Common
Musical isomorphism
Isomorphism between the tangent and cotangent bundles of a manifold
Musical isomorphism
In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of a Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. These isomorphisms are global versions of the canonical isomorphism between an inner product space and its dual. The term musical refers to the use of the musical notation symbols ♭ {\displaystyle \flat } (flat) and ♯ {\displaystyle \sharp } (sharp). In the notation of Ricci calculus and mathematical physics, the idea is expressed...
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