Proposition: Geometrically twisted transverse vectors

With the componentwise twisted Friedrichs extension, Theorems thm:green, thm:speed, and thm:closed apply to each sector (eq:helicity). For transverse vectors, ${\beta }_{+1}=\alpha$. In that sector:

The latter mode is deficient exactly when $\mu <\pi$. The opposite transverse helicity has the same criterion.

Proof. The operator is the scalar flat-connection operator in each parallel component. The deficiency statements are Lemma lem:deficiency with $\beta =\alpha$ and with the opposite character.