Musical Isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle $\mathrmM$ and the cotangent bundle $\mathrm^* M$ of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term ''musical'' refers to the use of the symbols $\flat$ (flat) and $\sharp$ (sharp).

In the notation of Ricci calculus, it is also known as raising and lowering indices.

Motivation

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space but not canonically isomorphic to it. On the other hand a finite-dimensional vector space $V$ endowed with a non-degenerate bilinear form $\langle\cdot,\cdot\rangle$, is canonically isomorphic to its dual, the isomorphism being given by:

$\beginV&\rightarrow &V^* \\ v &\mapsto& \langle v,\cdot\rangle\end$

An example is where $V$ is a Euclidean space, and $\langle\cdot,\cdot\rangle$ is its inner product.

Musical isomorphisms are the global version of this isomorphism and its inverse, for the tangent bundle and cotangent bundle of a pseudo-Riemannian manifold $(M,g)$. They are isomorphisms of vector bundles which are at any point $x \in M$ the above isomorphism applied to the (pseudo-)Euclidean space $\mathrm_p M$ (the tangent space of at point ) endowed with the inner product $g_p$. More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.

Because every paracompact manifold can be endowed with a (pseudo-)Riemannian metric, the musical isomorphisms allow to show that on those spaces a vector bundle is always isomorphic to its dual (but not canonically unless it is a without a pseudo-Riemannian metric being associated with the manifold).

Discussion

Let be a pseudo-Riemannian manifold. Suppose is a moving tangent frame (see also smooth frame) for the ''tangent bundle'' with, as dual frame (see also dual basis), the moving coframe (a ''moving tangent frame'' for the ''cotangent bundle'' $\mathrm^*M$; see also coframe) . Then, locally, we may express the pseudo-Riemannian metric (which is a -covariant tensor field that is symmetric and nondegenerate) as (where we employ the Einstein summation convention).

Given a vector field and denoting , we define its flat by
$\flat : \mathrmM \to \mathrm^*M : X \mapsto g_ X^i \, \mathbf^j = X_j \, \mathbf^j .$

This is referred to as ''lowering an index''. Using angle bracket notation for the bilinear form defined by , we obtain the somewhat more transparent relation
$X^\flat (Y) = \langle X, Y \rangle$
for any vector fields and .

In the same way, given a covector field and denoting , we define its sharp by
$\sharp : \mathrm^*M \to \mathrmM : \omega \mapsto g^ \omega_i \mathbf_j = \omega_j \mathbf_j ,$
where are the components of the inverse metric tensor (given by the entries of the inverse matrix to ). Taking the sharp of a covector field is referred to as ''raising an index''. In angle bracket notation, this reads
$\bigl \langle \omega^\sharp, Y \bigr \rangle = \omega(Y),$
for any covector field and any vector field .

Through this construction, we have two mutually inverse isomorphisms
$\flat: M \to ^* M, \qquad \sharp:^* M \to M.$

These are isomorphisms of vector bundles and, hence, we have, for each in , mutually inverse vector space isomorphisms between and .

Extension to tensor products

The musical isomorphisms may also be extended to the bundles
$\bigotimes ^k M, \qquad \bigotimes ^k ^* M .$

Which index is to be raised or lowered must be indicated. For instance, consider the -tensor field . Raising the second index, we get the -tensor field
$X^\sharp = g^ X_ \, ^i \otimes _k .$

Extension to ''k''-vectors and ''k''-forms

In the context of exterior algebra, an extension of the musical operators may be defined on and its dual , which with minor abuse of notation may be denoted the same, and are again mutual inverses:
$\flat : V \to ^* V , \qquad \sharp : ^* V \to V ,$
defined by
$(X \wedge \ldots \wedge Z)^\flat = X^\flat \wedge \ldots \wedge Z^\flat , \qquad (\alpha \wedge \ldots \wedge \gamma)^\sharp = \alpha^\sharp \wedge \ldots \wedge \gamma^\sharp .$

In this extension, in which maps ''p''-vectors to ''p''-covectors and maps ''p''-covectors to ''p''-vectors, all the indices of a totally antisymmetric tensor are simultaneously raised or lowered, and so no index need be indicated:
$Y^\sharp = ( Y_ \mathbf^i \otimes \dots \otimes \mathbf^k)^\sharp = g^ \dots g^ \, Y_ \, \mathbf_r \otimes \dots \otimes \mathbf_t .$

Trace of a tensor through a metric tensor

Given a type tensor field , we define the trace of through the metric tensor by
$\operatorname_g ( X ) := \operatorname ( X^\sharp ) = \operatorname ( g^ X_ \, ^i \otimes _k ) = g^ X_ .$

Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

See also

*Duality (mathematics)
*Raising and lowering indices
*
*Hodge star operator
*Vector bundle
*Flat (music) and Sharp (music) about the signs and

Citations

References

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{{Manifolds

Differential geometry
Riemannian geometry
Riemannian manifolds
Symplectic geometry