Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle ${\displaystyle \mathrm {T} M}$ and the cotangent bundle ${\displaystyle \mathrm {T} ^{*}M}$ of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols ${\displaystyle \flat }$ (flat) and ${\displaystyle \sharp }$ (sharp).

In the notation of Ricci calculus, it is also known as raising and lowering indices.