Shape Differentiation and Anisotropy Onset of Quadratic Invariant-Manifold Jets in Surface Navier–Stokes Flow

For a $C^1$ family of smooth closed genus-zero surfaces, we study the quadratic normal jet of unforced surface Navier–Stokes flow near the zero equilibrium. Under an isolated rank-three Riesz slow block and a three-weight slow–fast gap, local $C^2$ invariant graphs share this quotient-valued second derivative after tangent-input identification, without requiring uniqueness of the full graph. A fixed-complement representative is $C^1$ in shape in a bilinear Sobolev-operator norm with infinite-dimensional output. An exact covariance law handles admissible comparisons, including spatial transports with derivative loss. At the unit sphere, the jet vanishes. For symmetric affine deformations, its first variation is a degree-three spherical-harmonic tensor with coefficient $-1/\left(45\nu \right)$, vanishing exactly for scalar deformation matrices. For each fixed nonscalar D, the jet is therefore nonzero for all sufficiently small $0<\left|\delta \right|<{\delta }_0\left(D\right)$. The onset is local and non-quantitative; no explicit value of ${\delta }_0\left(D\right)$ is obtained.