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. An E-valued differential form of degree p is a smooth section of the tensor product bundle of E with Λp, the p-th exterior power of the cotangent bundle of M. The space of such forms is denoted by Because Γ is a strong monoidal functor, this can also be interpreted as where the latter two tensor products are the tensor product of modules over the ring Ω0 of smooth R-valued functions on M. 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 p is a differential form of degree p with values in the trivial bundleM × V. The space of such forms is denoted Ωp. 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 where the first tensor product is of vector spaces over R, is an isomorphism.
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 -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
Wedge product
Just as for ordinary differential forms, one can define a wedge product of vector-valued forms. The wedge product of an E1-valued p-form with an E2-valued q-form is naturally an -valued -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 p-form with an E-valued q-form is naturally an E-valued -form. For ω ∈ Ωp and η ∈ Ωq 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 -valued form. However, if E is an algebra bundle 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-commutativeassociative algebra. If the fibers of E are not commutative then Ω 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 is a basis for V then the differential of a V-valued p-form ω = ωαeα 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. 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 uniquecovariant 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.
Let E → M be a smooth vector bundle of rank k over M and let π : F → M be the frame bundle of E, which is a principal GLk bundle over M. The pullback of E by π is canonically isomorphic to F ×ρRk via the inverse of →u, where ρ is the standard representation. Therefore, the pullback by π of an E-valued form on M determines an Rk-valued form on F. It is not hard to check that this pulled back form is right-equivariantwith respect to the natural action of GLk on F × Rk and vanishes on vertical vectors. Such vector-valued forms on F are important enough to warrant special terminology: they are called basic or tensorial forms on F. Let π : P → M be a principal G-bundle and let V be a fixed vector space together with a representationρ : G → GL. 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 vi are vertical.
Here Rg denotes the right action of G on P for some g ∈ G. Note that for 0-forms the second condition is vacuously true.
Given P and ρ as above one can construct the associated vector bundleE = P ×ρV. 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 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 In particular, there is a natural isomorphism of vector spaces
Example: Let E be the tangent bundle of M. Then identity bundle map idE: 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 idE. 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 differentiationD on 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.
