Musical isomorphism

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. (January 2007)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

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.

It is also known as raising and lowering indices.

[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

Failed to parse (Cannot write to or create math output directory): \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: sharp raises indices, flat lowers them.

[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.