Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Exact characteristic coordinates

Tensor conventions and separated characteristic geometry › Exact characteristic coordinates · lines 573–610

Em01-F@3#eq:defect

Let

N = {r = u = v = ξ = 0}, S_{resp} = {r = 0, u = 0} .

Their codimensions are eight and five. Along $N$ ,

d q_{R} = 4 β d u, d q_{1} = 4 α d v, d q_{2} = 4 (β - α) (d u - d v) .

The three differentials have rank two. Their crossed restrictions to $S_{resp}$ have rank one along $N$ . Also, throughout the chart,

\frac{q _{1}}{α} + \frac{q _{2}}{β - α} = \frac{q _{R}}{β} - \frac{β}{α ( β - α )} ∣ ξ ∣^{2} .

Choose a small relatively compact patch about

a_{*} = z_{*} = (0, 0, 0, 0), y_{*} = a_{*} + 2 ℓ_{(0, 0, 1)}, b_{*} = a_{*} + 4 ℓ_{(0, 0, 1)} .

We require throughout its closure

b^{0} > y^{0} > a^{0}, b^{0} > z^{0},

and all positive separation bounds above. In particular $b \neq = a$ and $b \neq = z$ . Strict separation of $a, y, b$ does not remove $z = a$ .

Displayed formulas in this passage

Em01-F@3#eq:defect · line 589
$\frac{q _{1}}{α} + \frac{q _{2}}{β - α} = \frac{q _{R}}{β} - \frac{β}{α ( β - α )} ∣ ξ ∣^{2} .$
unlabeled · line 576
$N = {r = u = v = ξ = 0}, S_{resp} = {r = 0, u = 0} .$
unlabeled · line 581
$d q_{R} = 4 β d u, d q_{1} = 4 α d v, d q_{2} = 4 (β - α) (d u - d v) .$
unlabeled · line 598
$a_{*} = z_{*} = (0, 0, 0, 0), y_{*} = a_{*} + 2 ℓ_{(0, 0, 1)}, b_{*} = a_{*} + 4 ℓ_{(0, 0, 1)} .$
unlabeled · line 604
$b^{0} > y^{0} > a^{0}, b^{0} > z^{0},$