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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]}

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

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 endowed with a non-degenerate bilinear form , is canonically isomorphic to its dual, the isomorphism being given by:

An example is where is a Euclidean space, and 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 . They are isomorphisms of vector bundles which are at any point the above isomorphism applied to the (pseudo-)Euclidean space (the tangent space of *M* at point *p*) endowed with the inner product . 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).

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} and denoting *g*_{ij} *X*^{i} = *X*_{j}, we define its **flat** by:

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

In the same way, given a covector field *ω* = *ω*_{i} **e**^{i} and denoting *g*^{ij} *ω*_{i} = *ω*^{j}, 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 angle bracket notation, this reads

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*.

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

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]}

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:

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.

- Lee, J. M. (2003).
*Introduction to Smooth manifolds*. Springer Graduate Texts in Mathematics. Vol. 218. ISBN 0-387-95448-1. - Lee, J. M. (1997).
*Riemannian Manifolds – An Introduction to Curvature*. Springer Graduate Texts in Mathematics. Vol. 176. 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.