When can a functional be written as the integral of a 1-form?

Let a real, smooth manifold $M$ be given. Let $\Gamma$ denote the set of all path segments on $M$, namely the set of all paths of the form $\gamma:[a,b]\to M$.

Let $Q:\Gamma\to\mathbb R$ be a functional, namely a function that assigns a real number to each such path segment. Suppose that $Q$ has the property that for any path $\gamma:[a,c]\to M$, if I divide this path into two segments $\gamma_1[a,b]\to M$ and $\gamma_2[b,c]\to M$ such that $\gamma_1(t) = \gamma(t)$ for all $t\in [a,b]$ and $\gamma_2(t) = \gamma(t)$ for all $t\in [b,c]$, then \begin{align} Q[\gamma] = Q[\gamma_1] + Q[\gamma_2]. \end{align} Under what additional assumptions (if any) can one then show that there exists a 1-form $\omega$ such that \begin{align} Q[\gamma] = \int_\gamma\omega \end{align} for all $\gamma\in \Gamma$?

Solution 1:

First of all, this is a very nice and natural question. Hassler Whitney addressed exactly this question (ok, a bit more generally, for linear functionals on $k$-chains, not just 1-chains) about 60 years ago in his book "Geometric Integration Theory". The answer is given in Theorem 10A, page 167. Whitney proves that, under suitable analytical conditions, such linear functionals are indeed represented by differential $k$-forms. (In your question, $k=1$.)