Affine Rigidity Without Integration

Abstract

Real analytic ($\mathcal{C}^\omega$) surfaces $S^2$ in $\mathbb{R}^3 \ni (x,y,u)$ graphed as $\big\{ u = F(x,y) \big\}$ with $F_{xx} \neq 0$ whose Gaussian curvature vanishes identically: \[ 0 \,\equiv\, F_{xx}\,F_{yy} - F_{xy}^2, \] possess, under the action of the affine transformation group ${\sf Aff}_3(\mathbb{R}) = {\sf GL}_3(\mathbb{R}) \ltimes \mathbb{R}^3$, a basic invariant analogous to $2$-nondegeneracy for $\mathcal{C}^\omega$ real hypersurfaces $M^5 \subset \mathbb{C}^3$: \[ S_{\sf aff} \,:=\, \frac{F_{xx}\,F_{xxy}-F_{xy}\,F_{xxx}}{ F_{xx}^2}. \] It is known (or easily recovered) that $S$ is affinely equivalent to $\big\{ u = x^2 \big\}$ if and only if $S_{\sf aff} \equiv 0$. Assuming that $S_{\sf aff} \neq 0$ 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 $\big\{ u = \frac{x^2}{1-y} \big\}$ if and only if $W_{\sf aff} \equiv 0 \equiv J_{\sf aff}$. As a direct corollary of the (brief) proof, affine rigidity of CR-flat $2$-nondegenerate $\mathcal{C}^\omega$ Levi rank $1$ hypersurfaces $M^5 \subset \mathbb{C}^3$ 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 $\mathcal{C}^\infty$ (even $\mathcal{C}^7$) $F(x,y)$ can be handled.