Given a smooth real manifold $M$, and a real vector space $V$, when we talk about a $V$-valued $k$-form on $M$, do we mean an element of $\Gamma(\wedge^{*k}M)\otimes V$, where $\Gamma$ denotes sections of a vector bundle and $\wedge^{*k}M$ is the $k$th exterior bundle, or do we actually mean there is a $\pi:V\rightarrow M$ vector bundle, and a $V$-valued $k$-form is an element of $\Gamma(\wedge^{*k}M\otimes V)$?
Basically, if we take a $V$-valued $k$-form, and we will in its arguments, then will the resulting object be the element of a "global" vector space that is completely independent of the manifold, or will it be a "field", which at point $p$ has its value in $V_p$ that is distinct from $V_q$?
Also the prior question might be badly worded in the sense that I can totally imagine a $V$-valued $k$-form depend on points of the manifold after its arguments are all plugged in, what I am mainly curious about is that is it possible to compare the value of a vector-valued $k$-form at point $p$ with its value at point $q$?
Best Answer
I don't think that there is a uniform use of terminology, although it seems more common to me that "vector valued differential forms" refers to the case of a fixed vector space, whereas in the other case, I would use the terminology "vector bundle valued differential forms". An important difference between the two cases is that in the former case, there is a natural version of the exterior derivative, while in the latter case, one has to choose a linear connection on the vector bundle in order to have an analog of the exterior derivative.
The case of $\mathfrak g$-valued forms on a principal bundle you mention in your comment is the case of a fixed vector space (and if you define connection forms in that way, you need the exterior derivative on vector valued forms in the definition of curvature). Principal bundles also provide a nice relation between the two cases: Suppose $P\to M$ is a principal $G$-bundle, $V$ is a representation of $G$ (so in particular a finite dimensional vector space). Then there is a the associated vector bundle $E=P\times_G V\to M$ and one can identify the space $\Omega^*(M,E)$ of vector bundle valued forms with a subspace of the space $\Omega^*(P,V)$ of vector valued differential forms. The elements in that subspace are exactly those forms, which are horizontal and $P$-equivariant.