Manuscript XIV-C@1 · XIV-C.tex · proposition · a34d13df

Proposition: Characteristic geometry

J2: the retained characteristic configuration › A codimension-four characteristic stratum · lines 965–1014 · proof included or linked

XIV-C@1#prop:geometry, XIV-C@1#eq:rank

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$N^{\circ}$ is a smooth embedded submanifold of the twelve-dimensional three-vertex space, of dimension eight and codimension four.

Let

q_{R} = (x - u)^{2}, q_{1} = (y - u)^{2}, q_{2} = (x - y)^{2},

where the square uses $η$ . On $N^{\circ}$ ,

- \frac{1}{β} d q_{R} + \frac{1}{α} d q_{1} + \frac{1}{β - α} d q_{2} = 0, rank {d q_{R}, d q_{1}, d q_{2}} = 2.

Proof. The parameterization by $(u, α, β, n)$ is smooth. It is injective, with smooth inverse on its image:

u = u, α = y^{0} - u^{0}, β = x^{0} - u^{0}, n = \frac{y - u}{y ^{0} - u ^{0}} .

Its differential is injective: the $u$ components determine $δ u$ , the temporal differences determine $δ α$ and $δ β$ , and then the spatial differences determine the two tangent components of $δ n$ . The dimension is $4 + 1 + 1 + 2 = 8$ .

Writing $p_{n} = η (ℓ_{n}, \cdot)$ gives

d q_{R} = 2 β p_{n} \cdot (d x - d u),

d q_{1} = 2 α p_{n} \cdot (d y - d u), d q_{2} = 2 (β - α) p_{n} \cdot (d x - d y) .

These equations give the relation in (eq:rank). The last two differentials are independent because their nonzero components involve distinct free endpoint variations. The relation makes the rank exactly two.

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XIV-C@1#eq:rank · line 975
$- \frac{1}{β} d q_{R} + \frac{1}{α} d q_{1} + \frac{1}{β - α} d q_{2} = 0, rank {d q_{R}, d q_{1}, d q_{2}} = 2.$
unlabeled · line 971
$q_{R} = (x - u)^{2}, q_{1} = (y - u)^{2}, q_{2} = (x - y)^{2},$
unlabeled · line 988
$u = u, α = y^{0} - u^{0}, β = x^{0} - u^{0}, n = \frac{y - u}{y ^{0} - u ^{0}} .$
unlabeled · line 1001
$d q_{R} = 2 β p_{n} \cdot (d x - d u),$
unlabeled · line 1004
$d q_{1} = 2 α p_{n} \cdot (d y - d u), d q_{2} = 2 (β - α) p_{n} \cdot (d x - d y) .$

Proposition: Characteristic geometry

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