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In mathematics, the musical isomorphism is an isomorphism between the tangent bundle $TM$ and the cotangent bundle $T^{*}M$ of a Riemannian manifold given by its metric.

Introduction

A metric g on a Riemannian manifold M is a tensor field $g\in {\mathcal {T}}_{2}(M)$. If we fix one parameter as a vector $v_{p}\in T_{p}M$, we have an isomorphism of vector spaces:

${\hat {g}}_{p}:T_{p}M\longrightarrow T_{p}^{*}M$

${\hat {g}}_{p}(v_{p})=g(v_{p},-)$

$<{\hat {g}}_{p}(v_{p}),\omega _{p}>=g_{p}(v_{p},\omega _{p})$

And globally,

${\hat {g}}:TM\longrightarrow T^{*}M$ is a diffeomorphism.

Motivation of the name

The isomorphism ${\hat {g}}$ and its inverse ${\hat {g}}^{-1}$ are called musical isomorphisms because they move up and down the indexes of the vectors. For instance, a vector of TM is written as $\alpha ^{i}{\frac {\partial }{\partial x}}$ and a covector as $\alpha _{i}dx^{i}$, so the index i is moved up and down in $\alpha$ just as the symbols sharp ($\sharp$) and flat ($\flat$) move up and down the pitch of a tone.

Gradient

The musical isomorphisms can be used to define the gradient of a smooth function over a manifold M as follows:

$\mathrm {grad} \;f={\hat {g}}^{-1}\circ df=(df)^{\sharp }$