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From Wikipedia, the free encyclopedia

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

[edit] 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.

[edit] 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 $α i d x i$ , so the index i is moved up and down in $α$ just as the symbols sharp ( $\sharp$ ) and flat ( $\flat$ ) move up and down the pitch of a tone.