2019/03/03 by Joël Merker, Merker, Joel
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1903.00889
openalex publication_date 2019/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Real analytic (Cω) surfaces S2 in ℝ3 \ni (x,y,u) graphed as \ u = F(x,y) \ with Fxx ≠ 0 whose Gaussian curvature vanishes identically: 0 ≡ Fxx Fyy - Fxy2, possess, under the action of the affine transformation group \sf Aff3(ℝ) = \sf GL3(ℝ) \ltimes ℝ3, a basic invariant analogous to 2-nondegeneracy for Cω real hypersurfaces M5 ⊂ ℂ3: S\sf aff := \fracFxx Fxxy-Fxy Fxxx Fxx2. It is known (or easily recovered) that S is affinely equivalent to \ u = x2 \ if and only if S\sf aff ≡ 0. Assuming that S\sf aff ≠ 0 everywhere, two deeper affine invariants inspired from Pocchiola's Ph.D. are W\sf aff and J\sf aff. Explicit expressions are given in this article. Theorem. S is affinely equivalent to \ u = (x2)/(1-y) \ if and only if W\sf aff ≡ 0 ≡ J\sf aff. As a direct corollary of the (brief) proof, affine rigidity of CR-flat 2-nondegenerate Cω Levi rank 1 hypersurfaces M5 ⊂ ℂ3 is deduced. The arguments rely on pure affine geometry, avoid any tool from Analysis, and simplify A.V. Isaev, J. Differential Geom. 104 (2016), 111--141. An independent article will show, in a more general context, how C^∞ (even C7) F(x,y) can be handled.