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\section{Introduction} \label{sec:introduction}
\footnote{The boldly typed questions are supposed to structure the introduction but should be removed once done.}
In 1888 the botanist Friedrich Reinitzer heated cholesterol, a fat, extracted from carrots.
At room temperature it is solid,
at \qty{145}{\celsius} it looked milky and appeared to be between solid and liquid.
After further heating the cholesterol became a clear liquid at \qty{179}{\celsius}.
The physicist and crystallographer Otto Lehmann then studied this phenomenon.
He described the double refraction observed in the cholesterol. \autocite[5 Carrots][76]{q45-LCD}

By this point double refraction was already well-known from solid crystals.
% \footnote{Understand that double refraction changes the polarity and why! (see \autocite[][82]{q45-LCD} "optical activity"}
\textbf{What are nematic uniaxial liquid crystals?}
Crystals are solid matter with a lot of structure.
Namely the molecule position and orientation is ordered.
This causes different indices of refraction depending on the polarity of the light.
The effect can be seen in \cref{fig:doublerefraction}.
\begin{figure}[htbp]
  \centering
  \includegraphics[height=3cm]{.maindir/zeichnungen/Crystal_on_graph_paper}
  \caption{Double refraction of a calcite crystal.
    The two images of a blue line come from the two parts of the light of orthogonal polarity which are refracted by a different amount.
    % Why are there only two images and not a lot?
  Image from \textcite{fig:double-refraction}.}
  \label{fig:doublerefraction}
\end{figure}
Lehmann saw the same effect in the heated cholesterol and deduced that this liquid also has some internal structure.
% Liquids do not have structure.
He therefore coined the term \newTerm{flüssiger Kristall}, in English \newTerm{liquid crystal}.
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.7\textwidth]{.maindir/zeichnungen/colorful-LC}
  \caption{Double refraction in a liquid crystal.
    The local order of direction causes the colors while over a bigger range this direction changes. Cross-polarized light is shining through this plate.
  Image from \textcite{q57-nice-image}.}
  \label{fig:schlieren}
\end{figure}

In a liquid crystal the molecules agree locally on their direction.
In some liquid crystals there is also a partial ordering in the position of the molecules.
Otherwise they are positioned isotropically.
Liquid crystals can move like a liquid but in many cases with a high viscosity.
\autocite[II.2][18]{q42-Handbook-LC-vol1}


% there is no order in the orientation and position of the molecules.
% Furthermore the position and orientation easily change over time.
% Liquid crystals are matter in a phase between the phases \enquote{solid} and \enquote{liquid}.
% They do have a long-range order in the orientation and some have a partial long-range order of the position.


Since Reinitzer's discovery, many substances between liquid and crystal
were found.
The different ways how they are structured are called \newTerm{mesophases}.
\enquote{Meso} means middle or in between.
The \enquote{phase} is the state matter can be in, usually liquid, gaseous or solid.
% They can be roughly divided into smectic, nematic and columnar.

The \newTerm{nematic} phase is the simplest and best-studied liquid crystal mesophase.
In a nematic liquid crystal the position of the molecules is isotropic.
That means that there is no structure in it.
% not correct for biaxial:
On the other hand the orientation of the molecules is very similar to their neighbours.
That means that there orientation is locally structured.

% folgender Absatz ist unklar. "Director"(=Pfeil) und "Direction" müssen auseinandergehalten werden
% Nematic liquid crystals can be uniaxial and biaxial.

The molecule orientation can be described with a single direction per point in case of \newTerm{uniaxial} liquid crystals and two orthogonal directions in case of \newTerm{biaxial} liquid crystals.
In this thesis only uniaxial cases are considered.
% On the other hand the molecule orientation in biaxial nematic liquid crystals can be described by two orthogonal directors \autocite[2.2.2][42]{q42-Handbook-LC-vol1}.
A direction is modelled with a vector of length $1$, called the \newTerm{director}.
In many uniaxial cases the molecules are rods that are much longer in one dimension than in the other two.
Then the director points from one end to the other end of the rod.
In other cases they are disc-shaped.
Then the director is orthogonal to the disc.

\textbf{What are examples and how are they used?}
Most liquid crystals are polymers.
Polymers are molecules that consist of many repeating parts mostly based on carbon.
Hence it is not surprising that a lot of current liquid crystal research is in biophysics and tries to understand how organic molecules like proteins assemble and form more complex structures.
% this sentence only works with the current citation style
One example is the work of \textcite{q54-selfassembly-Joshi}.
The authors study polysaccharides.
The liquid crystal structure lets the molecules self-assemble and create a membrane upon evaporation of the water.
This is an example of a liquid crystal that forms upon dissolving a substance in a suitable solvent.
This is a difference to the cholesterol mentioned at the beginning which becomes a liquid crystal at the suitable temperature.

