@@ -140,7 +140,7 @@ The director field minimizes the \enquote{Frank-Oseen energy}.

\end{figure}

The Frank-Oseen model is simple but has shortcomings.

The paper of \textcite{q4-Ball2011} and this thesis address one of the shortcomings:

% The paper of \textcite{q4-Ball2011} and this thesis address one of the shortcomings: % die haben sich das ja nicht ausgedacht

The two ends of the rod-like molecules are equal so that the opposite orientation is actually the same.

Therefore it is more appropriate to use a \newTerm{line field} instead of a unit vector field.

\begin{figure}[htbp]

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@@ -166,7 +166,7 @@ We call a line field \newTerm{orientable} if a unit vector field exists that poi

\cref{fig:unorientable_line_field} shows an example of a continuous line field that is not orientable as explained in the caption.

In this example we already see that the hole in the domain plays a crucial role.

Without it we could not construct a non-orientable continuous line field.

Without it every continuous line field is orientable.

More generally the question of orientability of continuous line fields depends purely on the topology of the domain and is solved by a algebraic topology result about liftings.

This is discussed in \cref{sec:continuous_q_lift}.

The idea is that the sphere in which the unit vector fields map is a double \newTerm{covering} of the real projective space in which the line fields map.

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@@ -177,16 +177,16 @@ As mentioned above the goal is to calculate how the molecules in a liquid crysta

These circumstances include among others the shape and topology of the domain, boundary values and electrical or magnetic fields.

They are modeled with the Frank-Oseen energy, a functional of the line or director field.

The orientations of the molecules than minimize this energy.

These minimization problems are usually not solved in the class of continuous functions but weakly differentiable (Sobolev) functions.

Such minimization problems are usually not solved in the class of continuous functions but weakly differentiable (Sobolev) functions.

Therefore we need to transfer the results about orientability of continuous line fields to weakly differentiable fields.

Thus we want to approximate weakly differentiable functions by continuous functions.

This is often not possible because the functions are mapping into manifolds, namely the sphere and the real projective space and continuous functions mapping into manifolds are in general not dense in the respective Sobolev spaces.

Consequently we want to approximate weakly differentiable functions by continuous functions.

This is in general not possible because the functions are mapping into manifolds, namely the sphere and the real projective space and continuous functions mapping into manifolds are in general not dense in the respective Sobolev spaces.

In two cases we have sufficient density results though:

and surfaces (two-dimensional domains) (\cref{sec:sobolev_orientability_on_surface}).

For flat domains \textcite{q4-Ball2011} discusses the presented problems.

They showed that on simply-connected two- and three-dimensional domains for $p ≥ 2$ all weakly differentiable $W^{1,p}$ line fields are orientable \parencite[Thm.~2][]{q4-Ball2011}.

They showed that on simply-connected two- and three-dimensional domains for $q ≥ 2$ all weakly differentiable $W^{1,q}$ line fields are orientable \parencite[Thm.~2][]{q4-Ball2011}.

For two-dimensional domains with holes they showed that orientability of a line field is equivalent to orientability on the boundaries of the holes \parencite[Prop.~7][]{q4-Ball2011}.

Furthermore they characterise orientability on a hole boundary with the winding number of an auxiliary unit vector valued map.

Intuitively spoken, this auxiliary map turns twice as fast as the line field and if its winding number is even, the line field is orientable.%

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@@ -197,7 +197,8 @@ Therefore this thesis generalises the results of \parencite{q4-Ball2011} to curv

In order to stay as general as possible we also manifolds in higher dimensions.

% As long as possible we do not limit ourselves to surfaces though.

As a consequence the definitions of unit vector fields, line fields and Sobolev spaces need to be reconsidered

because the directions need to be tangent to the manifold.

because the directions need to be tangent to the manifold and the tangent spaces at different points are distinct.

In the flat case on the other hand all tangent spaces can be identified.

As the first step we define the unit vector fields and line fields as vector fields tangent to the domain manifold.

The line fields are defined as special 2-tensors because it is a special case of the \newTerm{$Q$-tensor} model discussed below.

\todo{actually discuss it}

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@@ -206,13 +207,13 @@ The line fields are defined as special 2-tensors because it is a special case of

% At this point we decide to define them as contravariant since this is the natural way of seeing directions.

% In most papers

Then in \cref{sec:sobolev_spaces} we define Sobolev spaces of tangent vector fields.

Proper discussion of Sobolev tangent vector fields is surprisingly rare in the literature probably because there are no major difficulties.

Proper discussion of Sobolev tangent vector fields is surprisingly rare in the literature, probably because there are no major difficulties.

Nevertheless we need to take care because there several possible definitions.

In \cref{sec:sobolev_tensor_fields} we define them intrinsically.

% while in \cref{sec:embedding_M} and \cref{sec:sobolev_field_spaces_on_parallelizable_manifolds} we use an embedding of the domain manifold or a global frame respectively\ask{do I need a comma here somewhere?} to

Then in \cref{sec:embedding_M} we use an embedding of the domain into a Euclidean space.

This definition is necessary for the proof of orientability of line fields on simply-connected domains.

On surfaces we have a global frame and use this to view tangent vector fields as $ℝ^m$-valued function for which Sobolev spaces definitions are clear.

Thirdly, on surfaces we have a global frame and use this to view tangent vector fields as $ℝ^m$-valued function for which Sobolev spaces definitions are clear.

In \cref{sec:sobolev_spaces} we show that all of these definitions are equivalent

which boils down to the equivalence of the norms and the density of smooth functions in all defined Sobolev spaces.

Here the smooth functions are dense because the codomains are in every point vector spaces.