@@ -16,16 +16,18 @@ Therefore we will use two specialised density results for the simply-connected c

\textinput{sobolev-lift-simply-connected}

\subsection{Orientability on surfaces}\label{sec:sobolev_orientability_on_surface}{

We are mostly interested in the special case $\dim M =2$.

2-dimensional manifolds are surfaces, many of them embedded in three-dimensional space and therefore the primary example in applications.

As mentioned in the \nameref{sec:introduction} liquid crystals exist in some cases like \parencite{q5-q3-vesicles} in very thin films that are best modelled as two-dimensional surfaces.

Therefore the case $\dim M =2$ is of special interest.

Section 4 of \parencite{q4-Ball2011} gives a roadmap how to transfer the orientability condition from continuous to Sobolev line fields.

In order to use those ideas we notice that the tangent bundle is trivial in the case of oriented surfaces with unit vector fields.\todo{somehow mention unorientable case}

This allows us to consider unit vector fields as maps into $𝕊^1$ as discussed in \cref{sec:sobolev_field_spaces_on_parallelizable_manifolds}.

\begin{lemma}[Triviality of Two Dimensions] \label{thm:triviality_of_two_dimensions}

Let $M$ be a compact two-dimensional manifold that is simply connected or orientable.

Let $M$ be a compact orientable two-dimensional manifold.

Then either $M$ admits no smooth unit tangent field or has a global orthonormal frame.

\end{lemma}% end lemma Triviality of Two Dimensions

The prominent example of a manifold without unit tangent field is the sphere by the Hairy Ball theorem \autocite{hairy-ball}.

The obvious other case are manifolds with a global chart like graphs of functions $f ⫶ ℝ^2 → ℝ$.

Having a global orthonormal frame is called \newTerm{parallizable}.\footnote{and I probably spell it incorrectly every time}

A manifold with a global frame is called \newTerm{parallelizable}.

Any global frame can be orthonormalized to an orthonormal global frame with the Gram-Schmidt process.

\begin{proof}% of lemma Triviality of Two Dimensions

If $M$ admits no smooth unit tangent field, we are done.