By using rods and discs in combination the created membrane is bridging \SI{8}{\milli\metre} instead of $\SI{1}{\milli\metre}$ in other experiments without discs.

Another instance of self-assembly was found in liver tissue by \textcite{q56-liver-self-organizing}.
These researchers were surprised to find a long-range liquid crystal order where the constituents were not molecules but cells.

% \footnote{Todo: check if that's actually correct.
% It sounds a bit unclear and is it really correct?}
% The most famous usage of nematic liquid crystals is found in LCDs--liquid crystal displays.
% This technology uses a special kind of nematic liquid crystals, namely chiral nematic ones.
% This means that the molecules are ordered in layers and their directions form a spiral.
% This turns the polarity of light passing through.
% This is used with a polarization filter in front and behind the liquid crystal layer.
% The molecules are dipoles and
% therefore an electric field can turn the molecules and depending
% on how close they are to the original spiral structure, more or less light passes trough.
% By combining a lot of those pieces, three per pixel for three colors,
% we see images. \autocite[1 Double Refraction][3]{q45-LCD}. \footnote{how to do this citation}

We see from those very different examples that the term \enquote{liquid crystal} describes a diverse set of substances.
That makes this mesophase so interesting and worthy of study.

% Currently I miss the connection to liquid crystals and a suitable reference.
% Another current research direction are \enquote{active fluids}.
% That is an umbrella term for all kinds of matter or groups in which the parts can move on their own.
% That can be birds, fish, bacteria or molecular motors.

\textbf{Why do we want to model them? What should be answered by the models?}
% In order to understand reality physicists derive models to describe the reality
% trying to keep the models simple and still accurate.
%For liquid crystals in particular
We want to know which temperature, pressure, electric field and other parameters are necessary to bring the liquid crystal into a desired state.
For example \textcite{q5-q3-vesicles} try to understand the activity of a nematic film on a shape-changing bubble%
to use it as molecular sized motors.

\textbf{Which model do we have? (Landau-de Gennes, Frank-Oseen)}
To describe the preferred orientation of the molecules in a nematic liquid crystal
the first model\autocite{q47-original-frank} is named after Carl Wilhelm Oseen who developed the model in 1933 and Frederick Charles Frank who refined it in 1958.
The preferred orientation is given by a director field, \ie a unit vector field.
A field is just a vector-valued function.
The value in each point describes the average orientation of the molecules close to that point.
% When regarding a sufficiently small patch of the domain all molecules have an orientation close to this direction.
The model further describes how to calculate how the molecules are oriented:
The director field minimizes the \enquote{Frank-Oseen energy}.
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.8\textwidth, height=5cm, keepaspectratio]{.maindir/zeichnungen/molecule-orientation}
  \caption{Preferred orientation of molecules in nematic liquid crystal. From \autocite[][3]{q46-fig-nematic-molecules}}
  \label{fig:nematic-molecules}
\end{figure}

The Frank-Oseen model is simple but has shortcomings.
We address one of them, namely that
% 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]
  \centering
  \includegraphics[width=0.8\textwidth,height=5cm,keepaspectratio]{.maindir/zeichnungen/unorientable-line-field}
  \caption{An unorientable line field on a domain with one hole.
    Intuitively it is clear that a director field that points upwards on the left of the hole has to point downwards on the right.
    Therefore it cannot agree with itself below the hole, creating an discontinuity where the line field is continuous.
  From \parencite[Fig.~1][]{q4-Ball2011}.}
  \label{fig:unorientable_line_field}
\end{figure}
\cref{fig:unorientable_line_field} illustrates how we can picture a line field on a flat domain.
Mathematically the codomain of the line field is the real projective space.
Since unit vector fields are easier and more common to handle mathematically the question is:
\begin{quote}
  Do we get a correct result if we work with unit vector fields or do we have to use line fields?
\end{quote}
As a main preparation we investigate:
\begin{quote}
  Is a given line field orientable?
\end{quote}
We call a line field \newTerm{orientable} if a unit vector field exists that points in every point in one of the two directions of the line and is continuous if the line field is continuous or weakly differentiable if the line field is weakly differentiable.
\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 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.
\imp{Covering} is the term from algebraic topology to describe that the two opposite directions are identified.
The theorem we use states that in every point one of the two vectors can be chosen in a continuous way (\enquote{a lifting}) if and only if this is possible on a set of loops that generate the fundamental group of the domain.

