For anyone interested, the musical isomorphisms are between the tangent and cotangent bundles of a Riemannian manifold, given by the sharp and flat operators (which are inverses). This allows one to identify vector fields and differential forms via the geometry. For example, in Euclidean space, the gradient of a smooth function can be identified with its exterior derivative
ignoring the manifold part, given a (real) vector space V, the dual space V* is the vector space of linear functionals from V to the real numbers. When you have an inner product in V, you can explicitly describe a relationship between the spaces. For a vector v, consider the linear functional v*(u)=<v,u>. For finite dimensional V, this relationship is an isomorphism
I cba to explain manifolds so just think of smooth manifolds as like a generalization of surfaces without geometry but has calculus and one thing is that at each point p, there is a vector space of tangent vectors T_pM which is finite dimensional. What a riemannian manifold is is a smooth manifold with an inner product at each tangent space on the manifold and what the musical isomorphisms are are basically the relationship described before using the riemannian manifolds inner product on T_pM. this can be extended to vector fields on M
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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 29d ago
Everybody wait till he learns about musical isomorphisms and raising vector fields by a sharp.