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arXiv · 2406.04798

Lewy curves in para-CR geometry

Wojciech Kryński·Omid Makhmali
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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.

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Wojciech Kryński, Omid Makhmali. 2024-06-07. Lewy curves in para-CR geometry. https://arxiv.org/abs/2406.04798

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