Musical isomorphism
Isomorphism between the tangent and cotangent bundles of a manifold. / From Wikipedia, the free encyclopedia
In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle
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
(flat) and
(sharp).[1][2]
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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.