Eli5: musical isomorphism (or canonical isomorphism) in differential geometry

Question

Big word make my bwain hurt!!!!😢

In: Mathematics

Answer 1

Ok there’s no real way to explain this to a 5 year old, but I can at least give a simple explanation. We have a vector space V (the tangent space). V* is the set of dual vectors to V. What’s a dual vector? Just a (linear) function f : V to R. In Riemannian geometry we have the metric g which is a (bilinear) function on V x V. If you give me a vector v then I can produce a dual vector f by defining f(w) = g(v,w). This gives me a recipe to turn vectors into dual vectors and this turns out to be an isomorphism.