= not below each other

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Felix Hilsky 2022-10-31 23:31:07 +01:00
parent e1842bd995
commit 1651b431ee

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@ -84,7 +84,10 @@ ${n ∈ Γ_{W^{1,q}}(𝕊M)}$ is also tangent to $M$.
It remains to be shown that $n$ is tangent to $M$ if $Q$ is tangent to $M$.
For the sake of contradiction, assume $n_p · η = n_p(η^T) \neq 0$ for some $p ∈ M$ and $η ⊥ T_pM$.
Then $P_N(n) (η^T, η^T) = Q(η^T, η^T) = (n ⊗ n)(η^T, η^T) = (n(η^T))^2 \neq 0$.
Then
\begin{equation*}
P_N(n) (η^T, η^T) = Q(η^T, η^T) = (n ⊗ n)(η^T, η^T) = (n(η^T))^2 \neq 0 \ .
\end{equation*}
If $Q$ is tangent to $M$, \ie $Q ∈ Γ_{W^{1,q}}(𝒬^{𝕊'}M)$, this is false \ae.
Hence $n ∈ Γ_{W^{1,q}}(𝕊M)$ is \ae tangent to $M$.
\end{proof} % of proposition Orientability preserved by weak convergence