Musical isomorphism

In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle $\mathrmM$ and the cotangent bundle $\mathrm^* M$ 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 $\flat$ (flat) and $\sharp$ (sharp).

In covariant and contravariant notation, it is also known as raising and lowering indices.

Motivation

In linear algebra, a finite-dimensional vector space is isomorphic to its dual but not canonically isomorphic to it. On the other hand a Euclidean vector space, i.e., a finite-dimensional vector space $E$ endowed with an inner product $\langle\cdot,\cdot\rangle$, is canonically isomorphic to its dual, the isomorphism being given by:

\left\} is a moving tangent frame (see also smooth frame) for the ''tangent bundle'' with, as dual frame (see also dual basis), the moving coframe (a ''moving tangent frame'' for the ''cotangent bundle'' $\mathrm^*M$; see also coframe) . Then, locally, we may express the