Vector field making Lie derivative injective

Question

I would like to know what conditions need to be put on a vector field $X$ so that the Lie derivative on differential forms is injective. That is, when does $L_X\alpha=L_X\beta$ imply $\alpha=\beta$, for arbitrary $k$-forms $\alpha$ and $\beta$? Here $k\geq 1$, since one could just add constants to functions without changing the Lie derivative.

Answer 1

Only when $X = 0$. Otherwise, in a neighborhood of a point where it's not zero you can find coordinates $x_1, \ldots, x_n$ such that $X = \partial/\partial x_1$, and then it's easy to write down forms of any degree which are preserved by the flow of $X$ (just make the formula independent of $x_1$).