Commit e8b0f26b authored by Felix Hilsky's avatar Felix Hilsky
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tiny improvements, tiny fix

parent 9e6877ec
......@@ -64,7 +64,7 @@ To use \cref{thm:sequentially_weak_density_of_smooth_manifold_maps} we need to s
⇒ ‾∇_v \hat n
&= (‾∇_v \hat Q)(\hat n^T, ·) \label{eq:∇ninQterms}
\end{align}
Hence $‾∇\hat n$ is in $L^p$. Since also $\abs{\hat n} = 1$, $\hat n ∈ W^{1,p}(M, 𝕊^{N-1})$ and the norm is bounded by the norm of $\hat Q$.
Hence $‾∇\hat n$ is in $L^p$. Since also $\abs{\hat n} = 1$, $\hat n ∈ W^{1,p}(M, 𝕊^{N-1})$, the norm of $\hat n$ is bounded by the norm of $\hat Q$.
Since $(Q_{(k)})_k$ converges weakly, it is a bounded sequence by the Uniform Boundedness Principle.
Therefore the previous calculation \eqref{eq:∇ninQterms} shows that $(n_{(k)})_k$ is bounded as well.
......@@ -84,10 +84,10 @@ If we only used the intrinsic definition of the tensor fields it would be unclea
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.
for $𝒬^{𝕊'}^N$- and $𝕊^{N-1}$-valued functions.
\begin{theorem}[Sobolev orientability on simply connected manifolds] \label{thm:sobolev_orientability_on_simply_connected_manifolds}
Let $M$ be a simply-connected and $ι \hookrightarrow^N$ an isometric embedding.
Let $M$ be a simply-connected and $ι ⫶ M \hookrightarrow^N$ an isometric embedding.
Let $q ∈ [2, ∞)$\footnote{$q=$ braucht irgendeine Sonderbehandlung?}
Let $Q ∈ \secsob(𝒬^{𝕊}M)$.
Then $Q$ is orientable, \ie there exists $n ∈ \secsob(𝕊M)$ with $P(n) = Q$.
......
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