User:ReiVaX/Sandbox

In mathematics, the musical isomorphism is an isomorphism between the tangent bundle $$TM$$ and the cotangent bundle $$T^*M$$ of a Riemannian manifold given by its metric.

Introduction
A metric g on a Riemannian manifold M is a tensor field $$g \in \mathcal{T}_2(M)$$. If we fix one parameter as a vector $$v_p \in T_p M$$, we have an isomorphism of vector spaces:


 * $$\hat{g}_p : T_p M \longrightarrow T^*_p M$$
 * $$\hat{g}_p(v_p) = g(v_p,-)$$
 * $$ < \hat{g}_p(v_p),\omega_p > = g_p(v_p,\omega_p)$$

And globally,

$$\hat{g} : TM \longrightarrow T^*M$$ is a diffeomorphism.

Motivation of the name
The isomorphism $$\hat{g}$$ and its inverse $$\hat{g}^{-1}$$ are called musical isomorphisms because they move up and down the indexes of the vectors. For instance, a vector of TM is written as $$\alpha^i \frac{\partial}{\partial x}$$ and a covector as $$\alpha_i dx^i$$, so the index i is moved up and down in $$\alpha$$ just as the symbols sharp ($$\sharp$$) and flat ($$\flat$$) move up and down the pitch of a tone.

Gradient
The musical isomorphisms can be used to define the gradient of a smooth function over a manifold M as follows:


 * $$\mathrm{grad}\;f=\hat{g}^{-1} \circ df = (df)^{\sharp}$$