Lewy curves in para-CR geometry

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.