Musical isomorphism

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In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle TM 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[edit]

Let (M, g) be a Riemannian manifold. Suppose {∂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 = gij dxidx j (where we employ the Einstein summation convention). Given a vector field X = X ii 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 ω = ωidxi we define its sharp by

\omega^\sharp :=g^{ij} \omega_i \partial_j = \omega^j \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". In inner product notation, this reads

\left \langle \omega^\sharp, Y \right \rangle = \omega(Y),

for ω an arbitrary covector and Y an arbitrary vector.

Through this construction we have two inverse isomorphisms

 \flat:TM \to T^*M, \qquad \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 ∗
p
M
.

The musical isomorphisms may also be extended to the bundles

 \bigotimes ^k TM, \qquad \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 = Xij dxidx 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[edit]

Given a (0, 2) tensor field X = Xij dxidx 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[edit]