### explain proof strategy for main results

`still missing: mention of counterexample on torus`
parent 3770ec8c
 ... ... @@ -202,25 +202,55 @@ As a consequence the definitions of unit vector fields, line fields and Sobolev 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} 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 several possible definitions. 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 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. 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. 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. ... ...
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