Affine Rigidity Without Integration

Abstract

Real analytic (${\mathcal{C}}^{\omega }$) surfaces $S^2$ in ${\mathbb{R}}^3\ni (x,y,u)$ graphed as $\{u=F(x,y)\}$ with $F_{xx}\ne 0$ whose Gaussian curvature vanishes identically: \[ 0 \,\equiv\, F_{xx}\,F_{yy} - F_{xy}^2, \] possess, under the action of the affine transformation group ${\mathsf{Aff}}_3(\mathbb{R})={\mathsf{GL}}_3(\mathbb{R})⋉{\mathbb{R}}^3$, 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 $\{u=x^2\}$ if and only if $S_{\mathsf{aff}}\equiv 0$. Assuming that $S_{\mathsf{aff}}\ne 0$ everywhere, two deeper affine invariants inspired from Pocchiola's Ph.D. are $W_{\mathsf{aff}}$ and $J_{\mathsf{aff}}$. Explicit expressions are given in this article. Theorem. $S$ is affinely equivalent to $\{u=\frac{x^2}{1-y}\}$ if and only if $W_{\mathsf{aff}}\equiv 0\equiv J_{\mathsf{aff}}$. 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.

28 pages, 2 figures

References in corpus (2)

• I: Lie symmetries of partial differential equations and CR geometry
• Structure equations of Levi degenerate CR hypersurfaces of uniform type

Cited by in corpus (1)

• Invariants of Surfaces in Three-Dimensional Affine Geometry