Vector-bundle-valued differential form

Let $E\to M$ be a smooth real or complex vector bundle and let $k\geq 0$ . An $E$ -valued differential $k$ -form is a smooth section

\omega\in\Omega^k(M;E):= \Gamma\!\left(\Lambda^kT^*M\otimes E\right).

Here the coefficient bundle is the tensor product of the exterior power of the cotangent bundle with $E$ .

Equivalently, $\omega$ assigns to vector fields $X_1,\ldots,X_k$ a section $\omega(X_1,\ldots,X_k)$ of $E$ , alternatingly and $C^\infty(M)$ -multilinearly. When $k=0$ , this is simply a smooth section of $E$ . When $E=M\times\mathbb R$ , the definition reduces to an ordinary differential $k$ -form.

Algebraic structure

A scalar form $\alpha\in\Omega^p(M)$ and an $E$ -valued form $\omega\in\Omega^q(M;E)$ have a wedge product $\alpha\wedge\omega\in\Omega^{p+q}(M;E)$ , defined in local frames using the scalar wedge product on coefficient forms. Thus $\Omega^\bullet(M;E)$ is a graded module over the algebra $\Omega^\bullet(M)$ .

A connection on $E$ extends the exterior derivative to the exterior covariant derivative

d_\nabla:\Omega^k(M;E)\longrightarrow\Omega^{k+1}(M;E).

Unlike the scalar exterior derivative, this operator depends on the chosen connection.

Local form and pullback

In a local frame $e_1,\ldots,e_r$ of $E|_U$ , every bundle-valued form has a unique expression

\omega=\sum_{a=1}^r\omega^a\otimes e_a, \qquad \omega^a\in\Omega^k(U).

The coefficient forms change with the frame, while $\omega$ does not. If $f:N\to M$ is smooth, then ordinary pullback on the form factor and bundle pullback on the value factor give

f^*\omega\in\Omega^k(N;f^*E).

The codomain is $f^*E$ , not $E$ , unless additional bundle data identify them.

Examples and scope

The curvature of a vector-bundle connection is an $\operatorname{End}(E)$ -valued $2$ -form. On a principal bundle, tensorial adjoint-valued forms descend to forms on the base with values in the adjoint Lie algebra bundle. These are instances of the same construction, but a Lie-algebra-valued form on the total space is not automatically a bundle-valued form on the base.

References

Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, tensorial forms and covariant differentiation.
Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. DOI record. Relevant: Chapter I, vector-bundle-valued differential forms.