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]

In the notation of Ricci calculus, it is also known as raising and lowering indices.

Motivation

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space but not canonically isomorphic to it. On the other hand a finite-dimensional vector space ${\displaystyle V}$ endowed with a non-degenerate bilinear form ${\displaystyle \langle \cdot ,\cdot \rangle }$, is canonically isomorphic to its dual, the isomorphism being given by:

$${\displaystyle {\begin{array}{rcl}V&\rightarrow &V^{*}\\v&\mapsto &\langle v,\cdot \rangle \end{array))}$$

An example is where ${\displaystyle V}$ is a Euclidean space, and ${\displaystyle \langle \cdot ,\cdot \rangle }$ is its inner product.

Musical isomorphisms are the global version of this isomorphism and its inverse, for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold ${\displaystyle (M,g)}$. They are isomorphisms of vector bundles which are at any point ${\displaystyle x\in M}$ the above isomorphism applied to the (pseudo-)Euclidean space ${\displaystyle \mathrm {T} _{p}M}$ (the tangent space of M at point p) endowed with the inner product ${\displaystyle g_{p))$. More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.

Because every paracompact manifold can be endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a paracompact manifold is always isomorphic to its dual (but not canonically unless a (pseudo-)Riemannian metric has been associated with the manifold).

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 and denoting g_ij Xⁱ = X_j, we define its flat by:

$${\displaystyle {\begin{aligned}\flat :\mathrm {T} M&\longrightarrow \mathrm {T} ^{*}M\\X&\longmapsto g_{ij}X^{i}\mathbf {e} ^{j}\\&\longmapsto X_{j}\mathbf {e} ^{j}\end{aligned))}$$

This is referred to as lowering an index. Using angle bracket notation for the bilinear form defined by g, we obtain the somewhat more transparent relation

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

In the same way, given a covector field ω = ω_i eⁱ and denoting g^ij ω_i = ω^j, we define its sharp by:

$${\displaystyle {\begin{aligned}\sharp :\mathrm {T} ^{*}M&\longrightarrow \mathrm {T} M\\\omega &\longmapsto g^{ij}\omega _{i}\mathbf {e} _{j}\\&\longmapsto \omega ^{j}\mathbf {e} _{j}\end{aligned))}$$

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 angle bracket notation, this reads

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

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:^[3]

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

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

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

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

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