The property of admitting a positive definite Hessian metric is not open under small affine deformations of a compact manifold. On the fixed torus T^2, the classical shear family with sole nonzero Christoffel coefficient Gamma^1_22=t consists of complete, volume-preserving affine connections and converges smoothly to the Euclidean connection. For an arbitrary hypothetical Hessian metric, one component of the Codazzi equation integrates to t times the integral of g_11 being zero, excluding every nonzero t. Products give the same obstruction in all dimensions at least two. We also give an averaging criterion for translation-invariant affine connections on tori. The affine family is due to the classical theory described by Baues and Goldman; this note supplies a self-contained application to Shimizu's stability question, without a novelty or priority claim. This is an unrefereed, self-audited preprint with AI assistance disclosed in the text.
