@@ -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

\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