Local tube realizations of CR-manifolds and maximal Abelian subalgebras

Fels, Gregor; kaup, Wilhelm

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 1, pp. 99-128.

For every real-analytic CR-manifold $M$ we give necessary and sufficient conditions that $M$ can be realized in a suitable neighbourhood of a given point $a\in M$ as a tube submanifold of some ${ℂ}^r$. We clarify the question of the ‘right’ equivalence between two local tube realizations of the CR-manifold germ $(M,a)$ by introducing two different notions of affine equivalence. One of our key results is a procedure that reduces the classification of equivalence classes to a purely algebraic manipulation in terms of Lie theory.

Classification : 32V05, 32V40, 32M25, 17B66

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