Proposition: Quadratic control on the defect plane

The copied quadratic expression induces the negative-definite form

on ${{\mathcal{D}}_{\tan }}_w$. Its Hessian is

under the Euclidean identification with ${\mathbf{n}}^{\perp }$.

On compact tangential sets inside $0<\alpha <\beta$, its nondegeneracy is uniform.

On the regular manifold $S_6$, the Hessian of $q_R$ in the relative angular directions is positive definite.

Proof. The copied expression (eq:copied-quadratic) depends on $v,w$ only through $d_{\perp }=\alpha v-\beta w$. Its kernel is the common angular tangent direction, so it descends to the quotient identified in Proposition prop:defect. Differentiation gives the displayed Hessian.

On a compact subset of $0<\alpha <\beta$, the positive denominator has positive lower and finite upper bounds. The Hessian eigenvalues are therefore bounded away from zero in absolute value.

For the last assertion, differentiate (eq:qR-on-S) twice at ${\mathbf{n}}_1={\mathbf{n}}_2$: for tangent angular variations,

It is positive definite on the two relative angular components and zero on the common angular tangent.