Manuscript XIV-F@1 · XIV-F.tex · section · bae5a40b

The selected conormal fiber product

F2: ambient space and the intrinsic conormal matching › The selected conormal fiber product · lines 992–1035

XIV-F@1#eq:fiberproduct

The four selected conormal incidences are

Λ_{R}^{-} Λ_{1}^{+} Λ_{2}^{+} Λ_{D} = {(w, - t_{R} d q_{R} (w)) : q_{R} (w) = 0, t_{R} > 0}, = {(w, t_{1} d q_{1} (w)) : q_{1} (w) = 0, t_{1} > 0}, = {(w, t_{2} d q_{2} (w)) : q_{2} (w) = 0, t_{2} > 0}, = {(w, κ_{μ} d d^{μ}) : d (w) = 0, κ \in R^{4}} .

Here $w$ denotes a point of $W$ , not the transverse displacement in (eq:copied-quadratic). The symbol $κ_{μ}$ in this section is an attachment covector coordinate, not the interaction coupling.

The zero attachment covector is included to accommodate the copied product-test selection. The conclusions below also hold on the punctured part $κ \neq = 0$ ; they do not depend on including zero in that factor.

Each cone hypersurface is smooth near the retained points: its separation vector is nonzero, so its defining differential is nonzero. The attachment conormal is likewise a smooth bundle. We therefore have smooth individual geometric relations before their matching is examined.

Let

F_{match} = Λ_{R}^{-} \times_{W} Λ_{1}^{+} \times_{W} Λ_{2}^{+} \times_{W} Λ_{D} .

In the common incidence coordinates,

F_{match} = N \times Θ, Θ = (0, \infty)^{3} \times R^{4} .

This parameterization names the set. It does not establish clean matching or a distributional product.

Displayed formulas in this passage

unlabeled · line 995
$Λ_{R}^{-} Λ_{1}^{+} Λ_{2}^{+} Λ_{D} = {(w, - t_{R} d q_{R} (w)) : q_{R} (w) = 0, t_{R} > 0}, = {(w, t_{1} d q_{1} (w)) : q_{1} (w) = 0, t_{1} > 0}, = {(w, t_{2} d q_{2} (w)) : q_{2} (w) = 0, t_{2} > 0}, = {(w, κ_{μ} d d^{μ}) : d (w) = 0, κ \in R^{4}} .$
XIV-F@1#eq:fiberproduct · line 1022
$F_{match} = Λ_{R}^{-} \times_{W} Λ_{1}^{+} \times_{W} Λ_{2}^{+} \times_{W} Λ_{D} .$
unlabeled · line 1029
$F_{match} = N \times Θ, Θ = (0, \infty)^{3} \times R^{4} .$

The selected conormal fiber product

Displayed formulas in this passage

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