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 : (\varphi^*\omega)_x(v_1,\cdots, v_p) = \omega_{\varphi(x)}(\mathrm d\varphi_x(v_1),\cdots,\mathrm d\varphi_x(v_p)).
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 (
E1⊗
E2)-valued (
p+
q)-form: :\wedge : \Omega^p(M,E_1) \times \Omega^q(M,E_2) \to \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 the
tensor product: :(\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)}). 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). In terms of local frames {
eα} and {
lβ} for
E1 and
E2 respectively, the wedge product of an
E1-valued
p-form
ω =
ωα eα, and an
E2-valued
q-form
η =
ηβ lβ is :\omega \wedge \eta = \sum_{\alpha, \beta} (\omega^\alpha \wedge \eta^\beta) (e_\alpha \otimes l_\beta), where
ωα ∧
ηβ is the ordinary wedge product of \mathbb{R}-valued forms. For ω ∈ Ω
p(
M) and η ∈ Ω
q(
M,
E) one has the usual commutativity relation: :\omega\wedge\eta = (-1)^{pq}\eta\wedge\omega. 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 :\Omega(M,E) = \bigoplus_{p=0}^{\dim M}\Omega^p(M,E) 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 {
eα} is a basis for
V then the differential of a
V-valued
p-form ω = ωα
eα is given by :d\omega = (d\omega^\alpha)e_\alpha.\, The exterior derivative on
V-valued forms is completely characterized by the usual relations: :\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} 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: :\nabla : \Omega^0(M,E) \to \Omega^1(M,E). If
E is equipped with a connection ∇ then there is a unique
covariant exterior derivative :d_\nabla: \Omega^p(M,E) \to \Omega^{p+1}(M,E) extending ∇. The covariant exterior derivative is characterized by
linearity and the equation :d_\nabla(\omega\wedge\eta) = d_\nabla\omega\wedge\eta + (-1)^p\,\omega\wedge d_\nabla\eta 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==