Vector-valued differential form

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.

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

==Definition==

Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle 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 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=https://math.stackexchange.com/q/492166 |website=math.stackexchange.com|access-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 Ω⁰(''M'') of smooth '''R'''-valued functions on ''M'' (see the seventh example 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, an ''E''-valued differential form can be defined as a bundle morphism :<math>TM\otimes\cdots\otimes TM \to E</math> which is totally skew-symmetric.

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'' &times; ''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 :<math>\Omega^p(M) \otimes_\mathbb{R} V \to \Omega^p(M,V),</math> where the first tensor product is of vector spaces over '''R''', is an isomorphism.<ref>Proof: 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 :<math>\Omega^p(M, V) = \Omega^0(M, V) \otimes_{\Omega^0(M)} \Omega^p(M),</math> and that because <math>\mathbb{R}</math> is a sub-ring of Ω⁰(''M'') via the constant functions, :<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></ref>

==Operations on vector-valued forms==

===Pullback===

One can define the pullback of vector-valued forms by smooth maps 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.

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 :<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>

===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&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_{\sigma\in S_{p+q}}\sgn(\sigma)\omega(v_{\sigma(1)},\cdots,v_{\sigma(p)})\otimes \eta(v_{\sigma(p+1)},\cdots,v_{\sigma(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''). 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''&otimes;''E'')-valued form. However, if ''E'' is an algebra bundle (i.e. a bundle of algebras 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, associative algebras 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.

===Exterior derivative===

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 of ''V''. Explicitly, if {''e''_α} is a basis for ''V'' then the differential of a ''V''-valued ''p''-form ω = ω^α''e''_α is given by :<math>d\omega = (d\omega^\alpha)e_\alpha.\,</math> The exterior derivative on ''V''-valued forms is completely characterized by the usual relations: :<math>\begin{align} &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{align}</math> 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''.

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 taking sections of ''E'' to ''E''-valued one forms: :<math>\nabla : \Omega^0(M,E) \to \Omega^1(M,E).</math> If ''E'' is equipped with a connection ∇ then there is a unique covariant exterior derivative :<math>d_\nabla: \Omega^p(M,E) \to \Omega^{p+1}(M,E)</math> 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_\nabla\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).

==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 of ''E'', which is a principal GL_''k''('''R''') bundle over ''M''. The pullback of ''E'' by ''π'' is canonically isomorphic to 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 right-equivariant with respect to the natural action of GL_''k''('''R''') on F(''E'') &times; '''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 #<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.

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.

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 ''q''-form <math>\overline{\phi}</math> on ''M'' with values in ''E'', define φ on ''P'' fiberwise by, say at ''u'', :<math>\phi = u^{-1}\pi^*\overline{\phi}</math> where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_{\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 isomorphism of vector spaces :<math>\Gamma(M, E) \simeq \{ 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.-->

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 :<math>\nabla \overline{\phi} = \overline{D \phi}.</math>

In particular for zero-forms, :<math>\nabla: \Gamma(M, E) \to \Gamma(M, T^*M \otimes E)</math>.

This is exactly the covariant derivative for the 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 Lemma 2.</ref>

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

==Notes== {{reflist}}

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

Category:Differential forms Category:Vector bundles