Musical isomorphism

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It has been suggested that this article be merged with Raising and lowering indices. (Discuss) Proposed since July 2024.

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 Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the musical notation symbols ${\displaystyle \flat }$ (flat) and ${\displaystyle \sharp }$ (sharp).^[1][2]

In the notation of Ricci calculus, the idea is expressed as the raising and lowering of indices.

In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points, and so, for these cases, is technically only a homomorphism.