Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Exact characteristic normal coordinates

Exact characteristic normal coordinates · lines 897–933

Em01-C@3#eq:N

Define

q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},

S_{resp} = {q_{R} = 0, a = z},

and

N = {z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β, ℓ_{n} = (1, n), n \in S^{2}} .

Fix $w_{0} \in N$ . On a sphere chart choose a smooth orthonormal frame $e_{1}, e_{2}$ of $n^{⊥}$ , and set

ℓ_{n} = (1, n), k_{n} = (1, - n), E_{A} = (0, e_{A}) .

Then

ℓ^{2} = k^{2} = 0, ℓ \cdot k = 2, E_{A} \cdot E_{B} = - δ_{A B} .

Displayed formulas in this passage

Em01-C@3#eq:N · line 908
$N = {z = a, y = a + α ℓ_{n}, b = a + β ℓ_{n}, 0 < α < β, ℓ_{n} = (1, n), n \in S^{2}} .$
unlabeled · line 901
$q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},$
unlabeled · line 904
$S_{resp} = {q_{R} = 0, a = z},$
unlabeled · line 923
$ℓ_{n} = (1, n), k_{n} = (1, - n), E_{A} = (0, e_{A}) .$
unlabeled · line 929
$ℓ^{2} = k^{2} = 0, ℓ \cdot k = 2, E_{A} \cdot E_{B} = - δ_{A B} .$