As mentioned above the goal is to calculate how the molecules in a liquid crystal are oriented under given circumstances.
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.
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.
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:
simply connected domains (\cref{sec:sobolev_orientability_on_simply_connected_manifolds})
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 $q ≥ 2$ 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.%
\ask{Vielleicht die letzten beiden Sätze weglassen? Das wird technisch so gemacht und es ist auch ein Grund, warum die Technik nicht auf höhere Dimensionen angewendet werden kann. Aber ich finde es nicht so klar verständlich und vielleicht auch nicht so wichtig.}

Some applications like \parencite{q5-q3-vesicles} study thin films of liquid crystals which are best modeled as two-dimensional surfaces.
Therefore this thesis generalises the results of \parencite{q4-Ball2011} to curved surfaces.
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 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 in \cref{sec:qtensor}.
% $Q$-tensors are a more general model for liquid crystals
% Here would be a good point to mention Q-Tensors but they and Landau de Gennes were not introduced yet.
% 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.
Nevertheless we need to take care because there exist several possible definitions.
In \cref{sec:sobolev_tensor_fields} we define them intrinsically.
The other two subsections define them differently and show that those definitions are equivalent.
% 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

The first of the main results of the thesis considers simply-connected domains.
In this case it was shown that all Sobolev functions can be weakly approximated by smooth functions.\ask{Is it clear that this density result is not from me but other peoples result and I just use it?}
As explained in \cref{note:choice_of_codomain} the only way to use this theorem is to have a common codomain for all points of the domain.
We combine all tangent spaces by embedding the domain into a Euclidean space.
The tangent vectors are then also elements of a common Euclidean space
and the Sobolev definition for $ℝ^N$-valued functions can be used.
\cref{sec:embedding_M} discusses this definition of Sobolev function and shows that it is equivalent with the intrinsic definition even though the norm is unequal.
The smooth weak approximations\ask{is it clear that weak refers to how they approximate and smooth refers to the smoothness of every approximating function?}
that the density result give are then line fields in the surrounding space, not necessarily tangent.
Fortunately the continuous orientability result holds for those as well.
The last bit to the proof of orientability of all Sobolev line fields on simply-connected manifold domains (\cref{thm:sobolev_orientability_on_simply_connected_manifolds}) is the stability of orientability under weak convergence (\cref{thm:orientability_preserved_by_weak_convergence_M_embedding}).

The other main result considers surfaces (two-dimensional manifolds)\ask{can I somehow avoid the ( ) and still explain that surface and two-dimensional manifold are synonyms?}.
We show that we can similarly to continuous line fields check orientability of Sobolev $W^{1,2}$ line fields on any set of loops that generate the fundamental group of the domain.
It turns out that all interesting orientable surfaces also have a trivial tangent bundle, \ie there exists a global frame.
This frame allows us to identify all tangent spaces and therefore consider vector and line fields as maps into $ℝ^2$.
The Sobolev function definition that this identification induces is discussed in
\cref{sec:sobolev_field_spaces_on_parallelizable_manifolds} where we also show that this definition is also equivalent to the intrinsic one.
The rest of the argument resembles the proof from Section 4 in \parencite{q4-Ball2011}:

The simple direction from orientability on the surface to orientability on loops is a classical trace result.
In the other direction we have to show that the approximations are orientable on the loops if the approximated Sobolev field is orientable.
For this we use the winding number which is an integer and depends continuously on the function.
The winding number says how often a circle valued function on a loop wraps around the circle and is originally defined in algebraic topology for continuous functions.
It got generalised for fractional $W^{1/2,2}$ Sobolev functions which is the suitable space of the traces of the line fields on surfaces.
To use it for line fields we introduce an auxiliary complex unit valued function that turns twice as fast as the line field.
If the auxiliary function has an even winding number a unit vector valued function that winds around the circle half as fast can be found and is an orientation of the line field.
In contrast to higher dimensions, on surfaces we have density of smooth functions in norm.
Since the winding number is an integer and depends continuously on the function, approximations that are close enough in the norm sense have the same winding number and are thus orientable on loops as well.
At this point the orientability criterion for continuous line fields is used and the convergence in norm gives the orientation of the approximated line field.\todo{Probably shorten this paragraph.}

\vskip 5cm

The mentioned density results are exploited in \cref{sec:orientability_of_sobolev_line_fields}.
If the domain is simply connected it was shown that all Sobolev functions can be weakly approximated by smooth functions.
We have this by embedding the domain manifold and 

