We prove that finite-range convolution on compact Riemannian manifolds generically produces gradient misalignment: the gradient fields of a scalar function and its smoothed counterpart are non-collinear at almost every point, unless the function is supported on a single eigenspace of the Laplace–Beltrami operator. We show that elliptic Green's functions provide canonical realisations of such convolution operators, with exponential correlation suppression at an intrinsic length scale. The resulting gradient pair carries exactly three independent scalar invariants under the isometry group, regardless of manifold dimension.
Ilja Schots (Sun,) studied this question.