Local tube realizations of CR-manifolds and maximal abelian subalgebras

Abstract

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 ∈ M as a tube submanifold of some C . 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. Mathematics Subject Classification Primary 32V05; Secondary 32V40, 32M25, 17B66