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Uniformly Levi degenerate CR manifolds: The 5-dimensional case. (English) Zbl 1020.32029

The problem studied in this paper is the existence of a CR diffeomorphism between two germs \((M,p)\) and \((M',p')\) of CR manifolds of hypersurface type. That question has already been solved for germs with a nondegenerate Levi form, and the author investigates here the case of 5-dimensional germs of hypersurface type with a degenerate Levi form of constant rank 1. Under a non-degeneracy condition he associates to any such germ \((M,p)\) a bundle \(Y\to M\) and a 1-form \(\omega\) on \(Y\) with value in \(\mathbb{R}^{\dim Y}\) giving an absolute parallelism on \(Y\), and he proves that two germs \((M,p)\) and \((M',p')\) are CR equivalent if and only if the associated pairs \((Y,\omega)\) and \((Y',\omega')\) are isomorphic.
Using this theorem he gives a unique continuation principle for CR mappings between two such germs and characterizes the germs isomorphic to the tube over the light cone in \(\mathbb{R}^3\).

MSC:

32V20 Analysis on CR manifolds
32V05 CR structures, CR operators, and generalizations
32V40 Real submanifolds in complex manifolds
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