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| In [[mathematics]], a '''vector-valued differential form''' on a [[manifold]] ''M'' is a [[differential form]] on ''M'' with values in a [[vector space]] ''V''. More generally, it is a differential form with values in some [[vector bundle]] ''E'' over ''M''. Ordinary differential forms can be viewed as '''R'''-valued differential forms. Vector-valued forms are natural objects in [[differential geometry]] and have numerous applications.
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| ==Formal definition==
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| Let ''M'' be a [[smooth manifold]] and ''E'' → ''M'' be a smooth [[vector bundle]] over ''M''. We denote the space of [[section (fiber bundle)|smooth section]]s of a bundle ''E'' by Γ(''E''). A '''''E''-valued differential form''' of degree ''p'' is a smooth section of the [[tensor product]] bundle of ''E'' with Λ<sup>''p''</sup>(''T''*''M''), the ''p''-th [[exterior power]] of the [[cotangent bundle]] of ''M''. The space of such forms is denoted by | |
| :<math>\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M).</math>
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| Because Γ is a [[monoidal functor]], this can also be interpreted as
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| :<math>\Gamma(E\otimes\Lambda^pT^*M) = \Gamma(E) \otimes_{\Omega^0(M)} \Gamma(\Lambda^pT^*M) = \Gamma(E) \otimes_{\Omega^0(M)} \Omega^p(M),</math>
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| where the latter two tensor products are the [[tensor product of modules]] over the [[ring_(mathematics)|ring]] Ω<sup>0</sup>(''M'') of smooth '''R'''-valued functions on ''M'' (see the fifth example [[module_(mathematics)#Examples|here]]). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is,
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| :<math>\Omega^0(M,E) = \Gamma(E).\,</math>
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| Equivalently, a ''E''-valued differential form can be defined as a [[vector bundle morphism|bundle morphism]]
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| :<math>TM\otimes\cdots\otimes TM \to E</math> | |
| which is totally [[skew-symmetric]].
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| Let ''V'' be a fixed [[vector space]]. A '''''V''-valued differential form''' of degree ''p'' is a differential form of degree ''p'' with values in the [[trivial bundle]] ''M'' × ''V''. The space of such forms is denoted Ω<sup>''p''</sup>(''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
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| :<math>\Omega^p(M) \otimes_\mathbb{R} V \to \Omega^p(M,V),</math>
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| where the first tensor product is of vector spaces over '''R''', is an isomorphism. One can verify this for ''p''=0 by turning a basis for ''V'' into a set of constant functions to ''V'', which allows the construction of an inverse to the above homomorphism. The general case can be proved by noting that
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| :<math>\Omega^p(M, V) = \Omega^0(M, V) \otimes_{\Omega^0(M)} \Omega^p(M),</math>
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| and that because <math>\mathbb{R}</math> is a sub-ring of Ω<sup>0</sup>(''M'') via the constant functions,
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| :<math>\Omega^0(M, V) \otimes_{\Omega^0(M)} \Omega^p(M) = (V \otimes_\mathbb{R} \Omega^0(M)) \otimes_{\Omega^0(M)} \Omega^p(M) = V \otimes_\mathbb{R} (\Omega^0(M) \otimes_{\Omega^0(M)} \Omega^p(M)) = V \otimes_\mathbb{R} \Omega^p(M).</math>
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| ==Operations on vector-valued forms==
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| ===Pullback===
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| One can define the [[pullback (differential geometry)|pullback]] of vector-valued forms by [[smooth map]]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]] of ''E'' by φ.
