}:\underset{i=1}{\overset{k}{\bigwedge }}{\mathrm{T}}^{\ast }M\to \underset{i=1}{\overset{k}{\bigwedge }}\mathrm{T}M,}$

More generally, musical isomorphisms always exist between a vector bundle endowed with abundle metric and its dual.

Given a(0, 2) tensorX =X_ijeⁱ ⊗e^j, we define thetrace ofX through the metric tensorg by$${\displaystyle {\mathrm{tr}}_g(X):=\mathrm{tr}(X^{\mathrm{♯}})=\mathrm{tr}(g^{jk}{X}_{ij}{\mathbf{e}}^i\otimes {\mathbf{e}}_k)=g^{ij}{X}_{ij}.}$$

Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

The trace of an${\displaystyle (r,s)}$ tensor can be taken in a similar way, so long as one specifies which two distinct indices are to be traced. This process is also called contracting the two indices. For example, ifX is an${\displaystyle (r,s)}$ tensor withr > 1, then the indices${\displaystyle i_1}$ and${\displaystyle i_2}$ can be contracted to give an${\displaystyle (r-2,s)}$ tensor with components

${\displaystyle X_{j_1{j}_2\cdots j_s}^{i_3{i}_4\cdots i_r}=g_{i_1{i}_2}{X}_{j_1{j}_2\cdots j_s}^{i_1{i}_2\cdots i_r}.}$

The covariant4-position is given by

${\displaystyle X_{\mu }=(-ct,x,y,z)}$

with components:

${\displaystyle X_0=-ct,X_1=x,X_2=y,X_3=z}$

(wherex,y,z are the usualCartesian coordinates) and theMinkowski metric tensor withmetric signature (− + + +) is defined as

in components:

${\displaystyle {\eta }_{00}=-1,{\eta }_{i0}={\eta }_{0i}=0,{\eta }_{ij}={\delta }_{ij}(i,j\ne 0).}$

To raise the index, multiply by the tensor and contract:

${\displaystyle X^{\lambda }={\eta }^{\lambda \mu }{X}_{\mu }={\eta }^{\lambda 0}{X}_0+{\eta }^{\lambda i}{X}_i}$

then forλ = 0:

${\displaystyle X^0={\eta }^{00}{X}_0+{\eta }^{0i}{X}_i=-X_0}$

and forλ =j = 1, 2, 3:

${\displaystyle X^j={\eta }^{j0}{X}_0+{\eta }^{ji}{X}_i={\delta }^{ji}{X}_i=X_j.}$

So the index-raisedcontravariant 4-position is:

${\displaystyle X^{\mu }=(ct,x,y,z).}$

This operation is equivalent to the matrix multiplication

Given two vectors,${\displaystyle X^{\mu }}$ and${\displaystyle Y^{\mu }}$, we can write down their (pseudo-)inner product in two ways:

${\displaystyle {\eta }_{\mu \nu }{X}^{\mu }{Y}^{\nu }.}$

By lowering indices, we can write this expression as

${\displaystyle X_{\mu }{Y}^{\mu }.}$

In matrix notation, the first expression can be written as

while the second is, after lowering the indices of${\displaystyle X^{\mu }}$,

For a (0,2) tensor,^[3] twice contracting with the inverse metric tensor and contracting in different indices raises each index:

${\displaystyle A^{\mu \nu }=g^{\mu \rho }{g}^{\nu \sigma }{A}_{\rho \sigma }.}$

Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:

${\displaystyle A_{\mu \nu }=g_{\mu \rho }{g}_{\nu \sigma }{A}^{\rho \sigma }}$

Let's apply this to the theory of electromagnetism.

Thecontravariantelectromagnetic tensor in the(+ − − −)signature is given by^[5]

In components,

${\displaystyle F^{0i}=-F^{i0}=-\frac{E^i}{c},F^{ij}=-{\epsilon }^{ijk}{B}_k}$

To obtain thecovariant tensorF_αβ, contract with the inverse metric tensor:

and sinceF⁰⁰ = 0 andF⁰ⁱ = −Fⁱ⁰, this reduces to

${\displaystyle F_{\alpha \beta }=\left({\eta }_{\alpha i}{\eta }_{\beta 0}-{\eta }_{\alpha 0}{\eta }_{\beta i}\right)F^{i0}+{\eta }_{\alpha i}{\eta }_{\beta j}{F}^{ij}}$

Now forα = 0,β =k = 1, 2, 3:

and by antisymmetry, forα =k = 1, 2, 3,β = 0:

${\displaystyle F_{k0}=-F^{k0}}$

then finally forα =k = 1, 2, 3,β =l = 1, 2, 3;

The (covariant) lower indexed tensor is then:

This operation is equivalent to the matrix multiplication

• Duality (mathematics)
• Dual space § Bilinear products and dual spaces
• Einstein notation
• Flat (music) andSharp (music) about the signs♭ and♯
• Hodge star operator
• Metric tensor
• Vector bundle

• Lee, J. M. (2003).Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics. Vol. 218.ISBN 0-387-95448-1.
• Lee, J. M. (1997).Riemannian Manifolds – An Introduction to Curvature. Springer Graduate Texts in Mathematics. Vol. 176. Springer Verlag.ISBN 978-0-387-98322-6.
• Vaz, Jayme; da Rocha, Roldão (2016).An Introduction to Clifford Algebras and Spinors.Oxford University Press.ISBN 978-0-19-878-292-6.

• Differential geometry
• Riemannian geometry
• Riemannian manifolds
• Symplectic geometry

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