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 $g \in \mathcal{T}_2^0(M)$ which is symmetric and positive-definite: thus g is a positive definite smooth section of the vector bundle $S^2T^*M\,$ of symmetric bilinear forms on the tangent bundle. At any point x∈M, $g_x\in S^2T^*_xM$ defines an isomorphism of vector spaces

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

(from the tangent space to the cotangent space) given by

$\widehat{g}_x(X_x) = g(X_x,-)$

for any tangent vector X_x in T_xM, i.e.,

$\widehat{g}_x(X_x)(Y_x) = g_x(X_x,Y_x).$

The collection of these linear isomorphisms define a bundle isomorphism

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

which is therefore, in particular, a diffeomorphism. This is called the musical isomorphism flat, and its inverse is called sharp.

[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.