Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Characteristic coordinates and the supported normal form

Characteristic coordinates and the supported normal form · lines 420–495

Em01-G@5#eq:chart, Em01-G@5#eq:density

On a sphere subchart choose

ℓ = (1, n), k = (1, - n), E_{A} = (0, e_{A}), A = 1, 2,

with

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

Use

z = a + r, b = a + β ℓ + u k, y = a + α ℓ + v k + ξ^{A} E_{A} .

On $U$ , assume positive lower bounds on $α, β - α, β - u$ , and strict order and separation

b^{0} > y^{0} > a^{0}, b^{0} > z^{0}, b \neq = a, b \neq = z .

The inverse is

n = \frac{b - a}{∣ b - a ∣}, β = \frac{b ^{0} - a ^{0} + ∣ b - a ∣}{2}, u = \frac{b ^{0} - a ^{0} - ∣ b - a ∣}{2},

α = \frac{y ^{0} - a ^{0} + n \cdot ( y - a )}{2}, v = \frac{y ^{0} - a ^{0} - n \cdot ( y - a )}{2}, ξ^{A} = e_{A} \cdot (y - a) .

The cone functions and density are

q_{R} = 4 β u, q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},

d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω_{n} d α d β d^{4} r d u d v d^{2} ξ .

The two null-coordinate Jacobians have absolute determinant two.

Set

Σ = {u = v = ξ = 0}, S_{resp} = {r = u = 0}, N_{mark} = {r = u = v = ξ = 0},

ϑ = (a, n, α, β), τ = (ϑ, v, ξ), d ϑ = d^{4} a d Ω_{n} d α d β .

Write $d τ = d ϑ d v d^{2} ξ$ .

For a residual multiindex $ζ$ , define

E_{ζ, j} (c) (φ) = (- 1)^{∣ ζ ∣ + j} \int c (τ) \partial_{r}^{ζ} \partial_{u}^{j} φ (τ, 0, 0) d τ .

Displayed formulas in this passage

Em01-G@5#eq:chart · line 434
$z = a + r, b = a + β ℓ + u k, y = a + α ℓ + v k + ξ^{A} E_{A} .$
Em01-G@5#eq:density · line 464
$d^{4} a d^{4} b d^{4} z d^{4} y = 4 (β - u)^{2} d^{4} a d Ω_{n} d α d β d^{4} r d u d v d^{2} ξ .$
unlabeled · line 424
$ℓ = (1, n), k = (1, - n), E_{A} = (0, e_{A}), A = 1, 2,$
unlabeled · line 429
$ℓ^{2} = k^{2} = 0, ℓ \cdot k = 2, E_{A} \cdot E_{B} = - δ_{A B} .$
unlabeled · line 441
$b^{0} > y^{0} > a^{0}, b^{0} > z^{0}, b \neq = a, b \neq = z .$
unlabeled · line 445
$n = \frac{b - a}{∣ b - a ∣}, β = \frac{b ^{0} - a ^{0} + ∣ b - a ∣}{2}, u = \frac{b ^{0} - a ^{0} - ∣ b - a ∣}{2},$
unlabeled · line 452
$α = \frac{y ^{0} - a ^{0} + n \cdot ( y - a )}{2}, v = \frac{y ^{0} - a ^{0} - n \cdot ( y - a )}{2}, ξ^{A} = e_{A} \cdot (y - a) .$
unlabeled · line 460
$q_{R} = 4 β u, q_{1} = 4 α v - ∣ ξ ∣^{2}, q_{2} = 4 (β - α) (u - v) - ∣ ξ ∣^{2},$
unlabeled · line 475
$Σ = {u = v = ξ = 0}, S_{resp} = {r = u = 0}, N_{mark} = {r = u = v = ξ = 0},$
unlabeled · line 480
$ϑ = (a, n, α, β), τ = (ϑ, v, ξ), d ϑ = d^{4} a d Ω_{n} d α d β .$
unlabeled · line 488
$E_{ζ, j} (c) (φ) = (- 1)^{∣ ζ ∣ + j} \int c (τ) \partial_{r}^{ζ} \partial_{u}^{j} φ (τ, 0, 0) d τ .$