introduction.tex 20.7 KB
 Felix Hilsky committed Oct 17, 2021 1 2 3 4 5 %! TEX program = lualatex \input{.maindir/tex/header/preamble-section} % inputs the preamble only if necessary \docStart  Felix Hilsky committed Jul 10, 2022 6 \section{Introduction} \label{sec:introduction}  Felix Hilsky committed Jul 08, 2022 7 \footnote{The boldly typed questions are supposed to structure the introduction but should be removed once done.}  Felix Hilsky committed Aug 01, 2022 8 In 1888 the botanist Friedrich Reinitzer heated cholesterol, a fat, extracted from carrots.  Felix Hilsky committed Aug 01, 2022 9 At room temperature it is solid,  Felix Hilsky committed Aug 01, 2022 10 at \qty{145}{\celsius} it looked milky and appeared to be between solid and liquid.  Felix Hilsky committed Aug 01, 2022 11 After further heating the cholesterol became a clear liquid at \qty{179}{\celsius}.  Felix Hilsky committed Aug 03, 2022 12 13 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}  Felix Hilsky committed Aug 01, 2022 14   Felix Hilsky committed Aug 03, 2022 15 By this point double refraction was already well-known from solid crystals.  Felix Hilsky committed Aug 05, 2022 16 % \footnote{Understand that double refraction changes the polarity and why! (see \autocite[][82]{q45-LCD} "optical activity"}  Felix Hilsky committed Jul 07, 2022 17 \textbf{What are nematic uniaxial liquid crystals?}  Felix Hilsky committed Jul 29, 2022 18 Crystals are solid matter with a lot of structure.  Felix Hilsky committed Aug 01, 2022 19 Namely the molecule position and orientation is ordered.  Felix Hilsky committed Aug 03, 2022 20 This causes different indices of refraction depending on the polarity of the light.  Felix Hilsky committed Aug 01, 2022 21 The effect can be seen in \cref{fig:doublerefraction}.  Felix Hilsky committed Aug 01, 2022 22 \begin{figure}[htbp]  Felix Hilsky committed Jul 08, 2022 23  \centering  Felix Hilsky committed Aug 01, 2022 24 25  \includegraphics[height=3cm]{.maindir/zeichnungen/Crystal_on_graph_paper} \caption{Double refraction of a calcite crystal.  Felix Hilsky committed Aug 01, 2022 26 27  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?  Felix Hilsky committed Aug 01, 2022 28 29  Image from \textcite{fig:double-refraction}.} \label{fig:doublerefraction}  Felix Hilsky committed Jul 08, 2022 30 \end{figure}  Felix Hilsky committed Aug 01, 2022 31 32 Lehmann saw the same effect in the heated cholesterol and deduced that this liquid also has some internal structure. % Liquids do not have structure.  Felix Hilsky committed Aug 01, 2022 33 34 35 36 37 38 39 40 41 42 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}  Felix Hilsky committed Aug 01, 2022 43 44 45 46 47 48 49 50 51 52 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}.  Felix Hilsky committed Aug 10, 2022 53 % They do have a long-range order in the orientation and some have a partial long-range order of the position.  Felix Hilsky committed Aug 01, 2022 54   Felix Hilsky committed Jul 07, 2022 55   Felix Hilsky committed Aug 01, 2022 56 Since Reinitzer's discovery, many substances between liquid and crystal  Felix Hilsky committed Aug 01, 2022 57 58 were found. The different ways how they are structured are called \newTerm{mesophases}.  Felix Hilsky committed Aug 01, 2022 59 \enquote{Meso} means middle or in between.  Felix Hilsky committed Aug 01, 2022 60 The \enquote{phase} is the state matter can be in, usually liquid, gaseous or solid.  Felix Hilsky committed Aug 01, 2022 61 62 % They can be roughly divided into smectic, nematic and columnar.  Felix Hilsky committed Aug 01, 2022 63 The \newTerm{nematic} phase is the simplest and best-studied liquid crystal mesophase.  Felix Hilsky committed Aug 01, 2022 64 In a nematic liquid crystal the position of the molecules is isotropic.  Felix Hilsky committed Jul 08, 2022 65 That means that there is no structure in it.  Felix Hilsky committed Aug 01, 2022 66 % not correct for biaxial:  Felix Hilsky committed Aug 01, 2022 67 68 On the other hand the orientation of the molecules is very similar to their neighbours. That means that there orientation is locally structured.  Felix Hilsky committed Aug 01, 2022 69 70 71 72 73  % 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.  Felix Hilsky committed Aug 05, 2022 74 In this thesis only uniaxial cases are considered.  Felix Hilsky committed Aug 01, 2022 75 76 77 78 79 80 % 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.  Felix Hilsky committed Jul 08, 2022 81 82  \textbf{What are examples and how are they used?}  Felix Hilsky committed Aug 01, 2022 83 Most liquid crystals are polymers.  Felix Hilsky committed Aug 03, 2022 84 Polymers are molecules that consist of many repeating parts mostly based on carbon.  Felix Hilsky committed Jul 29, 2022 85 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.  Felix Hilsky committed Aug 01, 2022 86 87 % this sentence only works with the current citation style One example is the work of \textcite{q54-selfassembly-Joshi}.  