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| 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
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| :<math> (\varphi^*\omega)_x(v_1,\cdots, v_p) = \omega_{\varphi(x)}(\mathrm d\varphi_x(v_1),\cdots,\mathrm d\varphi_x(v_p)).</math>
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| ===Wedge product===
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| Just as for ordinary differential forms, one can define a [[wedge product]] of vector-valued forms. The wedge product of an ''E''<sub>1</sub>-valued ''p''-form with an ''E''<sub>2</sub>-valued ''q''-form is naturally an (''E''<sub>1</sub>{{unicode|⊗}}''E''<sub>2</sub>)-valued (''p''+''q'')-form:
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| :<math>\wedge : \Omega^p(M,E_1) \times \Omega^q(M,E_2) \to \Omega^{p+q}(M,E_1\otimes E_2).</math>
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| The definition is just as for ordinary forms with the exception that real multiplication is replaced with the [[tensor product]]:
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| :<math>(\omega\wedge\eta)(v_1,\cdots,v_{p+q}) = \frac{1}{p!q!}\sum_{\pi\in S_{p+q}}\sgn(\pi)\omega(v_{\pi(1)},\cdots,v_{\pi(p)})\otimes \eta(v_{\pi(p+1)},\cdots,v_{\pi(p+q)}).</math>
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| 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]] to ''E''). For ω ∈ Ω<sup>''p''</sup>(''M'') and η ∈ Ω<sup>''q''</sup>(''M'', ''E'') one has the usual commutativity relation:
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| :<math>\omega\wedge\eta = (-1)^{pq}\eta\wedge\omega.</math>
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| In general, the wedge product of two ''E''-valued forms is ''not'' another ''E''-valued form, but rather an (''E''{{unicode|⊗}}''E'')-valued form. However, if ''E'' is an [[algebra bundle]] (i.e. a bundle of [[algebra over a field|algebra]]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 algebra|commutative]], [[associative algebra]]s then, with this modified wedge product, the set of all ''E''-valued differential forms
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| :<math>\Omega(M,E) = \bigoplus_{p=0}^{\dim M}\Omega^p(M,E)</math>
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| becomes a [[graded-commutative]] associative algebra. If the fibers of ''E'' are not commutative then Ω(''M'',''E'') will not be graded-commutative.
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| ===Exterior derivative===
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| For any vector space ''V'' there is a natural [[exterior derivative]] on the space of ''V''-valued forms. This is just the ordinary exterior derivative acting component-wise relative to any [[basis (linear algebra)|basis]] of ''V''. Explicitly, if {''e''<sub>α</sub>} is a basis for ''V'' then the differential of a ''V''-valued ''p''-form ω = ω<sup>α</sup>''e''<sub>α</sub> is given by
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| :<math>d\omega = (d\omega^\alpha)e_\alpha.\,</math>
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| The exterior derivative on ''V''-valued forms is completely characterized by the usual relations:
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| :<math>\begin{align}
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| &d(\omega+\eta) = d\omega + d\eta\\
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| &d(\omega\wedge\eta) = d\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta\qquad(p=\deg\omega)\\
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| &d(d\omega) = 0.
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| \end{align}</math>
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| 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]] of ''E''.
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| 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 (vector bundle)|connection]] on ''E''. A connection on ''E'' is a linear [[differential operator]] taking sections of ''E'' to ''E''-valued one forms:
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| :<math>\nabla : \Omega^0(M,E) \to \Omega^1(M,E).</math>
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| If ''E'' is equipped with a connection ∇ then there is a unique [[covariant exterior derivative]]
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| :<math>d_\nabla: \Omega^p(M,E) \to \Omega^{p+1}(M,E)</math>
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| extending ∇. The covariant exterior derivative is characterized by [[linearity]] and the equation
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| :<math>d_\nabla(\omega\wedge\eta) = d_\nabla\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta</math>
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| where ω is a ''E''-valued ''p''-form and η is an ordinary ''q''-form. In general, one need not have ''d''<sub>∇</sub><sup>2</sup> = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing curvature).
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| ==Lie algebra-valued forms==
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| An important case of vector-valued differential forms are '''Lie algebra-valued forms'''. These are <math>\mathfrak g</math>-valued forms where <math>\mathfrak g</math> is a [[Lie algebra]]. Such forms have important applications in the theory of [[connection (principal bundle)|connections]] on a [[principal bundle]] as well as in the theory of [[Cartan connection]]s.
