Lewy curves in para-CR geometry

TL;DR

This paper introduces Lewy curves in para-CR geometry, characterizes their role in defining path geometries, and establishes conditions under which they coincide with chains, especially in flat structures and higher dimensions.

Contribution

It defines Lewy curves in para-CR geometry, provides a characterization in dimension 3, and links their properties to the flatness of the para-CR structure in higher dimensions.

Findings

Lewy curves characterize para-CR structures in dimension 3.

Conditions for Lewy curves to coincide with chains are established.

Lewy curves define a path geometry only in flat para-CR structures.

Abstract

We define a class of curves, referred to as Lewy curves, in para-CR geometry, following H. Lewy's original definition in CR geometry. We give a characterization of path geometries defined by para-CR Lewy curves. In dimension 3 our characterization is given by a set of necessary and sufficient conditions which, with the exception of one condition, are easily computationally verifiable. Furthermore, we show that Lewy curves of a para-CR 3-manifold coincide with chains of some (para-)CR 3-manifold if and only if it is flat. Subsequently, it follows that Lewy curves determine the para-CR structure up to the sign of the almost para-complex structure. In higher dimensions we show that para-CR Lewy curves define a path geometry if and only if the para-CR structure is flat, in which case chains and Lewy curves coincide.