Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle ${\displaystyle \mathrm {T} M}$ and the cotangent bundle ${\displaystyle \mathrm {T} ^{*}M}$ 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 ${\displaystyle \flat }$ (flat) and ${\displaystyle \sharp }$ (sharp).[1][2] The exact origin of this notation is not known, but the term musicality in this context would be due to Marcel Berger.[3]

In covariant and contravariant notation, it is also known as raising and lowering indices.

Discussion

Let (M, g) be a pseudo-Riemannian manifold. Suppose {ei} is a moving tangent frame (see also smooth frame) for the tangent bundle TM with, as dual frame (see also dual basis), the moving coframe (a moving tangent frame for the cotangent bundle ${\displaystyle \mathrm {T} ^{*}M}$. See also coframe) {ei}. Then, locally, we may express the pseudo-Riemannian metric (which is a 2-covariant tensor field that is symmetric and nondegenerate) as g = gij eiej (where we employ the Einstein summation convention).

Given a vector field X = Xi ei , we define its flat by

${\displaystyle X^{\flat }:=g_{ij}X^{i}\,\mathbf {e} ^{j}=X_{j}\,\mathbf {e} ^{j}.}$

This is referred to as "lowering an index". Using the traditional diamond bracket notation for the inner product defined by g, we obtain the somewhat more transparent relation

${\displaystyle X^{\flat }(Y)=\langle X,Y\rangle }$

for any vector fields X and Y.

In the same way, given a covector field ω = ωi ei , we define its sharp by

${\displaystyle \omega ^{\sharp }:=g^{ij}\omega _{i}\mathbf {e} _{j}=\omega ^{j}\mathbf {e} _{j},}$

where gij are the components of the inverse metric tensor (given by the entries of the inverse matrix to gij ). Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads

${\displaystyle {\bigl \langle }\omega ^{\sharp },Y{\bigr \rangle }=\omega (Y),}$

for any covector field ω and any vector field Y.

Through this construction, we have two mutually inverse isomorphisms

${\displaystyle \flat :{\rm {T}}M\to {\rm {T}}^{*}M,\qquad \sharp :{\rm {T}}^{*}M\to {\rm {T}}M.}$

These are isomorphisms of vector bundles and, hence, we have, for each p in M, mutually inverse vector space isomorphisms between Tp M and T
p
M
.

Extension to tensor products

The musical isomorphisms may also be extended to the bundles

${\displaystyle \bigotimes ^{k}{\rm {T}}M,\qquad \bigotimes ^{k}{\rm {T}}^{*}M.}$

Which index is to be raised or lowered must be indicated. For instance, consider the (0, 2)-tensor field X = Xij eiej. Raising the second index, we get the (1, 1)-tensor field

${\displaystyle X^{\sharp }=g^{jk}X_{ij}\,{\rm {e}}^{i}\otimes {\rm {e}}_{k}.}$

Extension to k-vectors and k-forms

In the context of exterior algebra, an extension of the musical operators may be defined on V and its dual

V
, which with minor abuse of notation, may be denoted the same, and are again mutual inverses:[4]

${\displaystyle \flat :{\bigwedge }V\to {\bigwedge }^{*}V,\qquad \sharp :{\bigwedge }^{*}V\to {\bigwedge }V,}$

defined by

${\displaystyle (X\wedge \ldots \wedge Z)^{\flat }=X^{\flat }\wedge \ldots \wedge Z^{\flat },\qquad (\alpha \wedge \ldots \wedge \gamma )^{\sharp }=\alpha ^{\sharp }\wedge \ldots \wedge \gamma ^{\sharp }.}$

In this extension, in which maps p-vectors to p-covectors and maps p-covectors to p-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:

${\displaystyle Y^{\sharp }=(Y_{i\dots k}\mathbf {e} ^{i}\otimes \dots \otimes \mathbf {e} ^{k})^{\sharp }=g^{ir}\dots g^{kt}\,Y_{i\dots k}\,\mathbf {e} _{r}\otimes \dots \otimes \mathbf {e} _{t}.}$

Trace of a tensor through a metric tensor

Given a type (0, 2) tensor field X = Xij eiej, we define the trace of X through the metric tensor g by

${\displaystyle \operatorname {tr} _{g}(X):=\operatorname {tr} (X^{\sharp })=\operatorname {tr} (g^{jk}X_{ij}\,{\bf {e}}^{i}\otimes {\bf {e}}_{k})=g^{ji}X_{ij}=g^{ij}X_{ij}.}$

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