Inexistence of Non-Product Hessian Rank 1 Affinely Homogeneous Hypersurfaces $H^n$ in $\mathbb{R}^{n+1}$ in Dimension $n \geqslant 5$

Abstract

Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. Then what about higher dimensions? We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian matrix $(F_{x_i x_j})$, a relative affine invariant, is, similarly, of constant rank $1$. Are there homogeneous models? Complete explorations were done by the author on a computer in dimensions $n = 2, 3, 4, 5, 6, 7$. The first, expected outcome, was to obtain a complete classification of homogeneous models in dimensions $n = 2, 3, 4$ (forthcoming article, case $n = 2$ already known). The second, unexpected outcome, was that in dimensions $n = 5, 6, 7$, there are no affinely homogenous models! (Except those that are affinely equivalent to a product of $\mathbb{R}^m$ with a homogeneous model in dimensions $2, 3, 4$.) The present article establishes such a non-existence result in every dimension $n \geqslant 5$, based on the production of a normal form for $\{ u = F(x_1, \dots, x_n) \}$ under ${\rm Aff} (\mathbb{R}^{n+1})$, up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$.