### 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 $𝒬^{𝕊'}ℝ^N$ and $𝕊^{N-1}$ valued functions. for $𝒬^{𝕊'}ℝ^N$- 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 $ι \hookrightarrow ℝ^N$ an isometric embedding. Let $M$ be a simply-connected and $ι ⫶ M \hookrightarrow ℝ^N$ 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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