Musical isomorphism on arvau

Wikipedia / Musical isomorphism

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 Riemannian or pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. These isomorphisms are global versions of the canonical isomorphism between an inner product space and its dual. The term musical refers to the use of the musical notation symbols \flat (flat) and \sharp (sharp).

In the notation of Ricci calculus and mathematical physics, the idea is expressed as the raising and lowering of indices. Raising and lowering indices are a form of index manipulation in tensor expressions.

In certain specialized applications, such as on Poisson manifolds, the relationship may fail to be an isomorphism at singular points, and so, for these cases, is technically only a homomorphism.

Motivation

from its exterior derivative.

In linear algebra, a finite-dimensional vector space is isomorphic to its dual space (the space of linear functionals mapping the vector space to its base field), but not canonically. Given a fixed basis for this vector space, there is a natural way to go back and forth between vectors and linear forms: vectors are represented in the basis by column vectors, linear forms are represented in the basis by row vectors, and the identification is done by transposition.

endowed with a non-degenerate bilinear form is given by

is the dot product.

be the dual basis using Einstein summation notation:

gives a vector with components

. This process is called raising the index.

being inverses:

is the Kronecker delta or identity matrix.

and its inverse for the tangent bundle and cotangent bundle. They are canonical isomorphisms of vector bundles and its inverse

Because every smooth manifold can be (non-canonically) endowed with a Riemannian metric, the musical isomorphisms show that a vector bundle on a smooth manifold is (non-canonically) isomorphic to its dual.

Abstractly, the musical isomorphisms can be understood as a currying of the bilinear form.

Discussion

, its flat is the covector

, it defines a morphism of smooth vector bundles at each point, characterized by

is called the sharp.

The flat and sharp maps can be applied to vector fields and covector fields is a covector field,

and

In a moving frame

Suppose 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; see also coframe) . Then the pseudo-Riemannian metric, which is a 2-covariant tensor field, can be written locally in this coframe as using Einstein summation notation.

Given a vector field and denoting , its flat is

This is referred to as lowering an index, because the components of are written with an upper index are written with a lower index .

In the same way, given a covector field and denoting , its sharp is

where are the components of the inverse metric tensor (given by the entries of the inverse matrix to ). Taking the sharp of a covector field is referred to as raising an index.

Extension to tensor products

The musical isomorphisms may also be extended, for each , to an isomorphism between the bundle

tensors. Here can be positive or negative, so long as and .

. Which index is to be raised or lowered must be indicated.

For instance, consider the tensor . Raising the second index, we get the tensor

are given by

tensor , all indices are raised by:

tensor , all indices are lowered by:

, all lower indices are raised and all upper indices are lowered by

is not.

Extension to k-vectors and k-forms

In the context of exterior algebra, an extension of the musical operators may be defined on and its dual , and are again mutual inverses:

defined by

In this extension, in which maps k-vectors to k-covectors and maps k-covectors to k-vectors, all the indices of a totally antisymmetric tensor

This works not just for k-vectors in the context of linear algebra but also for k-forms in the context of a (pseudo-)Riemannian manifold:

Vector bundles with bundle metrics

More generally, musical isomorphisms always exist between a vector bundle endowed with a bundle metric and its dual.

Trace of a tensor

Given a tensor , we define the trace of through the metric tensor

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

tensor can be taken in a similar way, so long as one specifies which two distinct indices are to be traced. This process is also called contracting the two indices. For example, if tensor with tensor with components

Example computations

In Minkowski spacetime

The covariant 4-position is given by

with components:

(where ,, are the usual Cartesian coordinates) and the Minkowski metric tensor with metric signature (− + + +) is defined as

in components:

To raise the index, multiply by the tensor and contract:

then for :

and for :

So the index-raised contravariant 4-position is:

This operation is equivalent to the matrix multiplication

, we can write down their (pseudo-)inner product in two ways:

By lowering indices, we can write this expression as

In matrix notation, the first expression can be written as

In electromagnetism

For a (0,2) tensor, twice contracting with the inverse metric tensor and contracting in different indices raises each index:

Similarly, twice contracting with the metric tensor and contracting in different indices lowers each index:

Let's apply this to the theory of electromagnetism.

The contravariant electromagnetic tensor in the signature is given by

In components,

To obtain the covariant tensor , contract with the inverse metric tensor:

and since and , this reduces to

Now for , :

and by antisymmetry, for , :

then finally for , ;

The (covariant) lower indexed tensor is then: