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Classification of Levi degenerate homogeneous CR-manifolds in dimension 5. (English) Zbl 1171.32023
We recall that, given a submanifold \(F \subset \mathbb R^n\), the tube manifold over \(F\) is the submanifold \(F + i \mathbb R^n = \{\;z = x + i y\;,\;x \in F\;\} \subset \mathbb C^n\).
In the first part of this paper, the authors consider the CR geometry of a tube domain \(M = F + i \mathbb R^n \subset \mathbb C^n\) showing how several properties, like \(k\)-nondegeneracy, local CR homogeneity or CR equivalences with other tube domains, correspond to precise properties of the second fundamental form of \(F \subset \mathbb R^n\). Secondly, they introduce a recipe for constructing 2-dimensional cones \(F\) in \( \mathbb R^n\), whose associated tubes are \((n+2)\)-dimensional homogeneous CR manifolds of CR dimension 2 and 2-nondegenerate. For such manifolds, they describe the full automorphism groups and give a simple criterion to determine when two of them are CR equivalent. Finally, they restrict to the case \(n = 3\) and determine a complete list of all mutually inequivalent, homogeneous, 2-nondegenerate, CR hypersurfaces in \(\mathbb C^3\) that can be constructed with this recipe.
In the second part, they prove that any 5-dimensional, locally homogeneous, 2-nondegenerate CR manifold is locally CR equivalent to one of the tube manifolds \(F + i \mathbb R^3 \subset \mathbb C^3\) of the list in the first part. This is obtained through the classification of the Lie CR-algebras \((\mathfrak g, \mathfrak q)\) which correspond to some locally homogeneous 5-dimensional CR manifold of hypersurface type and 2-nondegenerate, and which have the minimal dimension amongst the Lie CR-algebras associated with the same manifold. The classification is quite long and consists of several lemmata. Roughly speaking, one may say that it is divided in two main steps. At first, they prove that if \((\mathfrak g, \mathfrak q)\), satisfies the hypothesis and \(\mathfrak g\) is non-solvable, then it is associated to a CR manifold that is locally equivalent to the tube over the so-called “light cone”. Secondly, they prove that if \((\mathfrak g, \mathfrak q)\) satisfies the hypotheses and \(\mathfrak g\) is solvable, then it is associated to a CR manifold which is locally CR equivalent to some of the other tube manifolds of the list.
From this result and Cartan’s classification of 3-dimensional Levi non-degenerate homogeneous CR manifolds, the classification of 5-dimensional locally homogeneous CR manifolds with degenerate Levi form is reached.

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