Musical isomorphism: Information from Answers.com

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

Let $(M, g)$ be a Riemannian manifold. Suppose $\{\partial_i\}$ is a local frame for the tangent bundle $T M$ with dual coframe ${d x i}$ . Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as $g=g_{ij}dx^i \otimes dx^j$ (where we employ the Einstein Summation Convention). Given a vector field $X=X^i \partial_i$ we define its flat by

$X^{\flat}:=g_{ij}X^idx^j=:X_jdx^j$ .

This is referred to as 'lowering an index'. Alternatively, given a covector field $ω = ω i d x i$ we definite its sharp by

$\omega^{\sharp}:=g^{ij}\omega_i \partial_j$

where $g i j$ are the elements of the inverse matrix to $g i j$ . Taking the sharp of a covector field is referred to as 'raising an index'.

Through this construction we have two inverse isomorphisms $\flat:TM \to T^*M$ and $\sharp:T^*M \to TM$ . These are isomorphisms of vector bundles and hence we have, for each $p \in M$ , inverse vector space isomorphisms between $T p M$ and $T^*_pM$ .

The musical isomorphisms may also be extended to the bundles $\bigotimes ^k TM$ and $\bigotimes ^k T^*M$ . It must be stated which index is to be raised or lowered. For instance, consider the (2,0) tensor field $X=X_{ij}dx^i \otimes dx^j$ . Raising the second index, we get the (1,1) tensor field $X^{\sharp} = g^{jk}X_{ij}dx^i \otimes \partial _k$ .

Trace of a tensor through a metric

Given a (2,0) tensor field $X=X_{ij}dx^i \otimes dx^j$ we define the trace of $X$ through the metric $g$ by

$\operatorname{tr}_g(X):=\operatorname{tr}(X^\sharp)=\operatorname{tr}(g^{jk}X_{ij})=g^{ji}X_{ij}=g^{ij}X_{ij}$ .

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

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Musical isomorphism

Discussion

Trace of a tensor through a metric

See also