@@ -191,4 +191,54 @@ We formulate the same in the language of \cref{sec:embedding_M} and allow fields

\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.

}

\end{example}% end example Circle

\todo{This paragraph about orientability on paths fits to continuous line lifting (this section) but is somehow more basic than the other theorems.

So I would expect it to come at the beginning but the introductory text there does not fit with it because it's not necessary for the other results.

Now it's here at the end but a better thought-through structure of the section would be good.}

Before we consider lifting a restricted $Q$-tensor field on any manifold, we first look at paths.

In the flat case this was done in \textcite[Lemma 3][505]{q4-Ball2011} and we will reduce the manifold case to this.%

% it is actually used when showing that in 2-D we have a global ON-frame

% \footnote{it's quite likely that we will not need \cref{thm:orientability_on_a_path}. So remove it if not used}

\begin{proposition}[Orientability on a path] \label{thm:orientability_on_a_path}

Let $M$ be a Riemannian $m$-dimensional manifold, $- ∞ < t_1 < t_2 < ∞$, $γ ⫶[t_1, t_2] → M$ be a continous injective path on $M$

and $Q ∈ Γ_C(\rest{𝒬^{𝕊}M}{γ([t_1, t_2])})$ be a continous restricted $Q$-tensor field along the path $γ$.

Then there exist exactly two continous maps (called \newTerm{liftings}\footnote{indeed new term? todo}) $n^+$ and $n^- ∈ Γ_C(𝕊M)$ so that

\begin{equation*}

Q_{γ(t)} = s ⸨ n^± ⊗ n^± - \frac1mg ⸩

\end{equation*}

and $n^± =- n^-$.

Equivalently, given either of the two possible initial orientations at $γ(t_1)$, there exists a unique continous lifting with this initial orientation.

Suppose in addition that $\overline{Q}= s ⸨ \overline n ⊗ \overline n -\frac1m g⸩ ∈ 𝒬_{γ(τ)}^{𝕊}M$, $n ∈ 𝕊_{γ(τ)}M$, $τ ∈ [t_1, t_2]$

and that

\begin{equation*}

\text{there is some } 0 < ε < √2 \text{ such that }\abs{ Q_{γ(t)} - P^γ_{τt}\overline Q }_g < \abs s ε \text{ for all } t ∈ [t_1, t_2]

\end{equation*}

Then one of the liftings, let us say $n^±$ satisfies

\begin{equation}

\abs{n^± - P^γ_{τt}\overline n}_g ≤ ε \text{ for all } t ∈ [t_1, t_2]

\end{equation}

and the other $n^-=- n^+$ satisfies

\begin{equation}

\abs{n^± + P^γ_{τt}\overline n}_g ≤ ε \text{ for all } t ∈ [t_1, t_2]

\end{equation}

Here $P^γ_{τt}$ is the parallel transport along $γ$ from $γ(τ)$ to $γ(t)$.

(For definition see \textcite[Theorem 4.32][107]{q3-IntroRiemannLee}.)

\end{proposition}% end proposition Orientability on a path

\begin{proof}% of proposition Orientability on a path

We use the parallel transport of $γ$ to transform $Q$ to a $Q$-tensor valued map agnostic of the manifold:

\begin{equation*}

\tilde Q (t) := P^γ_{tt_0} Q_{γ(t)}

\end{equation*}

Since $P^γ_{tt_0}$ is a linear isometry, it maps $𝒬_{γ(t)}^{𝕊}M$ to $𝒬_{γ(t_0)}^{𝕊}M$.

Then $\tilde Q ⫶ [t_1, t_2] → 𝒬_{γ(t_0)}^{𝕊}M$.

By choosing an orthonormal basis for $𝒬_{γ(t_0)}^{𝕊}M$ we can identify it with the space $Q$ as it is used in \textcite[Lemma 3][505]{q4-Ball2011}.

Furthermore the parallel transport $P^γ_{tt_0}$ commutes with the projection $P$%

\footnote{Please excuse the different uses of the letter $P$.}

which is clear when writing $n$ and $P(n)$ in coordinates with respect to an orthonormal frame transported by $P^γ_{t_0t}$.

With this construction the statement follows directly from \textcite[Lemma 3][505]{q4-Ball2011}.