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Musical isomorphism
https://en.wikipedia.org/wiki/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.

Contents

1 Discussion
2 Trace of a tensor through a metric
3 See also
4 References

See also

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