Commit 4649a5ce authored by Felix Hilsky's avatar Felix Hilsky
Browse files

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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