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.

Discussion[link]

Let Failed to parse (Missing texvc executable; please see math/README to configure.): (M,g)

be a Riemannian manifold.  Suppose Failed to parse (Missing texvc executable; please see math/README to configure.):  \{\partial_i\} 
is a local frame for the tangent bundle Failed to parse (Missing texvc executable; please see math/README to configure.):  TM 
with dual coframe Failed to parse (Missing texvc executable; please see math/README to configure.):  \{dx^i\} 

. Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as Failed to parse (Missing texvc executable; please see math/README to configure.): g=g_{ij}\,dx^i \otimes dx^j

(where we employ the Einstein summation convention).  Given a vector field Failed to parse (Missing texvc executable; please see math/README to configure.):  X=X^i \partial_i 
we define its flat by 

Failed to parse (Missing texvc executable; please see math/README to configure.): X^\flat := g_{ij} X^i \, dx^j=:X_j \, dx^j.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): X^\flat (Y) = \langle X, Y \rangle

for all vectors X and Y.

Alternatively, given a covector field Failed to parse (Missing texvc executable; please see math/README to configure.): \omega=\omega_i \, dx^i

we define its sharp by

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega^\sharp :=g^{ij} \omega_i \partial_j

where Failed to parse (Missing texvc executable; please see math/README to configure.): g^{ij}

are the elements of the inverse matrix to Failed to parse (Missing texvc executable; please see math/README to configure.):  g_{ij} 

. Taking the sharp of a covector field is referred to as 'raising an index'.

Through this construction we have two inverse isomorphisms Failed to parse (Missing texvc executable; please see math/README to configure.): \flat:TM \to T^*M

and Failed to parse (Missing texvc executable; please see math/README to configure.):  \sharp:T^*M \to TM 

. These are isomorphisms of vector bundles and hence we have, for each Failed to parse (Missing texvc executable; please see math/README to configure.): p \in M , inverse vector space isomorphisms between Failed to parse (Missing texvc executable; please see math/README to configure.): T_pM

and Failed to parse (Missing texvc executable; please see math/README to configure.):  T^*_pM 

.

The musical isomorphisms may also be extended to the bundles Failed to parse (Missing texvc executable; please see math/README to configure.): \bigotimes ^k TM

and Failed to parse (Missing texvc executable; please see math/README to configure.):  \bigotimes ^k T^*M 

. It must be stated which index is to be raised or lowered. For instance, consider the (2,0) tensor field Failed to parse (Missing texvc executable; please see math/README to configure.): X=X_{ij} \, dx^i \otimes dx^j . Raising the second index, we get the (1,1) tensor field Failed to parse (Missing texvc executable; please see math/README to configure.): X^\sharp = g^{jk}X_{ij} \, dx^i \otimes \partial _k.

Trace of a tensor through a metric[link]

Given a (2,0) tensor field Failed to parse (Missing texvc executable; please see math/README to configure.): X=X_{ij} \, dx^i \otimes dx^j

we define the trace of Failed to parse (Missing texvc executable; please see math/README to configure.):  X 
through the metric Failed to parse (Missing texvc executable; please see math/README to configure.):  g 
by

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{tr}_g(X):=\operatorname{tr}(X^\sharp)=\operatorname{tr}(g^{jk}X_{ij}) = g^{ji}X_{ij} = g^{ij}X_{ij}.

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

See also[link]

• Duality (mathematics)
• Raising and lowering indices
• Bilinear products and dual spaces
• Vector bundle
• Flat (music) and Sharp (music) about the signs ♭ and ♯