Inmathematics, avector-valued differential form on amanifoldM is adifferential form onM with values in avector spaceV. More generally, it is a differential form with values in somevector bundleE overM. Ordinary differential forms can be viewed asR-valued differential forms.

An important case of vector-valued differential forms areLie algebra-valued forms. (Aconnection form is an example of such a form.)

LetM be asmooth manifold andE →M be a smoothvector bundle overM. We denote the space ofsmooth sections of a bundleE by Γ(E). AnE-valued differential form of degreep is a smooth section of thetensor product bundle ofE with Λ^p(T ^∗M), thep-thexterior power of thecotangent bundle ofM. The space of such forms is denoted by

${\displaystyle {\mathrm{\Omega }}^p(M,E)=\mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^p{T}^{\ast }M).}$

Because Γ is astrong monoidal functor,^[1] this can also be interpreted as

${\displaystyle \mathrm{\Gamma }(E\otimes {\mathrm{\Lambda }}^p{T}^{\ast }M)=\mathrm{\Gamma }(E){\otimes }_{{\mathrm{\Omega }}^0(M)}\mathrm{\Gamma }({\mathrm{\Lambda }}^p{T}^{\ast }M)=\mathrm{\Gamma }(E){\otimes }_{{\mathrm{\Omega }}^0(M)}{\mathrm{\Omega }}^p(M),}$

where the latter two tensor products are thetensor product of modules over thering Ω⁰(M) of smoothR-valued functions onM (see the seventh examplehere). By convention, anE-valued 0-form is just a section of the bundleE. That is,

${\displaystyle {\mathrm{\Omega }}^0(M,E)=\mathrm{\Gamma }(E).}$

Equivalently, anE-valued differential form can be defined as abundle morphism

${\displaystyle TM\otimes \cdots \otimes TM\to E}$

which is totallyskew-symmetric.

LetV be a fixedvector space. AV-valued differential form of degreep is a differential form of degreep with values in thetrivial bundleM ×V. The space of such forms is denoted Ω^p(M,V). WhenV =R one recovers the definition of an ordinary differential form. IfV is finite-dimensional, then one can show that the natural homomorphism

${\displaystyle {\mathrm{\Omega }}^p(M){\otimes }_{\mathbb{R}}V\to {\mathrm{\Omega }}^p(M,V),}$

where the first tensor product is of vector spaces overR, is an isomorphism.^[2]

One can define thepullback of vector-valued forms bysmooth maps just as for ordinary forms. The pullback of anE-valued form onN by a smooth map φ :M →N is an (φ*E)-valued form onM, where φ*E is thepullback bundle ofE by φ.

The formula is given just as in the ordinary case. For anyE-valuedp-form ω onN the pullback φ*ω is given by

${\displaystyle ({\phi }^{\ast }\omega {)}_x(v_1,\cdots ,v_p)={\omega }_{\phi (x)}(\mathrm{d}{\phi }_x(v_1),\cdots ,\mathrm{d}{\phi }_x(v_p)).}$

Just as for ordinary differential forms, one can define awedge product of vector-valued forms. The wedge product of anE₁-valuedp-form with anE₂-valuedq-form is naturally an (E₁⊗E₂)-valued (p+q)-form:

${\displaystyle \wedge :{\mathrm{\Omega }}^p(M,E_1)\times {\mathrm{\Omega }}^q(M,E_2)\to {\mathrm{\Omega }}^{p+q}(M,E_1\otimes E_2).}$

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

${\displaystyle (\omega \wedge \eta )(v_1,\cdots ,v_{p+q})=\frac{1}{p!q!}\sum _{\sigma \in S_{p+q}}\mathrm{sgn}(\sigma )\omega (v_{\sigma (1)},\cdots ,v_{\sigma (p)})\otimes \eta (v_{\sigma (p+1)},\cdots ,v_{\sigma (p+q)}).}$

In particular, the wedge product of an ordinary (R-valued)p-form with anE-valuedq-form is naturally anE-valued (p+q)-form (since the tensor product ofE with the trivial bundleM ×R isnaturally isomorphic toE). In terms of local frames {e_α} and {l_β} forE₁ andE₂ respectively, the wedge product of anE₁-valuedp-formω =ω^αe_α, and anE₂-valuedq-formη =η^βl_β is

${\displaystyle \omega \wedge \eta =\sum _{\alpha ,\beta }({\omega }^{\alpha }\wedge {\eta }^{\beta })(e_{\alpha }\otimes l_{\beta }),}$

whereω^α ∧η^β is the ordinary wedge product of ℝ-valued forms.For ω ∈ Ω^p(M) and η ∈ Ω^q(M,E) one has the usual commutativity relation:

${\displaystyle \omega \wedge \eta =(-1{)}^{pq}\eta \wedge \omega .}$

In general, the wedge product of twoE-valued forms isnot anotherE-valued form, but rather an (E⊗E)-valued form. However, ifE is analgebra bundle (i.e. a bundle ofalgebras rather than just vector spaces) one can compose with multiplication inE to obtain anE-valued form. IfE is a bundle ofcommutative,associative algebras then, with this modified wedge product, the set of allE-valued differential forms

${\displaystyle \mathrm{\Omega }(M,E)=\underset{p=0}{\overset{\dim M}{⨁}}{\mathrm{\Omega }}^p(M,E)}$

becomes agraded-commutative associative algebra. If the fibers ofE are not commutative then Ω(M,E) will not be graded-commutative.

