Abstract

We study para-CR geometry in dimension 2n + 1. For a given para-CRstructure, we introduce a canonically associated family of curves, which we refer toas Lewy curves by analogy to a construction in CR geometry studied in [F]. We showthat for n > 1, the curves define a path geometry exclusively in the flat case (in whichthey coincide with chains). We prove that in the generic case, the Lewy curves aresolutions to a system of ODEs of order n. However, orders between 2 and n are alsopossible. Furthermore, we discuss the sub-maximal cases and provide a characterizationof the Lewy curves in terms of invariants of ODEs. We also discuss relations tothe twistor construction given in [D], generalized to higher dimensions. The talk isbased on joint work with O. Makhmali [KM].[F] J. J. Faran, Lewy’s curves and chains on real hypersurfaces, Trans. AMS ., 1981.[D] M. Dunajski. Twistor theory of dancing paths, SIGMA , 2022.[KM] W. Krynski, O. Makhmali, Lewy curves in para-CR geometry, arXiv:2406.04798,2024.