Musical isomorphism

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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_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:

[\hat_p : T_p M \longrightarrow T^_p M]

given by

[\hat_p(v_p) = g(v_p,-)]

i.e.

[ \langle\hat_p(v_p),\omega_p\rangle = g_p(v_p,\omega_p).]

Globally the map

[\hat : TM \longrightarrow T^M]

is a diffeomorphism.

Motivation of the name

The isomorphism [\hat] and its inverse [\hat^] 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] 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 Riemannian manifold M as follows:

[\mathrm\;f=\hat^ \circ df = (df)^]

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