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 {e_i} 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 = g_ij eⁱ ⊗ e^j (where we employ the Einstein summation convention).

Given a vector field X = Xⁱ e_i , 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 = ω_i eⁱ , we define its sharp by

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

where g^ij are the components of the inverse metric tensor (given by the entries of the inverse matrix to g_ij ). 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 T_p M and T^∗
_pM.

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 = X_ij eⁱ ⊗ e^j. 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 = X_ij eⁱ ⊗ e^j, 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.

See also

• Duality (mathematics)
• Raising and lowering indices
• Dual space § Bilinear products and dual spaces
• Hodge dual
• Vector bundle
• Flat (music) and Sharp (music) about the signs ♭ and ♯

Citations

1. Lee 2003, Chapter 11.
2. Lee 1997, Chapter 3.
3. see this thread
4. Vaz & da Rocha 2016, pp. 48, 50.

References

• Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics. 218. ISBN 0-387-95448-1.
• Lee, J. M. (1997). Riemannian Manifolds – An Introduction to Curvature. Springer Graduate Texts in Mathematics. 176. New York · Berlin · Heidelberg: 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.