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| Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. This operation is denoted by <math>[\omega\wedge\eta]</math> to indicate both operations involved, or often just <math>[\omega, \eta]</math>.
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| For example, if <math>\omega</math> and <math>\eta</math> are Lie algebra-valued one forms, then one has
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| :<math>[\omega\wedge\eta](v_1,v_2) = [\omega(v_1),\eta(v_2)] - [\omega(v_2),\eta(v_1)].</math>
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| With this operation the set of all Lie algebra-valued forms on a manifold ''M'' becomes a [[graded Lie superalgebra]].
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| The operation <math>[\omega\wedge\eta]</math> can also be defined as the bilinear operation on <math>\Omega(M, \mathfrak g)</math> satisfying
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| :<math>[(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta)</math> | |
| for all <math>g, h \in \mathfrak g</math> and <math>\alpha, \beta \in \Omega(M, \mathbb R)</math>.
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| The alternative notation <math>[\omega, \eta]</math>, which resembles a [[Commutator#Ring_theory|commutator]], is justified by the fact that if the Lie algebra <math>\mathfrak g</math> is a matrix algebra then <math>[\omega\wedge\eta]</math> is nothing but the [[graded commutator]] of <math>\omega</math> and <math>\eta</math>, i. e. if <math>\omega \in \Omega^p(M, \mathfrak g)</math> and <math>\eta \in \Omega^q(M, \mathfrak g)</math> then
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| :<math>[\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega,</math>
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| where <math>\omega \wedge \eta,\ \eta \wedge \omega \in \Omega^{p+q}(M, \mathfrak g)</math> are wedge products formed using the matrix multiplication on <math>\mathfrak g</math>.
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| ==Basic or tensorial forms on principal bundles==
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| Let ''E'' → ''M'' be a smooth vector bundle of rank ''k'' over ''M'' and let ''π'' : F(''E'') → ''M'' be the ([[associated bundle|associated]]) [[frame bundle]] of ''E'', which is a [[principal bundle|principal]] GL<sub>''k''</sub>('''R''') bundle over ''M''. The [[pullback bundle|pullback]] of ''E'' by ''π'' is isomorphic to the trivial bundle F(''E'') × '''R'''<sup>''k''</sup>. Therefore, the pullback by ''π'' of an ''E''-valued form on ''M'' determines an '''R'''<sup>''k''</sup>-valued form on F(''E''). It is not hard to check that this pulled back form is [[equivariant|right-equivariant]] with respect to the natural [[group action|action]] of GL<sub>''k''</sub>('''R''') on F(''E'') × '''R'''<sup>''k''</sup> and vanishes on [[vertical bundle|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'').
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| Let ''π'' : ''P'' → ''M'' be a (smooth) [[principal bundle|principal ''G''-bundle]] and let ''V'' be a fixed vector space together with a [[group representation|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
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| #<math>(R_g)^*\omega = \rho(g^{-1})\omega\,</math> for all ''g'' ∈ ''G'', and
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| #<math>\omega(v_1, \ldots, v_p) = 0</math> whenever at least one of the ''v''<sub>''i''</sub> are vertical (i.e., d''π''(''v''<sub>''i''</sub>) = 0).
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| Here ''R''<sub>''g''</sub> denotes the right action of ''G'' on ''P'' for some ''g'' ∈ ''G''. Note that for 0-forms the second condition is [[vacuously true]].
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| Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' ×<sub>''ρ''</sub> ''V''. Tensorial forms on ''P'' are in one-to-one correspondence with ''E''-valued forms on ''M''. As in the case of the principal bundle F(''E'') above, ''E''-valued forms on ''M'' pull back to ''V''-valued forms on ''P''. These are precisely the basic or tensorial forms on ''P'' of type ''ρ''. Conversely given any tensorial form on ''P'' of type ''ρ'' one can construct the associated ''E''-valued form on ''M'' in a straightforward manner.
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| [[Category:Differential forms]]
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| [[Category:Vector bundles]]
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| {{Unreferenced|date=December 2007}}
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