Musical isomorphism

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Musical isomorphism" – news · newspapers · books · scholar · JSTOR (April 2015) (Learn how and when to remove this message)

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).^[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.