# Musical isomorphism

In mathematics, the **musical isomorphism** (or **canonical isomorphism**) is an isomorphism between the tangent bundle *TM* and the cotangent bundle *T***M* of a Riemannian manifold given by its metric. There are similar isomorphisms on symplectic manifolds. The term *musical* refers to the use of the symbols ♭ and ♯.^{[1]}

It is also known as raising and lowering indices.

## Discussion

Let (*M*, *g*) be a Riemannian manifold. Suppose {∂_{i}} is a local frame for the tangent bundle *TM* with dual coframe {*dx ^{i}*}. Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as

*g*=

*g*⊗

_{ij}dx^{i}*dx*(where we employ the Einstein summation convention). Given a vector field

^{ j}*X*=

*X*∂

^{ i}_{i}we define its flat by

This is referred to as "lowering an index". Using the traditional diamond bracket notation for inner product defined by g, we obtain the somewhat more transparent relation

for all vectors X and Y.

Alternatively, given a covector field *ω* = *ω _{i}*

*dx*we define its sharp by

^{i}where *g ^{ij}* are the elements of the inverse matrix to

*g*. Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads

_{ij}for ω an arbitrary covector and Y an arbitrary vector.

Through this construction we have two inverse isomorphisms

These are isomorphisms of vector bundles and hence we have, for each p in M, inverse vector space isomorphisms between *T _{p}M* and

*T*∗

*p*

*M*.

The musical isomorphisms may also be extended to the bundles

It must be stated which index is to be raised or lowered. For instance, consider the (0, 2) tensor field *X* = *X _{ij} dx^{i}* ⊗

*dx*. Raising the second index, we get the (1, 1) tensor field

^{ j}## Trace of a tensor through a metric

Given a (0, 2) tensor field *X* = *X _{ij} dx^{i}* ⊗

*dx*, we define the trace of X through the metric g by

^{ j}Observe that the definition of trace is independent of the choice of index we raise since the metric tensor is symmetric.

## See also

- Duality (mathematics)
- Raising and lowering indices
- Bilinear products and dual spaces
- Vector bundle
- Flat (music) and Sharp (music) about the signs ♭ and ♯