Musical isomorphism

Musical isomorphism

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 TM with dual coframe {dxi}. 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^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

X^\flat (Y) = \langle X, Y \rangle

for all vectors X and Y.

Alternatively, given a covector field  \omega=\omega_i \, dx^i we define its sharp by

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

where gij are the elements of the inverse matrix to gij. 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 TpM 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.

See also


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