Five-dimensional para-CR manifolds and contact projective geometry in dimension three

TL;DR

This paper investigates five-dimensional para-CR structures with specific degeneracy properties, linking classical invariants to contact projective geometries and providing new geometric interpretations of these invariants.

Contribution

It introduces a geometric interpretation of the lowest order invariant of para-CR structures and establishes conditions for contact projective geometries on quotient spaces.

Findings

Vanishing of M(G) characterizes conic containment.

Vanishing of W(H) characterizes Lorentzian geometry from 3rd order ODEs.

Vanishing of N(G,H) characterizes contact projective structures on quotients.

Abstract

We study invariant properties of $5$-dimensional para-CR structures whose Levi form is degenerate in precisely one direction and which are $2$-nondegenerate. We realize that two, out of three, primary (basic) para-CR invariants of such structures are the classical differential invariants known to Monge (1810) and to Wuenschmann (1905) \[ M(G) := 40G_{ppp}^3-45G_{pp}G_{ppp}G_{pppp}+9G_{pp}^2G_{ppppp}, \quad W(H) := 9D^2H_r-27DH_p-18H_rDH_r+18H_pH_r+4H_r^3+54H_z. \] The vanishing $M(G) \equiv 0$ provides a local necessary and sufficient condition for the graph of a function in the $(p,G)$-plane to be contained in a conic, while the vanishing $W(H) \equiv 0$ gives an if-and-only-if condition for a 3rd order ODE to define a natural Lorentzian geometry on the space of its solutions. Mainly, we give a geometric interpretation of the third basic invariant of our class of para-CR structures, the simplest one, of lowest order, and of mixed nature $N(G,H):=2G_{ppp}+G_{pp}H_{rr}$. We establish that the vanishing $N(G,H) \equiv 0$ gives an if-and-only-if condition for the two $3$-dimensional quotients of the para-CR manifold by its two canonical integrable rank-$2$ distributions, to be equipped with contact projective geometries. A curious transformation between the Wuenschmann invariant and the Monge invariant, first noted by us in arXiv:2003.08166, is also discussed, and its mysteries are further revealed.