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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.
Let be a Riemannian manifold. Suppose is a local frame for the tangent bundle with dual coframe . Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as (where we employ the Einstein summation convention). Given a vector field we define its flat by
This is referred to as 'lowering an index'. Using the traditional diamond bracket notation for inner product defined by g, we obtain the somewhat more transparent relation
for all vectors X and Y.
Alternatively, given a covector field we define its sharp by
where are the elements of the inverse matrix to . Taking the sharp of a covector field is referred to as 'raising an index'.
Through this construction we have two inverse isomorphisms and . These are isomorphisms of vector bundles and hence we have, for each , inverse vector space isomorphisms between and .
The musical isomorphisms may also be extended to the bundles and . It must be stated which index is to be raised or lowered. For instance, consider the (2,0) tensor field . Raising the second index, we get the (1,1) tensor field
Given a (2,0) tensor field we define the trace of through the metric by
Observe that the definition of trace is independent of the choice of index we raise since the metric tensor is symmetric.