Definitions

# Musical isomorphism

In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle $T^\left\{*\right\}M$ of a Riemannian manifold given by its metric.

It is also known as raising and lowering indices.

## Introduction

A metric g on a Riemannian manifold M is a tensor field $g in mathcal\left\{T\right\}_2^0\left(M\right)$ 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 xM, $g_xin S^2T^*_xM$ defines a linear isomorphism of vector spaces
$widehat\left\{g\right\}_x : T_x M longrightarrow T^\left\{*\right\}_x M$
(from the tangent space to the cotangent space) given by
$widehat\left\{g\right\}_x\left(X_x\right) = g\left(X_x,-\right)$
for any tangent vector Xx in TxM, i.e.,
$widehat\left\{g\right\}_x\left(X_x\right)\left(Y_x\right) = g_x\left(X_x,Y_x\right).$

The collection of these linear isomorphisms define a bundle isomorphism

$widehat\left\{g\right\} : TM longrightarrow T^\left\{*\right\}M$
which is therefore, in particular, a diffeomorphism and is linear on each tangent space. This is called the musical isomorphism flat, and its inverse is called sharp: sharp raises indices, flat lowers them.

## Motivation of the name

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

The musical isomorphisms can be used to define the gradient, divergence and curl of smooth functions on $mathbb\left\{R\right\}^3$ as follows:


begin{array}{rcl} nabla f &=& left({mathbf d} f right)^sharp nabla cdot F &=& star {mathbf d} left(star F^flat right) nabla times F &=& left[star left({mathbf d} F^flat right) right]^sharp end{array}

where $star$ is the Hodge star operator. (Note that the first equation is also valid in a more general context of smooth functions on Riemannian manifolds.) Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine Contributor Dominique Hulin, Jacques Lafontaine (2004). Riemannian Geometry. Springer. ISBN 3540204938

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