In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the musical isomorphism (or canonical isomorphism) is an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 between the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

 TM and the cotangent bundle

Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...

 T^*M of a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 given by its metric. There are similar isomorphisms on symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

s.

It is also known as raising and lowering indices

Raising and lowering indices

In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a type tensor to a tensor or to a tensor...

.

Discussion

Let be a Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

. Suppose is a local frame for the tangent bundle with dual coframe . Then, locally, we may express the Riemannian metric (which is a 2-covariant tensor field which is symmetric and positive-definite) as (where we employ the Einstein summation convention). Given a vector field we define its flat by
This is referred to as 'lowering an index'. Using the traditional diamond bracket notation for inner product defined by g, we obtain the somewhat more transparent relation
for all vectors X and Y.

Alternatively, given a covector field we define its sharp by
where are the elements of the inverse matrix to . Taking the sharp of a covector field is referred to as 'raising an index'.

Through this construction we have two inverse isomorphisms and . These are isomorphisms of vector bundles and hence we have, for each , inverse vector space isomorphisms between and .

The musical isomorphisms may also be extended to the bundles and . It must be stated which index is to be raised or lowered. For instance, consider the (2,0) tensor field . Raising the second index, we get the (1,1) tensor field

Trace of a tensor through a metric

Given a (2,0) tensor field we define the trace of through the metric by

Observe that the definition of trace is independent of the choice of index we raise since the metric tensor is symmetric.

See also

• Duality (mathematics)
  Duality (mathematics)
  In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
  
  
• Raising and lowering indices
  Raising and lowering indices
  In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a type tensor to a tensor or to a tensor...
  
  
• Bilinear products and dual spaces
• Vector bundle
  Vector bundle
  In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
  
  
• Flat (music) and Sharp (music)
  Sharp (music)
  In music, sharp, dièse , or diesis means higher in pitch and the sharp symbol raises a note by a half tone. Intonation may be flat, sharp, or both, successively or simultaneously...
  
  about the signs and