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Locally homogeneous finitely nondegenerate CR-manifolds. (English) Zbl 1155.32027
In the first part of this paper, the author deals with germs of real analytic CR manifolds admitting a local transitive action of a Lie group \(G\) of CR automorphisms. Modulo local equivalence, any CR manifold \(M\) of this kind is completely determined by the corresponding CR algebra, a notion introduced by C. Medori and M. Nacinovich [J. Algebra 287, No.1, 234–274 (2005; Zbl 1132.32013)]. This is a pair \((\mathfrak g, \mathfrak q)\) formed by the Lie algebra \(\mathfrak g = Lie\;G\) and by a complex subalgebra \(\mathfrak q \subset \mathfrak g^{\mathbb C}\), which is uniquely determined by the \(G\)-invariant CR structure of \(M\). The author points out the perfect correspondence between the category of CR algebras and the category of germs of real analytic locally homogeneous CR manifolds and gives the Lie algebraic descriptions of various notions in CR geometry, like e.g. the property of \(k\)-nondegeneracy.
In the second part, he provides an example of a homogeneous hypersurface in a \(7\)-dimensional complex manifold, which is uniformly 3-nondegenerate. The example is a \(G\)-orbit in a flag manifold \(G^{\mathbb C}/P\) of a real form \(G \subset G^{\mathbb C}\). The author determines also an upper bound for the order of degeneracy of non-degenerate hypersurface orbits in flag manifolds and other information concerning orbits in compact Hermitian symmetric spaces.

32V40 Real submanifolds in complex manifolds
53C30 Differential geometry of homogeneous manifolds
22F30 Homogeneous spaces
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