Vector-valued differential form: Difference between revisions

Revision as of 00:56, 15 December 2014

Latest revision as of 07:37, 12 April 2025

(49 intermediate revisions by 14 users not shown)

Line 1:

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.

An important case of vector-valued differential forms are [[Lie algebra-valued forms]]. (A [[connection form]] is an example of such a form.)

==Formal definition==

==Definition==

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^''p''(''T''*''M''), the ''p''-th [[exterior power]] of the [[cotangent bundle]] of ''M''. The space of such forms is denoted by

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''). AAn '''''E''-valued differential form''' of degree ''p'' is a smooth section of the [[tensor product]] bundle]] of ''E'' with Λ^''p''(''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>

Because Γ is a [[strong monoidal functor]],<ref name=gamma_monoidal>{{cite web|title=Global sections of a tensor product of vector bundles on a smooth manifold|url=httphttps://math.stackexchange.com/questionsq/492166/global-sections-of-a-tensor-product-of-vector-bundles-on-a-smooth-manifold |website=math.stackexchange.com|accessdateaccess-date=27 October 2014|ref=monoidal}}</ref> this can also be interpreted as

:<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>

where the latter two tensor products are the [[tensor product of modules]] over the [[ring (mathematics)|ring]] Ω⁰(''M'') of smooth '''R'''-valued functions on ''M'' (see the fifthseventh example [[module (mathematics)#Examples|here]]). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is,

:<math>\Omega^0(M,E) = \Gamma(E).\,</math>

Equivalently, aan ''E''-valued differential form can be defined as a [[vector bundle morphism|bundle morphism]]

:<math>TM\otimes\cdots\otimes TM \to E</math>

which is totally [[skew-symmetric matrix|skew-symmetric]].

Line 31 ⟶ 33:

===Wedge product===

Just as for ordinary differential forms, one can define a [[wedge product]] 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{{unicode|&otimes;}}''E''₂)-valued (''p''+''q'')-form:

:<math>\wedge : \Omega^p(M,E_1) \times \Omega^q(M,E_2) \to \Omega^{p+q}(M,E_1\otimes E_2).</math>

The definition is just as for ordinary forms with the exception that real multiplication is replaced with the [[tensor product]]:

:<math>(\omega\wedge\eta)(v_1,\cdots,v_{p+q}) = \frac{1}{p! q!}\sum_{\pisigma\in S_{p+q}}\sgn(\pisigma)\omega(v_{\pisigma(1)},\cdots,v_{\pisigma(p)})\otimes \eta(v_{\pisigma(p+1)},\cdots,v_{\pisigma(p+q)}).</math>

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'' &times; '''R''' is [[naturally isomorphic]] to ''E''). For ω ∈ Ω^''p''(''M'') and η ∈ Ω^''q''(''M'', ''E'') one has the usual commutativity relation:

In terms of local frames {''e''_''α''} and {''l''_''β''} 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

:<math>\omega \wedge \eta = \sum_{\alpha, \beta} (\omega^\alpha \wedge \eta^\beta) (e_\alpha \otimes l_\beta),</math>

where ''ω''^''α'' ∧ ''η''^''β'' is the ordinary wedge product of <math>\mathbb{R}</math>-valued forms.

For ω ∈ Ω^''p''(''M'') and η ∈ Ω^''q''(''M'', ''E'') one has the usual commutativity relation:

:<math>\omega\wedge\eta = (-1)^{pq}\eta\wedge\omega.</math>

In general, the wedge product of two ''E''-valued forms is ''not'' another ''E''-valued form, but rather an (''E''{{unicode|&otimes;}}''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

:<math>\Omega(M,E) = \bigoplus_{p=0}^{\dim M}\Omega^p(M,E)</math>

becomes a [[graded-commutative]] associative algebra. If the fibers of ''E'' are not commutative then Ω(''M'',''E'') will not be graded-commutative.

Line 60 ⟶ 66:

extending ∇. The covariant exterior derivative is characterized by [[linearity]] and the equation

:<math>d_\nabla(\omega\wedge\eta) = d_\nabla\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta</math>

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 form|curvature]]).

==LieBasic algebra-valuedor 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 bundle|associated]]) [[frame bundle]] of ''E'', which is a [[principal bundle|principal]] GL_''k''('''R''') bundle over ''M''. The [[pullback bundle|pullback]] of ''E'' by ''π'' is canonically isomorphic to the trivial bundle F(''E'') &times;_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 [[equivariant|right-equivariant]] with respect to the natural [[groupGroup action (mathematics)|action]] of GL_''k''('''R''') on F(''E'') &times; '''R'''^''k'' 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'').

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.

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'' whichthat is '''equivariant''' and '''horizontal''' in the sense that

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>.

