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Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

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Abstract

The classification of CR hypersurfaces \(M^{2n+1} \subset \mathbb {C}^{n+1}\) up to biholomorphic equivalences, notably the homogeneous ones, is a vast problem, especially in dimension 5, i.e. for \(n = 2\), even with the assistance of all existing mostly sophisticated mathematical tools: Lie-theoretic algebras of differential invariants; Exterior differential systems; Cartan connections; Parabolic geometries; Poincaré-Moser normal forms. As understood by e.g. Lie, Tresse, Segre, Cartan, such classification problems are tightly linked with point equivalences of completely integrable systems of partial differential equations in \(n \ge 1\) independent variables and 1 dependent variable, over \(\mathbb {C}\) or \(\mathbb {R}\), so that those PDE systems that are associated to CR structures can rightly be called ‘para-CR structures’. In particular, the 3-dimensional case, i.e. \(n = 1\), is linked with the well understood geometry of second order ODEs \(y_{xx} = F(x, y, y_x)\). The present survey article: (1)  focuses considerations on the study of (para-)CR structures in dimensions 3 and 5; (2)  sketches relationships with affinely homogeneous submanifolds and their tubifications; (3)  provides several concrete classification lists of various Lie symmetry algebras; (4)  describes recent achievements due to Loboda and to Doubrov-Medvedev-The about nondegenerate homogeneous (para-)CR structures in 5D; (5)  concludes by reviewing the recent classification arXiv:2003.08166, due to the two authors, of degenerate homogeneous para-CR structures in 5D, which is based on Cartan’s method of equivalence and which is coherent with the CR classification due to Fels-Kaup.

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Authors are grateful to an anonymous referee for a careful reading, and for pointing out an inaccuracy in a classification list.

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  1. Laboratoire de Mathématiques d’Orsay, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Joël Merker

  2. Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668, Warszawa, Poland

    Paweł Nurowski

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Merker, J., Nurowski, P. Homogeneous CR and Para-CR Structures in Dimensions 5 and 3. J Geom Anal 34, 27 (2024). https://doi.org/10.1007/s12220-023-01461-0

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