Commit 4649a5ce by Felix Hilsky

### introduce b

parent 2186dcaa
 ... ... @@ -87,6 +87,13 @@ Q = s(n ⊗ n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê - for $Q ∈ W^{1-\frac1p, p}(∂Ω, 𝒬)$ orientable to $n ∈ W^{1-\frac1p, p}(Ω, 𝕊^2)$ $ℋ^{d-1}$-almost everywhere - $v_{,k}$ is the variable $v$ differentiated in the direction $k$. $Q_{ij,k}$ is the $i-j$'th component of the matrix field $Q$ differentiated in the direction $k$. This is not mentioned anywhere!! - Indeces which appear twice, are summed over, without mentioning it once!! Even when both are lower indeces. - $b:𝒬 → ℝP^2$ (and $b:𝒬_2 → ℝP^1$) is isometry between $𝒬$ and $RP^2$ by math b(s(n ⊗ n - \tfrac13 \operatorname{Id})) = \{n, -n\} ∈ ℝP^2  $⇒ 𝒬$ is Riemannian manifold (p. 12 (=504) section "Orientability Issues" With this identification, $P$ is a covering map (details p. 12) - in chapter 3.1 firstly for continous $Q$ since that's standard from topology - *Theorem 1*: $𝒬_2$ (one-dim), orientable on boundary of holes, $Q$ continous $⇒$ orientable (long proof with a lot of fiddling!) - chapter 3.2 Theorem 2: $Ω$ simply connected, $Q ∈ W^{1, p}$, $p \geq 2$ (!) $⇒$ orientable with Sobolev-Seminorm estimate. Counterexample for $p < 2$ ... ...
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