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

For a C1 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 C2 invariant graphs share this quotient-valued second derivative after tangent-input identification, without requiring uniqueness of the full graph. A fixed-complement representative is C1 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/(45ν), vanishing exactly for scalar deformation matrices. For each fixed nonscalar D, the jet is therefore nonzero for all sufficiently small 0<|δ|<δ0(D). The onset is local and non-quantitative; no explicit value of δ0(D) is obtained.