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- W2918656465 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." @default.
- W2918656465 created "2019-03-11" @default.
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- W2918656465 date "2019-03-03" @default.
- W2918656465 modified "2023-10-15" @default.
- W2918656465 title "Affine Rigidity Without Integration." @default.
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