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 {∂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 dxidxj (where we employ the Einstein summation convention). Given a vector field X = Xii we define its flat by

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

for all vectors X and Y.

Alternatively, given a covector field ω = ωidxi we define its sharp by

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

for ω an arbitrary covector and Y an arbitrary vector.

Through this construction we have two inverse isomorphisms

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

It must be stated which index is to be raised or lowered. For instance, consider the (0, 2) tensor field X = Xij dxidxj. Raising the second index, we get the (1, 1) tensor field

Trace of a tensor through a metric

Given a (0, 2) tensor field X = Xij dxidxj, we define the trace of X through the metric g by

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