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Uniqueness forces an extended axial field to commute

Statement

Let a relative datum as in Relative stationary deformation and extension obstruction have the following uniqueness property: two global Killing fields of $\tilde{g}$ agreeing on a nonempty open set agree on all of $U$ . If $Z$ has any global Killing extension $Y$ , then $[T, Y] = 0$ on $U$ . Consequently, for this uniqueness property, a commuting-extension obstruction excludes every global Killing extension of $Z$ , not just those initially assumed to commute.

Facts & Assumptions

[F1]

On $O$ the relative datum has $[T, Z] = 0$ . Relative stationary deformation and extension obstruction

[F2]

A bracket of smooth Killing fields is Killing. The bracket of two Killing fields is Killing

[A1]

The stated uniqueness property is an explicit premise of this local implication, not an assertion proved here.

Proof

Given: The datum, uniqueness property and a global Killing extension $Y$ .

Goal: Show the extension commutes with $T$ .

1.1F1F2

The global field $W = [T, Y]$ is Killing by [F2]. Since $Y ∣_{O} = Z$ , its bracket with $T$ vanishes on the open set $O$ . Equality of restrictions on an open set includes the derivatives defining the bracket.

2.1A1step 1.1∎

Apply the stated uniqueness property to the global Killing fields $W$ and zero: they agree on $O$ , hence on $U$ . Thus $[T, Y] = 0$ . If a commuting extension is forbidden, the existence of any Killing extension contradicts that prohibition.

Depends on

Used by · 1

A commuting-extension obstruction excludes arbitrary analytic-model isometries

Dependency tree · 2 levels

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