# Musical isomorphism

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In mathematics—more specifically, in differential geometry—the **musical isomorphism** (or **canonical isomorphism**) is an isomorphism between the tangent bundle and the cotangent bundle 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 (flat) and (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[edit]

Let (*M*, *g*) be a pseudo-Riemannian manifold. Suppose {**e**_{i}} is a moving tangent frame (see also smooth frame) for the *tangent bundle* T*M* with, as dual frame (see also dual basis), the moving coframe (a *moving tangent frame* for the *cotangent bundle* . See also coframe) {**e**^{i}}. 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 }**e**^{i} ⊗ **e**^{j} (where we employ the Einstein summation convention).

Given a vector field *X* = *X*^{i }**e**_{i} , we define its **flat** by

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

for any vector fields X and Y.

In the same way, given a covector field *ω* = *ω*_{i }**e**^{i} , we define its **sharp** by

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

for any covector field ω and any vector field Y.

Through this construction, we have two mutually inverse isomorphisms

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^{∗}_{p}*M*.

### Extension to tensor products[edit]

The musical isomorphisms may also be extended to the bundles

Which index is to be raised or lowered must be indicated. For instance, consider the (0, 2)-tensor field *X* = *X*_{ij }**e**^{i} ⊗ **e**^{j}. Raising the second index, we get the (1, 1)-tensor field

### Extension to *k*-vectors and *k*-forms[edit]

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]}

defined by

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:

## Trace of a tensor through a metric tensor[edit]

Given a type (0, 2) tensor field *X* = *X*_{ij }**e**^{i} ⊗ **e**^{j}, we define the **trace of** X **through the metric tensor** g by

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

## See also[edit]

- 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[edit]

**^**Lee 2003, Chapter 11.**^**Lee 1997, Chapter 3.**^**see this thread**^**Vaz & da Rocha 2016, pp. 48, 50.

## References[edit]

- Lee, J. M. (2003).
*Introduction to Smooth manifolds*. Springer Graduate Texts in Mathematics.**218**. ISBN 0-387-95448-1.CS1 maint: ref=harv (link) - 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.CS1 maint: ref=harv (link) - Vaz, Jayme; da Rocha, Roldão (2016).
*An Introduction to Clifford Algebras and Spinors*. Oxford University Press. ISBN 978-0-19-878-292-6.CS1 maint: ref=harv (link)