For example, if #<math>(R_g)^*\omega</math> and= <math>\etarho(g^{-1})\omega\,</math> arefor Lieall algebra-valued''g'' one∈ forms''G'', then one hasand

#<math>\omega(v_1, \ldots, v_p) = 0</math> whenever at least one of the ''v''_''i'' are vertical (i.e., d''π''(''v''_''i'') = 0).

:<math>[\omega\wedge\eta](v_1,v_2) = [\omega(v_1),\eta(v_2)] - [\omega(v_2),\eta(v_1)].</math>

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]].

With this operation the set of all Lie algebra-valued forms on a manifold ''M'' becomes a [[graded Lie superalgebra]].

Example: If ρ is the [[adjoint representation]] of ''G'' on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated [[curvature form]] Ω satisfies both; hence Ω is a tensorial form of adjoint type. The "difference" of two connection forms is a tensorial form.

The operation <math>[\omega\wedge\eta]</math> can also be defined as the bilinear operation on <math>\Omega(M, \mathfrak g)</math> satisfying

:<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>.

Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' &times;_''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 ''Eq''-valuedform forms<math>\overline{\phi}</math> on ''M'' pullwith backvalues toin ''VE''-valued, formsdefine φ on ''P''. Explicitlyfiberwise by, givensay aat ''qu''-form <math>\overline{\phi}</math>, define φ fiberwise by

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

:<math>[\omega\wedge\eta]phi = \omega\wedge\eta - (u^{-1)^{pq}\etapi^*\wedgeoverline{\omega,phi}</math>

where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_xE_{\pi(u)} = (\pi^*E)_u, v \mapsto [u, 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 <math>\overline{\phi}</math> on ''M'' (cf. the [[Chern–Weil homomorphism]].) In particular, there is a natural bijectionisomorphism of vector spaces

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>.

:<math>\Gamma(M, E) \tosimeq \{ f: P \to V | f(ug) = \rho(g)^{-1}f(u) \}, \, \overline{f} \leftrightarrow f</math>.

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]] 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.<!--Mention this somewhere else: The [[exterior covariant derivative]] of θ, Θ = ''D''θ is called a [[torsion form]].-->

==Basic or tensorial forms on principal bundles==

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

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_''k''('''R''') bundle over ''M''. The [[pullback bundle|pullback]] of ''E'' by ''π'' is isomorphic to the trivial bundle F(''E'') &times; '''R'''^''k''. 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 [[equivariant|right-equivariant]] with respect to the natural [[group action|action]] of GL_''k''('''R''') on F(''E'') &times; '''R'''^''k'' 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'').

:<math>\nabla \overline{\phi} = u^{-1}\pi^*\overline{D \phi}.</math>

In particular for zero-forms,

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

:<math>\nabla: \Gamma(M, E) \to \Gamma(M, T^*M \otimes E)</math>.

#<math>(R_g)^*\omega = \rho(g^{-1})\omega\,</math> for all ''g'' ∈ ''G'', and

#<math>\omega(v_1, \ldots, v_p) = 0</math> 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]].

This is exactly the [[covariant derivative]] for the [[connection (vector bundle)|connection on the vector bundle ''E'']].<ref>Proof: <math>D (f\phi) = Df \otimes \phi + f D\phi</math> for any scalar-valued tensorial zero-form ''f'' and any tensorial zero-form φ of type ρ, and ''Df'' = ''df'' since ''f'' descends to a function on ''M''; cf. this [[Chern–Weil homomorphism#Definition of the homomorphism|Lemma 2]].</ref>

Example: If ρ is the [[adjoint representation]] of ''G'' on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated [[curvature form]] Ω satisfies both; hence Ω is a tensorial form of adjoint type.

==Examples==

Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' &times;_''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, ''E''-valued forms on ''M'' pull back to ''V''-valued forms on ''P''. Explicitly, given a ''q''-form <math>\overline{\phi}</math>, define φ fiberwise by

[[Siegel modular form]]s arise as vector-valued differential forms on [[Siegel modular variety|Siegel modular varieties]].<ref>{{cite journal|title=The Geometry of Siegel Modular Varieties |last1=Hulek |first1=Klaus |last2=Sankaran |first2=G. K. |journal=Advanced Studies in Pure Mathematics |volume=35 |year=2002 |pages=89–156}}</ref>

:<math>\phi = u^{-1}\pi^*\overline{phi}</math>

where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_x</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 <math>\overline{\phi}</math> on ''M''. In particular, there is a natural bijection

==Notes==

:<math>\Gamma(M, E) \to \{ f: P \to V | f(ug) = g^{-1}f(u) \}</math>.

{{reflist}}

==References==

* [[Shoshichi Kobayashi]] and [[Katsumi Nomizu]] (1963) [[Foundations of Differential Geometry]], Vol. 1, [[Wiley Interscience]].

{{Unreferenced|date=December 2007}}

{{Reflist}}

[[Category:Differential forms]]