Musical isomorphism
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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, it is also known as raising and lowering indices.