Commit e69749d9 authored by Felix Hilsky's avatar Felix Hilsky
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make paragraph a remark to be able to refer to it with a number

parent 1a670fef
......@@ -79,12 +79,13 @@ To use \cref{thm:sequentially_weak_density_of_smooth_manifold_maps} we need to s
In the tangent case, \ie $Q ∈ Γ_{W^{1,p}}(𝒬^{𝕊'}M)$, this is false \ae.
Hence $n ∈ Γ_{W^{1,p}}(𝕊M)$ is tangent to $M$.
\end{proof} % of proposition Orientability preserved by weak convergence
If we only used the intrinsic definition of the tensor fields it would be unclear how to use \cref{thm:sequentially_weak_density_of_smooth_manifold_maps}.
If we used it with $N = 𝒬M$, we would not have the guarantee that $u^{(k)}_p ∈ T_pM$. If we used it with $N = 𝒬_pM$ we have to somehow identify all tangent spaces which is -- in the general case -- not possible in a smooth way.
By embedding $M$ and considering non-tangent tensor fields we solve this problem but need \cref{thm:orientability_on_simply_connected_manifolds} and
\cref{thm:orientability_preserved_by_weak_convergence_M_embedding}
for $𝒬^{𝕊'}^N$- and $𝕊^{N-1}$-valued functions.
\begin{note}[Choice of codomain] \label{note:choice_of_codomain}
If we only used the intrinsic definition of the tensor fields it would be unclear how to use \cref{thm:sequentially_weak_density_of_smooth_manifold_maps}.
If we used it with $N = 𝒬M$, we would not have the guarantee that $u^{(k)}_p ∈ T_pM$. If we used it with $N = 𝒬_pM$ we have to somehow identify all tangent spaces which is -- in the general case -- not possible in a smooth way.
By embedding $M$ and considering non-tangent tensor fields we solve this problem but need \cref{thm:orientability_on_simply_connected_manifolds} and
\cref{thm:orientability_preserved_by_weak_convergence_M_embedding}
for $𝒬^{𝕊'}^N$- and $𝕊^{N-1}$-valued functions.
\end{note} % end note Choice of codomain
\begin{theorem}[Sobolev orientability on simply connected manifolds] \label{thm:sobolev_orientability_on_simply_connected_manifolds}
Let $M$ be a simply-connected and $ι ⫶ M \hookrightarrow^N$ an isometric embedding.
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