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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''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 eve ...
''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
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 may ...
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 eve ...
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 , (over the same space ) is a vector bundle, denoted by , whose fiber over each point is the tensor product of vector spaces .To construct a tensor-product bundle over a paracompact ...
of ''E'' with Λ''p''(''T''''M''), the ''p''-th
exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
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 m ...
of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambda^pT^*M) = \Gamma(E) \otimes_ \Gamma(\Lambda^pT^*M) = \Gamma(E) \otimes_ \Omega^p(M), 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 produ ...
over the
ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
Ω0(''M'') of smooth R-valued functions on ''M'' (see the seventh example
here Here may refer to: Music * ''Here'' (Adrian Belew album), 1994 * ''Here'' (Alicia Keys album), 2016 * ''Here'' (Cal Tjader album), 1979 * ''Here'' (Edward Sharpe album), 2012 * ''Here'' (Idina Menzel album), 2004 * ''Here'' (Merzbow album), ...
). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is, :\Omega^0(M,E) = \Gamma(E).\, Equivalently, an ''E''-valued differential form can be defined as a bundle morphism :TM\otimes\cdots\otimes TM \to E 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
. 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 ( ''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 :\Omega^p(M) \otimes_\mathbb V \to \Omega^p(M,V), 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: ...
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 (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...
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 : (\varphi^*\omega)_x(v_1,\cdots, v_p) = \omega_(\mathrm d\varphi_x(v_1),\cdots,\mathrm d\varphi_x(v_p)).


Wedge product

Just as for ordinary differential forms, one can define a
wedge product A wedge is a triangular shaped tool, 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 converting a fo ...
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: :\wedge : \Omega^p(M,E_1) \times \Omega^q(M,E_2) \to \Omega^(M,E_1\otimes E_2). 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 V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
: :(\omega\wedge\eta)(v_1,\cdots,v_) = \frac\sum_\sgn(\sigma)\omega(v_,\cdots,v_)\otimes \eta(v_,\cdots,v_). 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''). In terms of local frames and for ''E''1 and ''E''2 respectively, the wedge product of an ''E''1-valued ''p''-form ''ω'' = ''ω''''α'' ''e''''α'', and an ''E''2-valued ''q''-form ''η'' = ''η''''β'' ''l''''β'' is :\omega \wedge \eta = \sum_ (\omega^\alpha \wedge \eta^\beta) (e_\alpha \otimes l_\beta), where ''ω''''α'' ∧ ''η''''β'' is the ordinary wedge product of \mathbb-valued forms. For ω ∈ Ω''p''(''M'') and η ∈ Ω''q''(''M'', ''E'') one has the usual commutativity relation: :\omega\wedge\eta = (-1)^\eta\wedge\omega. 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 ...
(i.e. a bundle of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
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. Perhaps most familiar as a pr ...
,
associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
s then, with this modified wedge product, the set of all ''E''-valued differential forms :\Omega(M,E) = \bigoplus_^\Omega^p(M,E) 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 , ''y'', ...
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 re ...
on the space of ''V''-valued forms. This is just the ordinary exterior derivative acting component-wise relative to any basis of ''V''. Explicitly, if is a basis for ''V'' then the differential of a ''V''-valued ''p''-form ω = ωα''e''α is given by :d\omega = (d\omega^\alpha)e_\alpha.\, The exterior derivative on ''V''-valued forms is completely characterized by the usual relations: :\begin &d(\omega+\eta) = d\omega + d\eta\\ &d(\omega\wedge\eta) = d\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta\qquad(p=\deg\omega)\\ &d(d\omega) = 0. \end 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 ( ''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 retur ...
taking sections of ''E'' to ''E''-valued one forms: :\nabla : \Omega^0(M,E) \to \Omega^1(M,E). If ''E'' is equipped with a connection ∇ then there is a unique covariant exterior derivative :d_\nabla: \Omega^p(M,E) \to \Omega^(M,E) extending ∇. The covariant exterior derivative is characterized by
linearity In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
and the equation :d_\nabla(\omega\wedge\eta) = d_\nabla\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta 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 that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
).


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)
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(E) associated with 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 naturally on ...
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: ...
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 (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
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'' that is equivariant and horizontal in the sense that #(R_g)^*\omega = \rho(g^)\omega\, for all ''g'' ∈ ''G'', and #\omega(v_1, \ldots, v_p) = 0 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. It is sometimes said that a s ...
. 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 \m ...
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 ''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 \overline on ''M'' with values in ''E'', define φ on ''P'' fiberwise by, say at ''u'', :\phi = u^\pi^*\overline where ''u'' is viewed as a linear isomorphism V \overset\to E_ = (\pi^*E)_u, v \mapsto , v/math>. φ is then a tensorial form of type ρ. Conversely, given a tensorial form φ of type ρ, the same formula defines an ''E''-valued form \overline on ''M'' (cf. the
Chern–Weil homomorphism In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing ...
.) In particular, there is a natural isomorphism of vector spaces :\Gamma(M, E) \simeq \, \, \overline \leftrightarrow f. Example: Let ''E'' be the tangent bundle of ''M''. Then identity bundle map id''E'': ''E'' →''E'' is an ''E''-valued one form on ''M''. The
tautological one-form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus pro ...
is a unique one-form on the frame bundle of ''E'' that corresponds to id''E''. Denoted by θ, it is a tensorial form of standard type. Now, suppose there is a connection on ''P'' so that there is an exterior covariant differentiation ''D'' on (various) vector-valued forms on ''P''. Through the above correspondence, ''D'' also acts on ''E''-valued forms: define ∇ by :\nabla \overline = \overline. In particular for zero-forms, :\nabla: \Gamma(M, E) \to \Gamma(M, T^*M \otimes E). This is exactly the
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
for the connection on the vector bundle ''E''.


Examples

Siegel modular form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s arise as vector-valued differential forms on Siegel modular varieties.


Notes

{{reflist


References

* Shoshichi Kobayashi and Katsumi Nomizu (1963)
Foundations of Differential Geometry ''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publis ...
, Vol. 1,
Wiley Interscience John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclope ...
. Differential forms Vector bundles