Manuscript III@2 · III.tex · section · 82dad3e5

The quotient being tested

The unchanged free-product obstruction › The quotient being tested · lines 575–599

Let

S_{g} = {h : G^{(1)} (h) = 0, h smooth and spacelike compact},

and

S_{ϕ} = {ϕ : P ϕ = 0, ϕ has compact smooth Cauchy data} .

Choose any linear subspace $G_{0} \subset S_{g}$ containing

{K ξ : ξ \in C_{c}^{\infty} (M; T^{*} M)} .

Define

P_{g} = S_{g} / G_{0}, P_{0} = P_{g} \times S_{ϕ}, q_{0} (h, ϕ) = ([h], ϕ) .

Only the metric is quotiented.

Displayed formulas in this passage

unlabeled · line 578
$S_{g} = {h : G^{(1)} (h) = 0, h smooth and spacelike compact},$
unlabeled · line 583
$S_{ϕ} = {ϕ : P ϕ = 0, ϕ has compact smooth Cauchy data} .$
unlabeled · line 588
${K ξ : ξ \in C_{c}^{\infty} (M; T^{*} M)} .$
unlabeled · line 592
$P_{g} = S_{g} / G_{0}, P_{0} = P_{g} \times S_{ϕ}, q_{0} (h, ϕ) = ([h], ϕ) .$