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JOURNALS // Ufimskii Matematicheskii Zhurnal // Archive

Ufimsk. Mat. Zh., 2023 Volume 15, Issue 1, Pages 56–121 (Mi ufa645)

This article is cited in 1 paper

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

J. Merker

Institut de Mathématique d’Orsay, CNRS, Université Paris-Saclay, Faculté des Sciences, 91405 Orsay Cedex, France

Abstract: Equivalences under the affine group $\mathrm{Aff}(\mathbb{R}^3)$ 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. In particular, homogeneous models and algebras of differential invariants in various branches were fully understood.
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 $1$. 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 $\mathrm{Aff}(\mathbb{R}^{n+1})$ up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$.

Keywords: Affine homogeneity, Normal forms, tangential vector fields.

UDC: 517.958

MSC: 53A55, 53B25, 53A15, 53A04, 53A05, 58K50, 16W22, 14R20, 22E05, 35B06

Received: 07.02.2022

Language: English


 English version:
Ufa Mathematical Journal, 2023, 15:1, 56–121

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© Steklov Math. Inst. of RAS, 2025