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.)

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

Because Γ is a strong monoidal functor,^{[1]} this can also be interpreted as

where the latter two tensor products are the tensor product of modules over the ring Ω^{0}(*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,

Equivalently, an *E*-valued differential form can be defined as a bundle morphism

which is totally skew-symmetric.

Let *V* be a fixed vector space. A ** V-valued differential form** of degree

where the first tensor product is of vector spaces over **R**, is an isomorphism.^{[2]}

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

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}⊗*E*_{2})-valued (*p*+*q*)-form:

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

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^{p}(*M*) and η ∈ Ω^{q}(*M*, *E*) one has the usual commutativity relation:

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 (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

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

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

The exterior derivative on *V*-valued forms is completely characterized by the usual relations:

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:

If *E* is equipped with a connection ∇ then there is a unique covariant exterior derivative

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

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).

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*) ×_{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*) × **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* which is **equivariant** and **horizontal** in the sense that

- for all
*g*∈*G*, and - 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* ×_{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 on *M* with values in *E*, define φ on *P* fiberwise by, say at *u*,

where *u* is viewed as a linear isomorphism . 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 on *M* (cf. the Chern–Weil homomorphism.) In particular, there is a natural isomorphism of vector spaces

- .

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.

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

In particular for zero-forms,

- .

This is exactly the covariant derivative for the connection on the vector bundle *E*.^{[3]}

Siegel modular forms arise as vector-valued differential forms on Siegel modular varieties.^{[4]}