: Mathematics: musical isomorphism

geometry > differential geometry > Riemannian geometry > musical isomorphism

geometry > differential geometry > symplectic geometry > musical isomorphism

category theory > morphism > isomorphism > musical isomorphism

algebra > abstract algebra > morphism > isomorphism > musical isomorphism

musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle ${\\displaystyle \\mathrm {T} M}$ and the cotangent bundle ${\\displaystyle \\mathrm {T} ^{*}M}$ 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 ${\\displaystyle \ lat }$ (flat) and ${\\displaystyle \\sharp }$ (sharp).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Musical_isomorphism)

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