Manuscript V@1 · V.tex · section · e7ddf37d

The full finite-plane annihilator

Restricted compact representatives on a two-dimensional TT plane › The full finite-plane annihilator · lines 1843–1881

V@1#eq:ann

For a compact tensor distribution $C$ , define

P_{Ann} C = C - ⟨ C, e ⟩ J_{1} - ⟨ C, r ⟩ J_{2} .

Then

Ann (V_{fin}) = {C : ⟨ C, e ⟩ = ⟨ C, r ⟩ = 0} = P_{Ann} E^{'} (S^{2} T M),

and

E^{'} (S^{2} T M) = Ann (V_{fin}) \oplus span_{R} {J_{1}, J_{2}} .

These identities follow immediately by evaluating on $v_{1}, v_{2}$ . In particular the annihilator has codimension two and $P_{Ann}$ is an idempotent projection. Moreover

supp (P_{Ann} C) \subset supp C \cup supp ρ .

All compact conserved representatives of the plane response form

J_{c} + (E \cap Ann (V_{fin})) .

The universal temporal, spatial-trace, gradient, and wave-equation annihilators do not exhaust the annihilator of a fixed plane; (eq:ann) does.

Figure fig:curvature-dual-plane shows the compact curvature-dual construction and the codimension-two annihilator decomposition.

Displayed formulas in this passage

V@1#eq:ann · line 1851
$Ann (V_{fin}) = {C : ⟨ C, e ⟩ = ⟨ C, r ⟩ = 0} = P_{Ann} E^{'} (S^{2} T M),$
unlabeled · line 1846
$P_{Ann} C = C - ⟨ C, e ⟩ J_{1} - ⟨ C, r ⟩ J_{2} .$
unlabeled · line 1858
$E^{'} (S^{2} T M) = Ann (V_{fin}) \oplus span_{R} {J_{1}, J_{2}} .$
unlabeled · line 1866
$supp (P_{Ann} C) \subset supp C \cup supp ρ .$
unlabeled · line 1871
$J_{c} + (E \cap Ann (V_{fin})) .$

Nearby in this section

figure: Compact representatives on the TT plane