For any vector spaceV there is a naturalexterior derivative on the space ofV-valued forms. This is just the ordinary exterior derivative acting component-wise relative to anybasis ofV. Explicitly, if {e_α} is a basis forV then the differential of aV-valuedp-form ω = ω^αe_α is given by

${\displaystyle d\omega =(d{\omega }^{\alpha })e_{\alpha }.}$

The exterior derivative onV-valued forms is completely characterized by the usual relations:

More generally, the above remarks apply toE-valued forms whereE is anyflat vector bundle overM (i.e. a vector bundle whose transition functions are constant). The exterior derivative is defined as above on anylocal trivialization ofE.

IfE is not flat then there is no natural notion of an exterior derivative acting onE-valued forms. What is needed is a choice ofconnection onE. A connection onE is a lineardifferential operator taking sections ofE toE-valued one forms:

${\displaystyle \mathrm{\nabla }:{\mathrm{\Omega }}^0(M,E)\to {\mathrm{\Omega }}^1(M,E).}$

IfE is equipped with a connection ∇ then there is a uniquecovariant exterior derivative

${\displaystyle d_{\mathrm{\nabla }}:{\mathrm{\Omega }}^p(M,E)\to {\mathrm{\Omega }}^{p+1}(M,E)}$

extending ∇. The covariant exterior derivative is characterized bylinearity and the equation

${\displaystyle d_{\mathrm{\nabla }}(\omega \wedge \eta )=d_{\mathrm{\nabla }}\omega \wedge \eta +(-1{)}^p\omega \wedge d\eta }$

where ω is aE-valuedp-form and η is an ordinaryq-form. In general, one need not haved_∇² = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishingcurvature).

LetE →M be a smooth vector bundle of rankk overM and letπ : F(E) →M be the (associated)frame bundle ofE, which is aprincipal GL_k(R) bundle overM. Thepullback ofE byπ is canonically isomorphic to F(E) ×_ρR^k via the inverse of [u,v] →u(v), where ρ is the standard representation. Therefore, the pullback byπ of anE-valued form onM determines anR^k-valued form on F(E). It is not hard to check that this pulled back form isright-equivariant with respect to the naturalaction of GL_k(R) on F(E) ×R^k and vanishes onvertical vectors (tangent vectors to F(E) which lie in the kernel of dπ). Such vector-valued forms on F(E) are important enough to warrant special terminology: they are calledbasic ortensorial forms on F(E).

Letπ :P →M be a (smooth)principalG-bundle and letV be a fixed vector space together with arepresentationρ :G → GL(V). Abasic ortensorial form onP of type ρ is aV-valued form ω onP that isequivariant andhorizontal in the sense that

1. ${\displaystyle (R_g{)}^{\ast }\omega =\rho (g^{-1})\omega }$ for allg ∈G, and
2. ${\displaystyle \omega (v_1,\dots ,v_p)=0}$ whenever at least one of thev_i are vertical (i.e., dπ(v_i) = 0).

HereR_g denotes the right action ofG onP for someg ∈G. Note that for 0-forms the second condition isvacuously true.

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

GivenP andρ as above one can construct theassociated vector bundleE =P ×_ρV. Tensorialq-forms onP are in a natural one-to-one correspondence withE-valuedq-forms onM. As in the case of the principal bundle F(E) above, given aq-form${\displaystyle \overline{\varphi }}$ onM with values inE, define φ onP fiberwise by, say atu,

${\displaystyle \varphi =u^{-1}{\pi }^{\ast }\overline{\varphi }}$

whereu is viewed as a linear isomorphism${\displaystyle V\stackrel{\simeq }{\to }{E}_{\pi (u)}=({\pi }^{\ast }E{)}_u,v\mapsto [u,v]}$. φ is then a tensorial form of type ρ. Conversely, given a tensorial form φ of type ρ, the same formula defines anE-valued form${\displaystyle \overline{\varphi }}$ onM (cf. theChern–Weil homomorphism.) In particular, there is a natural isomorphism of vector spaces

${\displaystyle \mathrm{\Gamma }(M,E)\simeq \{f:P\to V|f(ug)=\rho (g{)}^{-1}f(u)\},\overline{f}\leftrightarrow f}$.

Example: LetE be the tangent bundle ofM. Then identity bundle map id_E:E →E is anE-valued one form onM. Thetautological one-form is a unique one-form on the frame bundle ofE that corresponds to id_E. Denoted by θ, it is a tensorial form of standard type.

Now, suppose there is a connection onP so that there is anexterior covariant differentiationD on (various) vector-valued forms onP. Through the above correspondence,D also acts onE-valued forms: define ∇ by

${\displaystyle \mathrm{\nabla }\overline{\varphi }=\overline{D\varphi }.}$

In particular for zero-forms,

${\displaystyle \mathrm{\nabla }:\mathrm{\Gamma }(M,E)\to \mathrm{\Gamma }(M,T^{\ast }M\otimes E)}$.

This is exactly thecovariant derivative for theconnection on the vector bundleE.^[3]

Siegel modular forms arise as vector-valued differential forms onSiegel modular varieties.^[4]

• Shoshichi Kobayashi andKatsumi Nomizu (1963)Foundations of Differential Geometry, Vol. 1,Wiley Interscience.

• Differential forms
• Vector bundles