In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
—more specifically, in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
—the musical isomorphism (or canonical isomorphism) is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
between the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
and the
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. It may be described also as the dual bundle to the tangent bundle. This may ...
of a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
induced by its
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
. There are similar isomorphisms on
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s. The term ''musical'' refers to the use of the symbols
(flat) and
(sharp).
In the notation of
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
, it is also known as
raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Math ...
.
Motivation
In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, a
finite-dimensional vector space
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to di ...
is isomorphic to its
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
but not canonically isomorphic to it. On the other hand a finite-dimensional vector space
endowed with a non-degenerate
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
, is canonically isomorphic to its dual, the isomorphism being given by:
An example is where
is a Euclidean space, and
is its
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
.
Musical isomorphisms are the global version of this isomorphism and its inverse, for the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
and
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. It may be described also as the dual bundle to the tangent bundle. This may ...
of a pseudo-Riemannian manifold
. They are isomorphisms of
vector bundles
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
which are at any point
the above isomorphism applied to the (pseudo-)Euclidean space
(the tangent space of at point ) endowed with the inner product
. More generally, musical isomorphisms always exist between a vector bundle endowed with a
bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.
Definition
If ''M'' is a topological manifold ...
and its dual.
Because every
paracompact manifold
In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
can be endowed with a (pseudo-)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 it is a without a pseudo-Riemannian metric being associated with the manifold).
Discussion
Let be a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. Suppose is a
moving tangent frame (see also
smooth frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay t ...
) for the ''tangent bundle'' with, as
dual frame (see also
dual basis
In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with th ...
), the
moving coframe (a ''moving tangent frame'' for the ''cotangent bundle''
; see also
coframe
In mathematics, a coframe or coframe field on a smooth manifold M is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M, one has a natural map from v_k:\bigoplus^kT^*M\to\big ...
) . Then,
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
, we may express the
pseudo-Riemannian metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
(which is a -covariant
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
that is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
) as (where we employ the
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
).
Given a
vector field and denoting , we define its flat by
This is referred to as ''lowering an index''. Using angle bracket notation for the
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
defined by , we obtain the somewhat more transparent relation
for any vector fields and .
In the same way, given a
covector
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
field and denoting , we define its sharp by
where are the
components
Circuit Component may refer to:
•Are devices that perform functions when they are connected in a circuit.
In engineering, science, and technology Generic systems
* System components, an entity with discrete structure, such as an assem ...
of the
inverse metric tensor (given by the entries of the
inverse matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
to ). Taking the sharp of a covector field is referred to as ''raising an index''. In angle bracket notation, this reads
for any covector field and any vector field .
Through this construction, we have two mutually
inverse isomorphisms
These are isomorphisms of
vector bundles
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
and, hence, we have, for each in , mutually inverse vector space isomorphisms between and .
Extension to tensor products
The musical isomorphisms may also be extended to the bundles
Which index is to be raised or lowered must be indicated. For instance, consider the -tensor field . Raising the second index, we get the -tensor field
Extension to ''k''-vectors and ''k''-forms
In the context of
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
, an extension of the musical operators may be defined on and its dual , which with minor
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
may be denoted the same, and are again mutual inverses:
defined by
In this extension, in which maps ''p''-vectors to ''p''-covectors and maps ''p''-covectors to ''p''-vectors, all the indices of a
totally antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates Sign (mathematics), sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must gener ...
are simultaneously raised or lowered, and so no index need be indicated:
Trace of a tensor through a metric tensor
Given a type tensor field , we define the trace of through the metric tensor by
Observe that the definition of trace is independent of the choice of index to raise, since the metric tensor is symmetric.
See also
*
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
*
Raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Math ...
*
*
Hodge star operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
*
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
*
Flat (music)
In music, flat (Italian bemolle for "soft B") means "lower in pitch". Flat is the opposite of sharp, which is a raising of pitch. In musical notation, flat means "lower in pitch by one semitone (half step)", notated using the symbol which is deri ...
and
Sharp (music)
In music, sharp, dièse (from French), or diesis (from Greek) means, "higher in pitch". More specifically, in musical notation, sharp means "higher in pitch by one semitone (half step)". Sharp is the opposite of flat, which is a lowering of pit ...
about the signs and
Citations
References
*
*
*
{{Manifolds
Differential geometry
Riemannian geometry
Riemannian manifolds
Symplectic geometry