Can all covector field be written as a product of a function and a differential of another function?

Question

Is it available to write every covector field

$$ \alpha = \sum_{i=1}^{n}\alpha^i\mathrm dx_i $$ on a manifold into the form $$ \alpha = f\mathrm{d}g, $$ where $f$ and $g$ are smooth functions? Or, can we do this at least locally?

Answer 1

Your question is about whether there is an integrating factor ($1/f$ in your notation). In the two-dimensional case, there is always an integrating factor (this is essentially equivalent to existence of solutions to the differential equation $\alpha=0$. But already in 3D you are in trouble. In that case, there is an integrating factor only if $\alpha\wedge d\alpha=0$. So the answer to your question would be: yes in 2D, not in general if $n\ge 3$.