Reconstruct a smooth vector field from direction field (Arnold's ODE)

Question

In Chapter 1, section 5.4 of Arnold's ODE book he is talking about diffeomorphisms on direction fields. To prove that integral curves of the field in $M$ are mapped to integral curves of the image of this field under diffeomorphism $g$ in $N$, the vector field needs to be reconstructed from the direction field.

He does it a paragraph earlier: for every $y$ in $N$, find $x=g^{-1}y$ in $M$. The direction field at $x$ is a line in $T_xM$. Take a vector $v\neq0$ along the line and get $w=g_{*x}v\neq0$ (as far as I understand, since $g$ is diffeomorphism, $w=0$ would violate smoothness of the reverse mapping $g^{-1}$) in $T_{g(x)}N$. Vector $w$ is independent of the choice of $v$ due to linearity of $g_{*x}v$ and will define the line of the image of the direction field.

So far, he hasn't been talking about the field being smooth. Now, in Problem 1 it is asked if a smooth vector field can be reconstructed from a smooth direction field and the answer says "No, if the region (I guess in a more general case, manifold) $M$ is not simply connected".

So, my first question is if I understand it correctly that crucial part here is about the field being smooth?

Secondly, I cannot see how the smoothness would be violated if the region $M$ has a hole in it as in Fig. 58, illustrating the answer.

PS. I am quoting from Russian edition of the book.

Answer 1

Consider, on a Mobius band, the line field that runs orthogonal to the central circle. You can pick a nonzero vector in at $\theta = 0$, but by the time you've come back around to $\theta = 2\pi$, the nonzero vector is pointing in the other direction, and there's no way to adjust it to make a smooth nonzero field.

I guess the answer to the question you actually asked is "no, the key thing is non-zero-ness," and for the second, it's not the smoothness that's violated, but the non-zero-ness.

Given your math.se login name (mobiuseng), there's a small irony in this answer. :)