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 ♯.

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 = g_ij dxⁱ ⊗ dx^j (where we employ the Einstein summation convention). Given a vector field X = Xⁱ∂_i 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 = ω_i dxⁱ we define its sharp by