Felix Hilsky committed Jul 29, 2022 88 The authors study polysaccharides.  Felix Hilsky committed Aug 01, 2022 89 The liquid crystal structure lets the molecules self-assemble and create a membrane upon evaporation of the water.  Felix Hilsky committed Aug 01, 2022 90 91 92 93 94 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.  Felix Hilsky committed Jul 29, 2022 95 Another instance of self-assembly was found in liver tissue by \textcite{q56-liver-self-organizing}.  Felix Hilsky committed Aug 01, 2022 96 These researchers were surprised to find a long-range liquid crystal order where the constituents were not molecules but cells.  Felix Hilsky committed Jul 29, 2022 97   Felix Hilsky committed Aug 05, 2022 98 99 % \footnote{Todo: check if that's actually correct. % It sounds a bit unclear and is it really correct?}  Felix Hilsky committed Aug 05, 2022 100 101 102 103 104 105 106 107 108 109 % 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}  Felix Hilsky committed Jul 08, 2022 110   Felix Hilsky committed Aug 01, 2022 111 112 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.  Felix Hilsky committed Jul 29, 2022 113 114 115 116 117  % 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.  Felix Hilsky committed Jul 07, 2022 118 119  \textbf{Why do we want to model them? What should be answered by the models?}  Felix Hilsky committed Aug 01, 2022 120 121 122 123 % 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.  Felix Hilsky committed Jul 09, 2022 124 125 126 127 128 129 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.  Felix Hilsky committed Aug 06, 2022 130 131 132 133 134 135 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]  Felix Hilsky committed Jul 09, 2022 136 137 138 139 140 141 142  \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.  Felix Hilsky committed Aug 10, 2022 143 % The paper of \textcite{q4-Ball2011} and this thesis address one of the shortcomings: % die haben sich das ja nicht ausgedacht  Felix Hilsky committed Aug 06, 2022 144 145 146 147 148 149 150 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.  Felix Hilsky committed Aug 10, 2022 151  Therefore it cannot agree with itself below the hole, creating an discontinuity where the line field is continuous.  Felix Hilsky committed Aug 06, 2022 152 153 154 155 156 157 158 159 160 161 162 163 164  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}  Felix Hilsky committed Aug 10, 2022 165 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.  Felix Hilsky committed Aug 06, 2022 166 167 168 \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.  Felix Hilsky committed Aug 10, 2022 169 Without it every continuous line field is orientable.  Felix Hilsky committed Aug 10, 2022 170 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.  Felix Hilsky committed Aug 06, 2022 171 This is discussed in \cref{sec:continuous_q_lift}.  Felix Hilsky committed Aug 10, 2022 172 173 174 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.  Felix Hilsky committed Aug 06, 2022 175 176 177 178  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.  Felix Hilsky committed Aug 10, 2022 179 The orientations of the molecules than minimize this energy.  Felix Hilsky committed Aug 10, 2022 180 Such minimization problems are usually not solved in the class of continuous functions but weakly differentiable (Sobolev) functions.  Felix Hilsky committed Aug 06, 2022 181 Therefore we need to transfer the results about orientability of continuous line fields to weakly differentiable fields.  Felix Hilsky committed Aug 10, 2022 182 183 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.  Felix Hilsky committed Aug 10, 2022 184 In two cases we have sufficient density results though:  Felix Hilsky committed Aug 06, 2022 185 186 187 simply connected domains (\cref{sec:sobolev_orientability_on_simply_connected_manifolds}) and surfaces (two-dimensional domains) (\cref{sec:sobolev_orientability_on_surface}).  Felix Hilsky committed Aug 10, 2022 188 For flat domains \textcite{q4-Ball2011} discusses the presented problems.  Felix Hilsky committed Aug 10, 2022 189 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}.  Felix Hilsky committed Aug 07, 2022 190 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}.  Felix Hilsky committed Aug 06, 2022 191 Furthermore they characterise orientability on a hole boundary with the winding number of an auxiliary unit vector valued map.  Felix Hilsky committed Aug 07, 2022 192 193 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.}  Felix Hilsky committed Aug 06, 2022 194   Felix Hilsky committed Aug 10, 2022 195 196 197 198 199 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  Felix Hilsky committed Aug 10, 2022 200 201 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.  