Commit 7650d6c3 authored by Felix Hilsky's avatar Felix Hilsky
Browse files

small fix in circle example

parent c266baa7
......@@ -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.
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