Musical isomorphism

From Wikipedia, the free encyclopedia

In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle $T * M$ of a Riemannian manifold given by its metric. There are similar isomorphisms on symplectic manifolds.

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 a linear isomorphism of vector spaces

$\widehat{g}_x : T_x M \longrightarrow T^{*}_x M$

(from the tangent space to the cotangent space) given by

$\widehat{g}_x(X_x) = g(X_x,\cdot)$

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 and is linear on each tangent space. $\widehat{g}$ is called the musical isomorphism flat ( $\flat$ ), and its inverse $\widehat{g}^{-1}$ is called sharp ( $\sharp$ ): sharp raises indices, flat lowers them (Gallot, Hullin & Lafontaine 2004, p. 75).

[edit] Motivation of the name

The isomorphism $\widehat{g}$ (i.e. $\flat$ ) and its inverse $\widehat{g}^{-1}$ (i.e. $\sharp$ ) 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 down and up in $α$ just as the symbols flat ( $\flat$ ) and sharp ( $\sharp$ ) move down and up the pitch of a semitone (Gallot, Hullin & Lafontaine 2004, p. 75).

[edit] Gradient, divergence and curl

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

$\begin{align} \nabla f &= \left( {\mathbf d} f \right)^\sharp &=\hat{g}^{-1} \circ df\\ \nabla \cdot F &= \star {\mathbf d} \left( \star F^\flat \right) \\ \nabla \times F &= \left[ \star \left( {\mathbf d} F^\flat \right) \right]^\sharp \end{align}$

where $\star$ is the Hodge star operator (Marsden & Raţiu 1999, p. 135). The first equation is also valid in a more general context of smooth functions on Riemannian manifolds. On symplectic manifold, the first equation defines a Hamiltonian vector field with the Hamiltonian f.

Moreover, it may be nice to notice that even the cross product can be defined using the Hodge star operator and the musical isomorphism. In fact, being v and w vector fields over $\mathbb{R}^3$ , it's easy to prove that:

$\mathbf{v}\times\mathbf{w} = \left[ \star \left( \mathbf{v}^\flat \wedge \mathbf{w}^\flat \right) \right]^\sharp$

[edit] References

Gallot, Sylvestre; Hullin, Dominique; Lafontaine, Jacques (2004), Riemannian Geometry (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-20493-0 .
Marsden, Jerrold E.; Raţiu, Tudor S. (1999), Introduction to Mechanics and Symmetry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98643-2 .

Musical isomorphism

From Wikipedia, the free encyclopedia

Contents

[edit] Introduction

[edit] Motivation of the name

[edit] Gradient, divergence and curl

[edit] References

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