Manuscript VII-B@1 · VII-B.tex · lemma · be632835

Lemma: Nondegeneracy of the boundary map

Exact sharp-Cauchy adjoint · lines 731–747 · proof included or linked

VII-B@1#lem:B-injective, VII-B@1#eq:B-components

The map $v \mapsto B (v)$ is injective. In the chosen coordinates,

B^{00} (v) = - \frac{1}{2} v_{0}, B^{0 i} (v) = \frac{1}{2} v_{i}, B^{ij} (v) = - \frac{1}{2} δ^{ij} v_{0} .

Proof. The displayed components follow directly from (eq:boundary-map). The $00$ and $0 i$ components recover all components of $v$ .

Displayed formulas in this passage

VII-B@1#eq:B-components · line 735
$B^{00} (v) = - \frac{1}{2} v_{0}, B^{0 i} (v) = \frac{1}{2} v_{i}, B^{ij} (v) = - \frac{1}{2} δ^{ij} v_{0} .$