In this question the asker defines 1-forms on a (real, smooth) manifold $M$ to be
$C^{\infty}(M)$-module homomorphism[s] from $Vec(M)$ to $C^{\infty}(M).\:\:\:\:(*)$
I'm wondering if this is correct.
I've previously seen 1-forms defined as being homomorphisms from $TM$ to $\mathbb{R}\times M$ as vector-bundles-over-$M$. It's easy to see that a 1-form in this sense defines a 1-form in the first sense, but I can't see why the converse is true.
It seems like an extra condition ought to be added to $(*)$ saying that given a supposed 1-form $\eta$ taking a vector $X$ to a scalar $f$, the value of $f$ at $x\in M$ should depend only on the value of $X$ at $x$. Or something like that.
- Are all $C^{\infty}(M)$-modules given by spaces of sections of vector bundles?
- Are all $C^{\infty}(M)$-homomorphisms between the spaces of sections of bundles given by corresponding maps between the bundles themselves?
At least when $M$ is finite-dimensional, the answer to your second question is yes. The proof involves localization via bump functions and local frames. One place where you can read a proof is Lemma 10.29 in my Introduction to Smooth Manifolds (2nd ed.).