Musical isomorphism


In mathematics—more specifically, in differential geometry—the musical isomorphism is an isomorphism between the tangent bundle and the cotangent bundle 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 and . The exact origin of this notation is not known, but the term musicality in this context would be due to Marcel Berger.
In covariant and contravariant notation, it is also known as raising and lowering indices.

Discussion

Let be a pseudo-Riemannian manifold. Suppose is a moving tangent frame for the tangent bundle with, as dual frame, the moving coframe . Then, locally, we may express the pseudo-Riemannian metric as .
Given a vector field , we define its flat by
This is referred to as "lowering an index". Using the traditional diamond bracket notation for the inner product defined by, we obtain the somewhat more transparent relation
for any vector fields and.
In the same way, given a covector field , we define its sharp by
where are the components of the inverse metric tensor. Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads
for any covector field and any vector field.
Through this construction, we have two mutually inverse isomorphisms
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
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

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:
defined by
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:

Trace of a tensor through a metric tensor

Given a type tensor field, we define the trace of through the metric tensor by
Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.

Citations