Felix Hilsky committed Aug 10, 2022 202 203 204 205 206 207 208 209 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} % $Q$-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.  Felix Hilsky committed Aug 10, 2022 210 Proper discussion of Sobolev tangent vector fields is surprisingly rare in the literature, probably because there are no major difficulties.  Felix Hilsky committed Aug 10, 2022 211 212 213 214 215 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.  Felix Hilsky committed Aug 10, 2022 216 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.  Felix Hilsky committed Aug 10, 2022 217 218 219 220 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.  Felix Hilsky committed Aug 06, 2022 221 222   Felix Hilsky committed Jul 09, 2022 223 Secondly it does not take into account how close the molecules are to the preferred direction.  Felix Hilsky committed Aug 01, 2022 224 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.  Felix Hilsky committed Aug 01, 2022 225 Thirdly the Frank-Oseen model assumes uniaxiality but biaxial nematic liquid crystals were found later.  Felix Hilsky committed Aug 01, 2022 226 227 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}  Felix Hilsky committed Jul 09, 2022 228 Instead of a unit vector, the orientation is described by a trace-free symmetric tensor.  Felix Hilsky committed Aug 01, 2022 229 230 One parameter of the $Q$-tensor is the order parameter. It describes how close the molecule directions are to the preferred direction.  Felix Hilsky committed Jul 07, 2022 231   Felix Hilsky committed Aug 01, 2022 232 In the following we restrict ourselves to the simpler case of constant order parameter and uniaxiality.  Felix Hilsky committed Aug 03, 2022 233 Therefore our restricted $Q$-tensor model treats opposite directors as equal but does not fix the other shortcomings of the Frank-Oseen model.  Felix Hilsky committed Jul 09, 2022 234 We therefore call these restricted $Q$-tensor fields \newTerm{line fields}.  Felix Hilsky committed Jul 07, 2022 235 236  \textbf{What was the work of Ball and Zarnescu?}  Felix Hilsky committed Aug 01, 2022 237 The Landau-de Gennes model is more accurate but harder to deal with than the Frank-Oseen model.  Felix Hilsky committed Aug 01, 2022 238 \textcite{q4-Ball2011} discusses under which conditions the Frank-Oseen model can be used. They limit their study to  Felix Hilsky committed Jul 09, 2022 239 \begin{itemize}  Felix Hilsky committed Aug 01, 2022 240  \item uniaxial nematic liquid crystals,  Felix Hilsky committed Jul 09, 2022 241 242  \item constant order parameter, no defects, \item two-dimensional domain with holes,  Felix Hilsky committed Aug 01, 2022 243  \item continuous and Sobolev $W^{1,q}$ line fields (some results only for $q=2$),  Felix Hilsky committed Jul 09, 2022 244 245  \item harmonic energy which is a special case of the Frank-Oseen energy. \end{itemize}  Felix Hilsky committed Jul 29, 2022 246 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.  Felix Hilsky committed Jul 09, 2022 247 248 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.  Felix Hilsky committed Jul 07, 2022 249 250  \textbf{What is my work about? Why is that an interesting question?}  Felix Hilsky committed Jul 09, 2022 251 252 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}.  Felix Hilsky committed Aug 03, 2022 253 254 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}.  Felix Hilsky committed Jul 07, 2022 255   Felix Hilsky committed Aug 01, 2022 256 \footnote{2 people asked if this paragraph is actually needed. Should I remove it or move it somewhere else?}  Felix Hilsky committed Jul 07, 2022 257 \textbf{What are my (main) results?}  Felix Hilsky committed Jul 09, 2022 258 There are three main results of this thesis.  Felix Hilsky committed Aug 01, 2022 259 Firstly we%  Felix Hilsky committed Aug 10, 2022 260 \footnote{is \enquote{we} actually suitable here?}  Felix Hilsky committed Aug 01, 2022 261 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}.  Felix Hilsky committed Jul 09, 2022 262 263 264 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.  Felix Hilsky committed Jul 07, 2022 265 266  \textbf{What is the structure of my thesis?}  Felix Hilsky committed Jul 09, 2022 267 268 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}.  Felix Hilsky committed Jul 11, 2022 269 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.  Felix Hilsky committed Aug 01, 2022 270 271 272 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}.  Felix Hilsky committed Jul 22, 2022 273 Since the results about Sobolev fields are achieved via approximation with continuous fields, the continuous case is studied in \cref{sec:continuous_q_lift}.  Felix Hilsky committed Jul 28, 2022 274 275 \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}.  Felix Hilsky committed Jul 09, 2022 276 Finally the example for non-orientability of the line field minimizer has its own \cref{sec:harmonic_energy_minimizer_orientation}.  Felix Hilsky committed Oct 17, 2021 277 \docEnd