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 ...
, a vector-valued differential form on a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''M'' is a
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
on ''M'' with values in a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V''. More generally, it is a differential form with values in some
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 ...
''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms.
An important case of vector-valued differential forms are
Lie algebra-valued forms
A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be inter ...
. (A
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
is an example of such a form.)
Definition
Let ''M'' be a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
and ''E'' → ''M'' be a smooth
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 ...
over ''M''. We denote the space of
smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the
tensor product bundle
In differential geometry, the tensor product of vector bundles ''E'', ''F'' (over same space X) is a vector bundle, denoted by ''E'' ⊗ ''F'', whose fiber over a point x \in X is the tensor product of vector spaces ''E'x'' ⊗ ''F'x''.To co ...
of ''E'' with Λ
''p''(''T''
∗''M''), the ''p''-th
exterior power
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 a ...
of 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 ''M''. The space of such forms is denoted by
:
Because Γ is a
strong monoidal functor
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ...
,
this can also be interpreted as
:
where the latter two tensor products are the
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produc ...
over the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
Ω
0(''M'') of smooth R-valued functions on ''M'' (see the seventh example
here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is,
:
Equivalently, an ''E''-valued differential form can be defined as a
bundle morphism
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. T ...
:
which is totally
skew-symmetric.
Let ''V'' be a fixed
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. A ''V''-valued differential form of degree ''p'' is a differential form of degree ''p'' with values in the
trivial bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
''M'' × ''V''. The space of such forms is denoted Ω
''p''(''M'', ''V''). When ''V'' = R one recovers the definition of an ordinary differential form. If ''V'' is finite-dimensional, then one can show that the natural homomorphism
:
where the first tensor product is of vector spaces over R, is an isomorphism.
Operations on vector-valued forms
Pullback
One can define the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
of vector-valued forms by
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
s just as for ordinary forms. The pullback of an ''E''-valued form on ''N'' by a smooth map φ : ''M'' → ''N'' is an (φ*''E'')-valued form on ''M'', where φ*''E'' is the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
of ''E'' by φ.
The formula is given just as in the ordinary case. For any ''E''-valued ''p''-form ω on ''N'' the pullback φ*ω is given by
:
Wedge product
Just as for ordinary differential forms, one can define a
wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...
of vector-valued forms. The wedge product of an ''E''
1-valued ''p''-form with an ''E''
2-valued ''q''-form is naturally an (''E''
1⊗''E''
2)-valued (''p''+''q'')-form:
:
The definition is just as for ordinary forms with the exception that real multiplication is replaced with the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
:
:
In particular, the wedge product of an ordinary (R-valued) ''p''-form with an ''E''-valued ''q''-form is naturally an ''E''-valued (''p''+''q'')-form (since the tensor product of ''E'' with the trivial bundle ''M'' × R is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to ''E''). For ω ∈ Ω
''p''(''M'') and η ∈ Ω
''q''(''M'', ''E'') one has the usual commutativity relation:
:
In general, the wedge product of two ''E''-valued forms is ''not'' another ''E''-valued form, but rather an (''E''⊗''E'')-valued form. However, if ''E'' is an
algebra bundle In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every al ...
(i.e. a bundle of
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
s rather than just vector spaces) one can compose with multiplication in ''E'' to obtain an ''E''-valued form. If ''E'' is a bundle of
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
,
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
s then, with this modified wedge product, the set of all ''E''-valued differential forms
:
becomes a
graded-commutative In Abstract algebra, algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy
:xy = (-1)^ yx,
where , ''x'' , and , ...
associative algebra. If the fibers of ''E'' are not commutative then Ω(''M'',''E'') will not be graded-commutative.
Exterior derivative
For any vector space ''V'' there is a natural
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
on the space of ''V''-valued forms. This is just the ordinary exterior derivative acting component-wise relative to any
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of ''V''. Explicitly, if is a basis for ''V'' then the differential of a ''V''-valued ''p''-form ω = ω
α''e''
α is given by
:
The exterior derivative on ''V''-valued forms is completely characterized by the usual relations:
:
More generally, the above remarks apply to ''E''-valued forms where ''E'' is any
flat vector bundle over ''M'' (i.e. a vector bundle whose transition functions are constant). The exterior derivative is defined as above on any
local trivialization
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
of ''E''.
If ''E'' is not flat then there is no natural notion of an exterior derivative acting on ''E''-valued forms. What is needed is a choice of
connection on ''E''. A connection on ''E'' is a linear
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
taking sections of ''E'' to ''E''-valued one forms:
:
If ''E'' is equipped with a connection ∇ then there is a unique
covariant exterior derivative
:
extending ∇. The covariant exterior derivative is characterized by
linearity
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
and the equation
:
where ω is a ''E''-valued ''p''-form and η is an ordinary ''q''-form. In general, one need not have ''d''
∇2 = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
).
Basic or tensorial forms on principal bundles
Let ''E'' → ''M'' be a smooth vector bundle of rank ''k'' over ''M'' and let ''π'' : F(''E'') → ''M'' be the (
associated Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
* Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associati ...
)
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natur ...
of ''E'', which is a
principal GL
''k''(R) bundle over ''M''. The
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
of ''E'' by ''π'' is canonically isomorphic to F(''E'') ×
ρ R
''k'' via the inverse of
'u'', ''v''→''u''(''v''), where ρ is the standard representation. Therefore, the pullback by ''π'' of an ''E''-valued form on ''M'' determines an R
''k''-valued form on F(''E''). It is not hard to check that this pulled back form is
right-equivariant with respect to the natural
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of GL
''k''(R) on F(''E'') × R
''k'' and vanishes on
vertical vectors (tangent vectors to F(''E'') which lie in the kernel of d''π''). Such vector-valued forms on F(''E'') are important enough to warrant special terminology: they are called ''basic'' or ''tensorial forms'' on F(''E'').
Let ''π'' : ''P'' → ''M'' be a (smooth)
principal ''G''-bundle and let ''V'' be a fixed vector space together with a
representation ''ρ'' : ''G'' → GL(''V''). A basic or tensorial form on ''P'' of type ρ is a ''V''-valued form ω on ''P'' which is equivariant and horizontal in the sense that
#
for all ''g'' ∈ ''G'', and
#
whenever at least one of the ''v''
''i'' are vertical (i.e., d''π''(''v''
''i'') = 0).
Here ''R''
''g'' denotes the right action of ''G'' on ''P'' for some ''g'' ∈ ''G''. Note that for 0-forms the second condition is
vacuously true
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
.
Example: If ρ is the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
of ''G'' on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
Ω satisfies both; hence Ω is a tensorial form of adjoint type. The "difference" of two connection forms is a tensorial form.
Given ''P'' and ''ρ'' as above one can construct the
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
''E'' = ''P'' ×
''ρ'' ''V''. Tensorial ''q''-forms on ''P'' are in a natural one-to-one correspondence with ''E''-valued ''q''-forms on ''M''. As in the case of the principal bundle F(''E'') above, given a ''q''-form
on ''M'' with values in ''E'', define φ on ''P'' fiberwise by, say at ''u'',
:
where ''u'' is viewed as a linear isomorphism