structural:scholar_edition, quality:correction, linguistic:translation
Musical isomorphism

In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its dual given by v ↦ ⟨ v, ⋅ ⟩ {\displaystyle v\mapsto \langle v,\cdot \rangle }, a Riemannian metric induces an isomorphism of bundles between the tangent bundle and the cotangent bundle. Namely, if g {\displaystyle g} is a Riemannian metric, then

is a isomorphism of smooth vector bundles from the tangent bundle T M {\displaystyle TM} to the cotangent bundle T ∗ M {\displaystyle T^{*}M}.