Article

    Ufa Mathematical Journal
    Volume 15, Number 1, pp. 56-121

    Inexistence of non-product Hessian rank 1 affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimension $n \geqslant 5$


    Merker J.

    DOI:10.13108/2023-15-1-56

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    Equivalences under the affine group ${\mathtt{Aff}}(\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called {\sl parabolic}, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. In particular, homogeneous models and algebras of differential invariants in various branches were fully understood. {\sl Then what is about higher dimensions?} We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\big\{ u = F(x_1, \dots, x_n) \big\}$ whose Hessian matrix $\big( F_{x_i x_j} \big)$, a relative affine invariant, is similarly of constant rank $1$. {\sl 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 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 $\big\{ u = F(x_1, \dots, x_n) \big\}$, under ${\mathtt{Aff}}(\mathbb{R}^{n+1})$ up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$.