$\newcommand{\R}{\mathbb{R}}$ I've been learning some basic differential geometry on $\R^n$ recently and feel like I have begun to grasp the concept of a $1$-form. To reiterate, let $T_p(\R^n)$ be the tangent space at a point $p\in\R^n$, and $T_p^{*}(\R^n) = (T_p(\R^n))^{\vee}$. Then a differential $1-$form $\omega$ is a map
$$\omega:U\rightarrow \bigcup_{p\,\in\, U}\,T_p^{*}(\R^n)$$
Where in particular,
$$\omega:p\mapsto \omega_p$$
Now a particular example of a $1$-form is one induced by a smooth function $f$. We call this the differential of $f$, and its action is essentially
$$(df):p\mapsto (df)_p$$
Where $(df)_p$ acts on a vector $v = \sum_i v_ie_i$ as
$$(df)_p(v) = \sum_{i+1}^n v_i\frac{\partial f}{\partial x^i}(p)$$
I've noticed that in a lot of theorems I've encountered there's been some sort of one-to-one equivalence between seemingly disparate objects. For example, the set of smooth vector fields $\mathfrak{X}$ defined on some subset $U$ of $\R^n$ has a one-to-one correspondence with the derivations on $C^{\infty}(U)$. This prompts me to ask the question:
- Is there some well-defined subclass of differential $1$-forms such that each one of its members can be rationalized as the differential of a smooth function?
- What are the necessary conditions for the above to happen?