An E-valued differential form of degree p is a smooth section of the tensor product bundle of E with Λp(T ∗M), 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,[1] this can also be interpreted as 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).
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.
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.
This is just the ordinary exterior derivative acting component-wise relative to any basis of V. Explicitly, if {eα} 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 (i.e. a vector bundle whose transition functions are constant).
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.
It is not hard to check that this pulled back form is right-equivariant with respect to the natural action of GLk(R) on F(E) × Rk and vanishes on vertical vectors (tangent vectors to F(E) which lie in the kernel of dπ).
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 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.
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).
on M with values in E, define φ on P fiberwise by, say at u, where u is viewed as a linear isomorphism
Denoted by θ, it is a tensorial form of standard type.