@@ -41,10 +41,10 @@ Since we want to lift from $𝒬^{𝕊}M$ to $𝕊M$, we need to check if $P ⫶
\end{proof}% of lemma Projection as a covering map
We want to use the theorem about lifting of continuous map from algebraic topology from \autocite[Proposition 1.33][61]{q4-q23-algebraic-topology}): % page 70 in pdf
\begin{proposition}[Lifting of continuous maps] \label{thm:lifting_of_continuous_maps}
\begin{externalresult}[Lifting of continuous maps] \label{thm:lifting_of_continuous_maps}
Suppose given a connected covering space $p ⫶(\tilde X, \tilde x_0) → (X, x_0)$ and a map $f⫶(Y, y_0) → (X,x_0)$ with $Y$ path-connected and locally path-connected.
Then a lift $\tilde f ⫶ (Y, y_0) → (\tilde X, \tilde x_0)$ of $f$ exists if and only if $f_* ⸨ π_1(Y, y_0) ⸩ ⊆ p_* ⸨ π_1(\tilde X, \tilde x_0) ⸩$.
\end{proposition}% end proposition Lifting of continuous maps
\end{externalresult}% end proposition Lifting of continuous maps
{% notation explanation
The spaces in this theorem are \newTerm{pointed spaces}.
They are equipped with one distinguished point which is used to define the fundamental group $π_1$.
@@ -98,9 +98,9 @@ Therefore we will use two specialised density results for the simply-connected c
For continuous line fields we already know that orientability can be checked on loops generationg the fundamental group.
In order to use this for Sobolev line fields it is useful if we can approximate them with smooth maps.
\begin{proposition}[Density of smooth maps on 2-manifolds] \label{thm:density_of_smooth_maps_on_2_manifolds} (from \textcite[Proposition in section 4][267]{q4-q34-approximation} ) Let $M$ be a 2-dimensional compact manifold. Let $N$ be a compact manifold without boundary. Then $C^∞(M, N)$ is dense in $W^{1,2}(M, N)$.
\begin{externalresult}[Density of smooth maps on 2-manifolds] \label{thm:density_of_smooth_maps_on_2_manifolds} (from \textcite[Proposition in section 4][267]{q4-q34-approximation} ) Let $M$ be a 2-dimensional compact manifold. Let $N$ be a compact manifold without boundary. Then $C^∞(M, N)$ is dense in $W^{1,2}(M, N)$.
Note that the authors denote $W^{1,2}$ by $L^2_1$.\footnote{todo: after formatting?}
\end{proposition}% end proposition Density of smooth maps on 2-manifolds
\end{externalresult}% end proposition Density of smooth maps on 2-manifolds
\footnote{todo: proof that this also applies to my norm for vector fields}
As the authors show this is in general not true for higher dimensions of the domain $M$.
Let $M$, $N$ be compact smooth manifolds with $M$ simply connected.
The Sobolev space $W^{1,2}(M, N)$ is defined as
$\setDef{f ∈ W^{1,2}(M, ℝ^d)}{f ∈ N \text{\ae}}$ where $N$ is embedded isometrically into $ℝ^d$.
Then for $u ∈ W^{1,2}(M, N)$ there exists a sequence $(u^{(k)})_k$ with $u^{(k)} ∈ C^∞(M, N)$ for all $k ∈ ℕ$ so that $u^{(k)}$ converges weakly to $u$.
\end{theorem}% end theorem Sequentially Weak Density of Smooth Manifold Maps
\end{externalresult}% end theorem Sequentially Weak Density of Smooth Manifold Maps
To use \cref{thm:sequentially_weak_density_of_smooth_manifold_maps} we need to show that orientability is preserved by weak convergence to transfer orientability of the continuous approximations to the approximated Sobolev function.
\begin{proposition}[Orientability preserved by weak convergence] \label{thm:orientability_preserved_by_weak_convergence_M_embedding}
% Note that in general we do not have $Γ_{W^{𝔓\!, q}}(E) ⊆ Γ_{W^{k,q}_{\loc}}(E)$ (with $k = \max \{k_1, …, k_s\}$) since there can be directions of differentiation that $𝔓$ does not include.
The following results about basic properties of $Γ_{W^{D,q}}(E)$ are proven in \textcite[Theorem I.19 and paragraph in front][29]{q14-sobolev-spaces-on-vectorbundles}:
\begin{proposition}\label{thm:ReflSep}
\begin{externalresult}[Seperability and Reflexivity] \label{thm:ReflSep}
\ask{What could be a good name (or markup/formatting?) of results that I do not prove but merely cite and use? I want to distinguish them because some should be called theorem because they are big results but I want that only my big results are theorems. Secondly it should be as clear as possible what is my work and what I merely copy from elsewhere.}
For any $D ∈ 𝒟(E, F)$, the spaces $Γ_{W^{D,q}}(E)$ are separable for all $q ∈ [1, ∞)$ and reflexive for all $q ∈ (1, ∞)$.
% removed Γ_{W^{D,q}_0}(E) removed since not mentioned beforehand and we are only looking at compact manifolds anyway.
\end{proposition}% end proposition
\begin{theorem}[Meyers-Serrin (density of smooth sections)] \label{thm:density_of_smooth_sections}
\end{externalresult}% end proposition
\begin{externalresult}[Meyers-Serrin (density of smooth sections)] \label{thm:density_of_smooth_sections}
\footnote{todo: Look closely again if we also get this up to the boundary%
\footnote{before look if we need it}}
In the situation of \cref{def:sobolev_space_of_vector_bundle_sections},
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@@ -201,7 +202,7 @@ The following results about basic properties of $Γ_{W^{D,q}}(E)$ are proven in
&\abs{f_n(x)} ≤ \norm f_{L^∞} ∈ [0, ∞] \text{ for all } x ∈ M, n ∈ ℕ \\
&\norm{f_n - f}_{W^{D, q}} → 0 \text{ as } n → ∞
\end{align*}
\end{theorem}% end theorem Meyers-Serrin
\end{externalresult}% end theorem Meyers-Serrin
% The proof uses approximations with mollifiers, reusing techniques from the Euclidean case.%
\footnote{Is it obvious that it also holds for manifolds with boundaries?
\textcite[Section 5.3.3 Theorem 3][266]{q15-pde-evans} sounds like it
Assume that $u ∈ W_{\loc}^{1,q}(Ω)$, $1 ≤ q ≤ ∞$ and let $Ω' ⋐ Ω$.
Then there exists $u^* ⫶ Ω → [−∞, ∞]$ such that $u^∗ = u$
almost everywhere in $Ω$ and $u^*$ is absolutely continuous on $(n − 1)$-dimensional
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@@ -100,7 +100,7 @@
coordinate axes and the classical partial derivatives of $u^∗$ coincide with the weak
partial derivatives of $u$ almost everywhere in $Ω$.
Conversely, if $u ∈ L_{\text{loc}}^q(Ω)$ and there exists $u^*$ as above such that $D i u^* ∈ L_{\text{loc}}^q(Ω)$, $i =1,…,\dim Ω$, then $u ∈ W_{\loc}^{1,q}(Ω)$.
\end{theorem}% end theorem Nikodym, ACL characterization
\end{externalresult}% end theorem Nikodym, ACL characterization
\begin{lemma}[Image of embedding based projection] \label{thm:image_of_embedding_based_projection}
Let $q ∈ [1, ∞]$. For $n ∈ W^{1,q}(M, 𝕊^{N-1})$, we have that its projection
