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