@@ -172,10 +172,8 @@ We formulate the same in the language of \cref{sec:embedding_M} and allow fields

In this simple form the converse is not true.

For this consider the circle $𝕊^1$. $T_p𝕊^1$ is one-dimensional at every $p∈𝕊^1$.

Therefore $𝕊^*𝕊^1$ consists of only two vectors at every base point and $𝒬^{𝕊}𝕊^1$ even only of one. Therefore there is only exactly one line field on the sphere and it is orientable.

\footnote{It appears plausible that the technique for the torus works for most manifolds of dimension 2 and higher since \enquote{turning around} on a loop seems easy and then we \enquote{only} need to extends this vector field to all of $M$. But this might not be trivial as the example of the sphere shows.

}

Therefore $𝕊𝕊^1$ consists of only two vectors at every base point and $𝒬^{𝕊}𝕊^1$ even only of one element.

Therefore there is only exactly one line field on the sphere and it is orientable.

\end{example}% end example Circle

We will also look at the one example of a boundaryless two-dimensional compact manifold with unit vector field and construct a non-orientable line field: a torus.