Context / intuition
I'll use Picard theorem to state the reasoning behind the question. The question itself is not related to this theorem.
In $\mathbb R^2$ with coordinates $(x,u)$.
Consider $\theta=du-f(x,u)dx$ such that $f(x,u)=du/dx$.
Consider curves $c:\mathbb R\rightarrow \mathbb R^2$ such that the pullbacks $c^*\theta=0$ and $c^*dx\neq0$ are valid.
Then (according to Picard), there exist a one-to-one correspondence between solutions of $du/dx=f(x,u)$ and the curves $c$.
I make the following observations:
Saying that a vector field $\theta$ is always normal to a curve $c$ is equivalent to saying that the dot product of the line tangent to the curve against the vector field is always zero. Now, $\theta$ looks exactly like the normal vector to a solutions of $f(x,u)=du/dx$, and in this case $c^*\theta=0$ would look like the corresponding vector product that should always be null.
Through the same reasoning, stating that $c^*dx\neq0$ is always (at least in a certain domain) would mean that the curve $c$ is never precisely perpendicular to the constant vector field $dx$.
So interpreting these pullbacks as dot products between the tangents of curve $c$ and the $\theta$ and $dx$ vector spaces, all existing in $\mathbb R^2$, makes sense in this setting/context.
Now, in a general setting/context (no longer Picard), I know pullbacks of differential forms by functions (each defined over some crazy due manifold) can have different intuitive interpretations.
The question
In a general setting/context what does it take for such pullbacks to actually correspond to dot products and, furthermore, is there any special relation between pullbacks and dot products?