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. The term musical refers to the use of the symbols ♭ and ♯.^[1]

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 ${dx i}$ . 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^{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 $ω = ω i dx i$ we define its sharp by

\omega ^{\sharp }:=g^{ij}\omega _{i}\partial _{j}=\omega ^{j}\partial _{j}

where $g ij$ are the elements of the inverse matrix to $g ij$ . 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 $T p M$ 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 (0, 2) 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 (0, 2) 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.

References

↑ http://mathoverflow.net/questions/69074/the-origin-of-the-musical-isomorphisms

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Musical isomorphism

Discussion

Trace of a tensor through a metric

See also

References