I'm trying to solve an exercise which asks to prove the following: there is a 1-distribution $\mathcal{D}$ on the Klein bottle $K$ which isn't of the form $X_m \mathbb{R}$ for some non vanishing vector field $X$ on $K$.
I don't know how to proceed, I guess one should start by giving an explicit distribution (maybe on the klein bottle in $\mathbb{R}^2$ with the usual identifications?) and prove it can't come from a vector field because $K$ is non-orientable.
Here's my idea on $\mathbb{R}^2$: first we consider the moebius strip, with the usual identifications (shown here): my idea would be to define a distribution by $\mathcal{D}((x,y))=\mathbb{R}x$ for every point $(x,y)$ in the rectangle $R$ which defines the identifications (so vertical lines). The point is that intuitively, if there was a vector field defining this distribution, when we cross the identifications the vector field would change sign. Is this a valid distribution? And how to prove formally that the vector field would be zero somewhere?
Now consider the klein bottle $K$ with the usual identifications (for example as shown here). My idea would be to just say that we define the same vector field we defined before (but this time horizontal lines) and get the same contradiction (the klein bottle contains mobius strips).
I'd be grateful to anyone who can help.