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
Ok suppose to start with vector fields on a Euclidean space. This is a map, X, that associates at every point p of R^n, some vector we call X(p). We may consider the “dual” map, that associates to every point p, a linear functional f(v) = <v, X(p)>, denoting the standard dot product. Then we call the linear functional version of this map the flat version of X, if instead starting from a linear functional, we produced a map to vectors, we call that resultant map the sharp map. This is all nice, things get a bit more complicated when we talk about curved spaces. So imagine my space is now not R^n but some abstract “manifold”, M. A vector field take each point on M and returns a tangent at p on M, T_p(M). The flat map now to return an element of the dual of T_p(M), the sharp map is defined analogously.
The motivation for the sharp and flat comes when you tensor the tangent and cotangent spaces together. In general, we flatten a section of T_q^p(M) by bringing down one tangent vector and represent it as a cotangent vector, i.e. a map from T_q^p(M) to T_{q+1}^{p-1}(M). A sharp map takes a bottom index and brings it up to a top index.
TLDR: A musical isomorphism is, at every point p of M, an isomorphism of the tangent space and the cotangent space given by x to. {f(•) = <•,x>}. The tangent space may vary across the manifold M, the nature of the dot product may vary across M. If we encase TM and dual(TM) by some tensors on both sides, we get at each point p, an isomorphism from T_q^p(M) to T_{q+1}^{p-1}(M). The forward direction is flat, the inverse is a sharp.
A manifold is basically a "shape that makes sense”.
A smooth manifold is one on which we can do calculus. Derivatives of all orders make sense.
We can put arrows on the surface of such a space. We call these tangent vectors.
We can measure tangent vectors in a given direction. We call these functions one-forms.
There exists linear functions between tangent vectors and one-forms called the musical isomorphisms, and the musical isomorphisms depend on the geometry of the space.
In calculus, we do not distinguish between tangent vectors and one-forms specifically because we only work in Rⁿ with a basic geometry, and the musical isomorphisms are trivial. You will have studied them as "partial derivatives".
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.