Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle $\mathrm{T}M$ and the cotangent bundle ${\mathrm{T}}^{*}M$ of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the musical notation symbols $\mathrm{♭}$ (flat) and $\mathrm{♯}$ (sharp).Template:SfnTemplate:Sfn

In the notation of Ricci calculus, the idea is expressed as the raising and lowering of indices.

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

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 $V$ endowed with a non-degenerate bilinear form $⟨\cdot ,\cdot ⟩$, is canonically isomorphic to its dual. The canonical isomorphism $V\to V^{*}$ is given by

$v\mapsto ⟨v,\cdot ⟩$.

The non-degeneracy of $⟨\cdot ,\cdot ⟩$ means exactly that the above map is an isomorphism.

An example is where $V={ℝ}^n$, and $⟨\cdot ,\cdot ⟩$ is the dot product.

The musical isomorphisms are the global version of this isomorphism and its inverse for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold $(M,g)$. They are canonical isomorphisms of vector bundles which are at any point Template:Math the above isomorphism applied to the tangent space of Template:Math at Template:Math endowed with the inner product $g_p$.

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

Discussion

Let Template:Math be a (pseudo-)Riemannian manifold. At each point Template:Mvar, the map Template:Math is a non-degenerate bilinear form on the tangent space Template:Math. If Template:Mvar is a vector in Template:Math, its flat is the covector

$v^{\mathrm{♭}}=g_p(v,\cdot )$

in Template:Math. Since this is a smooth map that preserves the point Template:Mvar, it defines a morphism of smooth vector bundles $\mathrm{♭}:\mathrm{T}M\to {\mathrm{T}}^{*}M$. By non-degeneracy of the metric, $\mathrm{♭}$ has an inverse $\mathrm{♯}$ at each point, characterized by

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

for Template:Mvar in Template:Math and Template:Mvar in Template:Math. The vector ${\alpha }^{\mathrm{♯}}$ is called the sharp of Template:Mvar. The sharp map is a smooth bundle map $\mathrm{♯}:{\mathrm{T}}^{*}M\to \mathrm{T}M$.

Flat and sharp are mutually inverse isomorphisms of smooth vector bundles, hence, for each Template:Mvar in Template:Mvar, there are mutually inverse vector space isomorphisms between Template:Math and Template:Math.

The flat and sharp maps can be applied to vector fields and covector fields by applying them to each point. Hence, if Template:Mvar is a vector field and Template:Mvar is a covector field,

$X^{\mathrm{♭}}=g(X,\cdot )$

and

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

In a moving frame

Suppose Template:Math is a moving tangent frame (see also smooth frame) for the tangent bundle Template:Math with, as dual frame (see also dual basis), the moving coframe (a moving tangent frame for the cotangent bundle ${\mathrm{T}}^{*}M$; see also coframe) Template:Math. Then the pseudo-Riemannian metric, which is a symmetric and nondegenerate Template:Math-covariant tensor field can be written locally in terms of this coframe as Template:Math using Einstein summation notation.

Given a vector field Template:Math and denoting Template:Math, its flat is

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

This is referred to as lowering an index.

In the same way, given a covector field Template:Math and denoting Template:Math, its sharp is

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

where Template:Math are the components of the inverse metric tensor (given by the entries of the inverse matrix to Template:Math). Taking the sharp of a covector field is referred to as raising an index.

Extension to tensor products

The musical isomorphisms may also be extended to the bundles $${⨂}^k\mathrm{T}M,{⨂}^k{\mathrm{T}}^{*}M.$$

Which index is to be raised or lowered must be indicated. For instance, consider the Template:Nowrap-tensor field Template:Math. Raising the second index, we get the Template:Nowrap-tensor field $$X^{\mathrm{♯}}=g^{jk}{X}_{ij}{\mathrm{e}}^i\otimes {\mathrm{e}}_k.$$

Extension to k-vectors and k-forms

In the context of exterior algebra, an extension of the musical operators may be defined on Template:Math and its dual Template:Math, which with minor abuse of notation may be denoted the same, and are again mutual inverses:Template:Sfn $$\mathrm{♭}:\bigwedge V\to {\bigwedge }^{*}V,\mathrm{♯}:{\bigwedge }^{*}V\to \bigwedge V,$$ defined by $$(X\wedge \dots \wedge Z{)}^{\mathrm{♭}}=X^{\mathrm{♭}}\wedge \dots \wedge Z^{\mathrm{♭}},(\alpha \wedge \dots \wedge \gamma {)}^{\mathrm{♯}}={\alpha }^{\mathrm{♯}}\wedge \dots \wedge {\gamma }^{\mathrm{♯}}.$$

In this extension, in which Template:Math maps p-vectors to p-covectors and Template:Math 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: $$Y^{\mathrm{♯}}=(Y_{i\dots k}{𝐞}^i\otimes \dots \otimes {𝐞}^k{)}^{\mathrm{♯}}=g^{ir}\dots g^{kt}{Y}_{i\dots k}{𝐞}_r\otimes \dots \otimes {𝐞}_t.$$

Vector bundles with bundle metrics

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

Trace of a tensor through a metric tensor

Given a type Template:Nowrap tensor field Template:Math, we define the trace of Template:Mvar through the metric tensor Template:Mvar by $${\mathrm{tr}}_g(X):=\mathrm{tr}(X^{\mathrm{♯}})=\mathrm{tr}(g^{jk}{X}_{ij}{𝐞}^i\otimes {𝐞}_k)=g^{ij}{X}_{ij}.$$

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

See also

• Duality (mathematics)
• Raising and lowering indices
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• Hodge star operator
• Vector bundle
• Flat (music) and Sharp (music) about the signs Template:Music and Template:Music

References

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