Equivalences of PDE systems associated to degenerate para-CR Structures: foundational aspects

Abstract

Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions $\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under split-diffeomorphisms $(x,y,a,b) \,\longmapsto\, \big( f(x,y),\,g(x,y),\,\varphi(a,b),\,\psi(a,b) \big)$. Two Levi forms exist, and have the same rank $r \leqslant \min (n,m)$. If $\mathcal{M}$ is $k$-nondegenerate with respect to parameters and $l$-nondegenerate with respect to variables, $\mbox{Aut}(\mathcal{M})$ is a local Lie group of dimension: \[ \dim\, \mbox{Aut} (\mathcal{M}) \,\,\leqslant\,\, {\textstyle{\binom{n+1+2k+2l}{2k+2l}}}\,\, \min\, \big\{ (n+1),\, (m+1) \big\}. \] Mainly, our goal is to set up foundational material addressed to CR geometers. We focus on $n = m = 2$, assuming $r = 1$. In coordinates $(x,y,z, a,b,c)$, a local equation is: \[ z \,=\, c + xa + \beta\,xxb + \underline{\beta}\,yaa + c\,{\rm O}_{x,y,a,b}(2) + {\rm O}_{x,y,a,b,c}(4), \] with $\beta$ and $\underline{\beta}$ representing the two $2$-nondegeneracy invariants at $0$. The associated para-CR PDE system: \[ z_y \,=\, \big(x,y,z,z_x,z_{xx}\big) \ \ \ \ \ \ \ \ \ \ \ \ \ \& \ \ \ \ \ \ \ \ \ \ \ \ \ z_{xxx} \,=\, H\big(x,y,z,z_x,z_{xx}\big), \] satisfies $F_{z_{xx}} \equiv 0$ from Levi degeneracy. We show in details that the hypothesis of $2$-nondegeneracy with respect to variables is equivalent to $F_{z_x z_x} \neq 0$. This gives CR-geometric meaning to the first two para-CR relative differential invariants encountered independently in arXiv:2003.08166 .