Suppose that $M$ is a manifold whose points are k-dimensional sub-manifolds of another n-manifold $N$ where $n > k$ is finite. Then define a bundle $E$ over $M$ whose fibers are $\Omega^k P$ for $P \in M$. That is, the fibers are all the $k$-forms on $P$.
Question: Is the bundle $E$ trivial? How do I know?
Guess: If the ambient manifold $N$ has a non-vanishing $k$-form $\alpha$, then $E$ will be trivial because the section $s:P \mapsto \alpha|_P$ will trivialize $E$. I am guessing this is if and only if but I don't know how to show that.
Secondary question: Where do I read about this kind of infinite-dimensional setting?