I am reading about geometric quantization and real polarizations, and it is claimed that there exist no real polarizations on the sphere $S^2 \subset \Bbb R^3$ because every tangent field on $S^2$ must vanish in some point ("real polarization" = Lagrangian distribution with respect to some given symplectic form - but Lagrangeanity does not seem to matter in the argument, so just ignore it).
I don't understand the given reason. I agree that, if $X$ is a tangent field on $S^2$, then $\Bbb R X$ cannot be a distribution of constant dimension on $S^2$ because it must have dimension $0$ at those points where $X$ vanishes. What I don't understand, though, is why every $1$-dimensional distribution must be of the form $\Bbb R X$ for some $X$, in order for the argument in the first paragraph to be applicable. Could you please explain this? Thank you.
It is clear that locally any one-dimensional distribution is of the form $\mathbb R\cdot X$ for a non-vanishing local vector field $X$ on $S^2$. To get the global result, you need the fact that any real line bundle over $S^2$ has to be trivial since $S^2$ is simply connected. Thus you get a global non-vanishing section.