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Concept
Musical isomorphism
Summary
In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle 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) and (sharp).
In the notation of Ricci calculus, it is also known as raising and lowering indices.
In linear algebra, a finite-dimensional vector space is isomorphic to its dual space but not canonically isomorphic to it. On the other hand a finite-dimensional vector space endowed with a non-degenerate bilinear form , is canonically isomorphic to its dual, the isomorphism being given by:
An example is where is a Euclidean space, and is its inner product.
Musical isomorphisms are the global version of this isomorphism and its inverse, for the tangent bundle and cotangent bundle of a (pseudo-)Riemannian manifold . They are isomorphisms of vector bundles which are at any point the above isomorphism applied to the (pseudo-)Euclidean space (the tangent space of M at point p) endowed with the inner product . More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.
Because every paracompact manifold can be endowed with a Riemannian metric, the musical isomorphisms allow to show that on those spaces a vector bundle is always isomorphic to its dual (but not canonically unless a (pseudo-)Riemannian metric has been associated with the manifold).
Let (M, g) be a pseudo-Riemannian manifold. Suppose {e_i} is a moving tangent frame (see also smooth frame) for the tangent bundle TM with, as dual frame (see also dual basis), the moving coframe (a moving tangent frame for the cotangent bundle ; see also coframe) {e^i}. Then, locally, we may express the pseudo-Riemannian metric (which is a 2-covariant tensor field that is symmetric and nondegenerate) as g = g_ij e^i ⊗ e^j (where we employ the Einstein summation convention).
Official source
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