2022/06/03 by Joël Merker, Merker, Joel
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2206.01449
openalex publication_date 2022/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous memoir 2202.03030, we showed that in every dimension n ≥ 5, there exists (unexpectedly) no affinely homogeneous hypersurface Hn ⊂ ℝn+1 having Hessian of constant rank 1 (and not being affinely equivalent to a product with ℝm \geqslant 1). The present work is devoted to determine all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces Hn ⊂ ℝn+1 in dimensions n = 2, 3, 4, the cases n = 1, 2 being known. With complete details in the case n = 2, we illustrate the main features of what can be termed the "Power Series Method of Equivalence". The gist is to capture invariants at the origin only, to create branches, and to infinitesimalize calculations. In dimension n = 3, we find a single homogeneous model: u = (1)/(3 z2) \ ( 1-2 y+y2-2 xz )3/2 - (1-y) ( 1-2 y+y2-3 xz ) \, the singularity (1)/(3z2) being illusory. In dimension n = 4, without reaching closed forms, we find two simply homogeneous models, differing by some ± sign.