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Local tube realizations of CR-manifolds and maximal abelian subalgebras. (English) Zbl 1229.32020
A tube CR-submanifold in $$\mathbb C^r$$ is a smooth submanifold of the form $$T_F:=\mathbb R^r+iF$$, where $$F\subset\mathbb R^r$$ is a real submanifold called the base of $$T_f$$. These submanifolds serve as test cases in several questions of CR geometry. The main aim of this article is to give necessary and sufficient conditions for a general submanifold $$M$$ to be, locally around a point $$a\in M$$, isomorphic to some tube CR-submanifold, and to study the question of uniqueness of such a realization. This is done in the analytic category.
The key ingredient in the study of these problems is that the translations in real directions evidently give rise to an $$r$$-dimensional commutative subalgebra in the Lie algebra of infinitesimal CR-automorphisms. The authors first characterize when such a subalgebra comes from a tube realization. In the uniqueness question, it turns out that there are two interesting notions of (local) equivalence for local tube realizations. It is shown that they correspond to conjugacy of the abelian subalgebras under two different groups. Several examples are studied in detail.
Reviewer: Andreas Cap (Wien)

##### MSC:
 32V05 CR structures, CR operators, and generalizations 32V40 Real submanifolds in complex manifolds 32M25 Complex vector fields, holomorphic foliations, $$\mathbb{C}$$-actions
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