Secondly it does not take into account how close the molecules are to the preferred direction.
This is also related to the fact that liquid crystals are in some cases forced to have \enquote{defects} where no director can be properly defined.
Thirdly the Frank-Oseen model assumes uniaxiality but biaxial nematic liquid crystals were found later.
In 1974 \textcite{q48-original-de-gennes} Pierre-Gilles de Gennes therefore formulated another model of so-called \newTerm{$Q$-tensors} that are described in \cref{sec:qtensor}.
It is based on the solid-liquid phase transition theory of Lev Landau. \autocite[Section 10][162]{q45-LCD}
Instead of a unit vector, the orientation is described by a trace-free symmetric tensor.
One parameter of the $Q$-tensor is the order parameter.
It describes how close the molecule directions are to the preferred direction.

In the following we restrict ourselves to the simpler case of constant order parameter and uniaxiality.
Therefore our restricted $Q$-tensor model treats opposite directors as equal but does not fix the other shortcomings of the Frank-Oseen model.
We therefore call these restricted $Q$-tensor fields \newTerm{line fields}.

\textbf{What was the work of Ball and Zarnescu?}
The Landau-de Gennes model is more accurate but harder to deal with than the Frank-Oseen model.
\textcite{q4-Ball2011} discusses under which conditions the Frank-Oseen model can be used. They limit their study to
\begin{itemize}
  \item uniaxial nematic liquid crystals,
  \item constant order parameter, no defects,
  \item two-dimensional domain with holes,
  \item continuous and Sobolev $W^{1,q}$ line fields (some results only for $q=2$),
  \item harmonic energy which is a special case of the Frank-Oseen energy.
\end{itemize}
They show that a continuous or $W^{1,q}$ line field is orientable if and only if it is orientable on the boundaries of the holes of the domain.
Orientable here means that we can replace the line field by a unit vector director field.
Furthermore they give a characterisation of the domains in which the line field minimizer of the harmonic energy is orientable.

\textbf{What is my work about? Why is that an interesting question?}
The following work is based on the ideas \textcite{q4-Ball2011} and generalises them to manifolds.
Many applications and numerical studies are taking place on curved surfaces as for example the mentioned \textcite{q5-q3-vesicles}.
The question of orientability is closely related to the topology of the underlying domain.
Since manifolds have a different topology than flat domains work is required to generalise the results of \textcite{q4-Ball2011}.

\footnote{2 people asked if this paragraph is actually needed. Should I remove it or move it somewhere else?}
\textbf{What are my (main) results?}
There are three main results of this thesis.
Firstly we%
\footnote{is \enquote{we} actually suitable here?}
show that on simply-connected manifolds on any dimension all $W^{1,q}$ line fields are orientable in \cref{thm:sobolev_orientability_on_simply_connected_manifolds}.
Secondly we give a characterisation of orientability of $W^{1,q}$ line fields on surfaces, \ie two-dimensional manifolds in \cref{thm:characterisation_of_orientability_of_sobolev_line_fields}.
Note that due to the topology there are no unit vector fields on surfaces without boundary except of the torus.
Thirdly we present on example of a torus with \enquote{boundary}\footnote{the torus has no boundary, so it's a different type of condition. But how to call it?} condition that has a line field minimizer of the harmonic energy that is not orientable.

\textbf{What is the structure of my thesis?}
In \cref{sec:qtensor} the $Q$-tensor model is adapted to manifolds.
In order to define what Sobolev ($W^{1,q}$) line and unit vector fields are, we define them intrinsically in \cref{sec:sobolev_tensor_fields}.
For \cref{thm:sobolev_orientability_on_simply_connected_manifolds} and \cref{thm:characterisation_of_orientability_of_sobolev_line_fields} we need two different definitions of Sobolev fields though and therefore need to show that those three definitions are equivalent.
This is done in \cref{sec:embedding_M} via an embedding of the base manifold and
\cref{sec:sobolev_field_spaces_on_parallelizable_manifolds} via a global frame respectively.
The rest of the thesis is structured similarly to \textcite{q4-Ball2011}.
Since the results about Sobolev fields are achieved via approximation with continuous fields, the continuous case is studied in \cref{sec:continuous_q_lift}.
\cref{thm:sobolev_orientability_on_simply_connected_manifolds} and \cref{thm:characterisation_of_orientability_of_sobolev_line_fields} are then proven in
\cref{sec:orientability_of_sobolev_line_fields}.
Finally the example for non-orientability of the line field minimizer has its own \cref{sec:harmonic_energy_minimizer_orientation}.
\docEnd