Musical isomorphism

Isomorphism between the tangent and cotangent bundles of a manifold.

In mathematics—more specifically, in differential geometry—the musical isomorphism is an isomorphism between the tangent bundle and the cotangent bundle 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 (flat) and (sharp).

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