Musical isomorphism

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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 $g \in \mathcal{T}_2(M)$ which is symmetric, nondegenerate and positive-definite. If we fix one parameter as a vector $v_p \in T_p M$ , we have an isomorphism of vector spaces:

$\widehat{g}_p : T_p M \longrightarrow T^{*}_p M$

given by

$\widehat{g}_p(v_p) = g(v_p,-)$

i.e.

$\langle\widehat{g}_p(v_p),\omega_p\rangle = g_p(v_p,\omega_p).$

Globally the map

$\widehat{g} : TM \longrightarrow T^{*}M$

is a diffeomorphism.

[edit] Motivation of the name

The isomorphism $\widehat{g}$ and its inverse $\widehat{g}^{-1}$ are called musical isomorphisms because they move up and down the indexes of the vectors. For instance, a vector of TM is written as $\alpha^i \frac{\partial}{\partial x^i}$ 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 semitone.