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(See Musical isomorphism).
The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism.
In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle TM and the cotangent bundle TM of a Riemannian manifold given by its metric.
Wörterbuch für Mathematik