2022/02/07 by Joël Merker, Joel Merker, Merker, Joel
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #math.AC #math.DG #math.GR
paper · pdf · doi:10.48550/arxiv.2202.03030
68 pages, 0 figure
arxiv created 2022/02/07 · openalex publication_date 2022/02/07 · arxiv updated 2022/02/08 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Equivalences under the affine group \rm Aff (ℝ3) of constant Hessian rank 1 surfaces S2 ⊂ ℝ3, 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 Hn ⊂ ℝn+1 graphed as \ u = F(x1, …, xn) \ whose Hessian matrix (Fxi xj), 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 ℝm 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(x1, …, xn) \ under \rm Aff (ℝn+1), up to order \leqslant n+5, valid in any dimension n \geqslant 2.