On degenerate para-CR structures: Cartan reduction and homogeneous models

TL;DR

This paper classifies homogeneous models of 5-dimensional para-CR structures with constant rank Levi forms, using Cartan's method to find explicit PDE representations and their symmetry groups.

Contribution

It extends Levi degenerate CR geometry to para-CR structures, providing a complete classification of homogeneous models via Cartan reduction.

Findings

Identified four classes of homogeneous models with explicit PDE forms.

Determined symmetry groups for each model.

Provided explicit conditions for integrability and degeneracy.

Abstract

Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider $5$-dimensional para-CR structures whose Levi forms are of constant rank $1$ and that are $2$-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs $z_y=F(x, y, z, z_x)$ $\,\,\&\,\,$ $z_{xxx}=H(x,y,z,z_x,z_{xx})$, with $F$ independent of $z_{xx}$ and $F_{z_xz_x} \neq 0$, that are completely integrable $D_x^3 F=\Delta_y H$, Performing at an advanced level Cartan's method of equivalence, we determine all concerned homogeneous models, together with their symmetries: (i) $z_y=\tfrac14 (z_x)^2\quad \&\quad z_{xxx}=0$; (ii) $z_y=\tfrac14 (z_x)^2\quad \& \quad z_{xxx}=(z_{xx})^3$; (iiia) $z_y=\tfrac14 (z_x)^b\,\, \& \,\,z_{xxx} = (2-b)\frac{(z_{xx})^2}{z_x}$ with $z_x>0$ for any real $b\in[1,2)$; (iiib) $z_y = f(z_x)\quad \& \quad z_{xxx}=h(z_x)\big(z_{xx}\big)^2$, where the function $f$ is determined by the implicit equation: \[ (z_x^2+f(z_x)^2)\, \mathrm{exp} \left( 2b\,\mathrm{arctan}\tfrac{bz_x-f(z_x)}{z_x+bf(z_x)} \right) = 1+b^2 \] and where: \[ h(z_x) := \frac{(b^2-3)z_x-4bf(z_x)}{(f(z_x)-bz_x)^2}, \] for any real $b>0$.