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

Introduction

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A metric g on a Riemannian manifold M is a tensor field  . If we fix one parameter as a vector  , we have an isomorphism of vector spaces:

 
 
 

And globally,

  is a diffeomorphism.

Motivation of the name

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The isomorphism   and its inverse   are called musical isomorphisms because they move up and down the indexes of the vectors. For instance, a vector of TM is written as   and a covector as  , so the index i is moved up and down in   just as the symbols sharp ( ) and flat ( ) move up and down the pitch of a tone.

Gradient

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