Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Switch and plateau

Cauchy problems and the admitted gauge domain › Switch and plateau · lines 373–396

VII-A@1#eq:switch, VII-A@1#eq:U, VII-A@1#eq:liealgebra

Fix

g \in C_{c}^{\infty} (M_{+}^{\circ}), 0 \leq g \leq 1, P_{g} := int {x : g (x) = 1} \neq = \emptyset .

Choose a nonempty relatively compact open set $U$ such that

U \subset P_{g} .

In particular, $U$ is separated from $Σ_{0}$ .

Let

g_{U} = C_{c}^{\infty} (U; T M) .

It is closed under the ordinary Lie bracket.

Displayed formulas in this passage

VII-A@1#eq:switch · line 376
$g \in C_{c}^{\infty} (M_{+}^{\circ}), 0 \leq g \leq 1, P_{g} := int {x : g (x) = 1} \neq = \emptyset .$
VII-A@1#eq:U · line 384
$U \subset P_{g} .$
VII-A@1#eq:liealgebra · line 391
$g_{U} = C_{c}^{\infty} (U; T M) .$