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 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.

CR-quadrics with a symmetry property

“…Since Q ∩ reg(κ −1 ) is connected by Lemma 2.2 in [5], also S ∩ reg(κ) is connected. Consequently S is a connected generic real-analytic CR-submanifold of E. Furthermore, κ induces an isomorphism between the real Lie algebras g = hol (Q) and s := hol (S).…”

“…There exists a complex manifold X, containing Q as generic real-analytic submanifold, in such a way that π extends to a holomorphic map π : X → X. This implies that A := π −1 (A) is complex-analytic in X and hence that π −1 (Q) = Q\ A is connected by Lemma 2.2 in [5]. Since Q is simply connected the covering map π : Q → Q must be a homeomorphism.…”

Regularization of Local CR-Automorphisms of Real-Analytic CR-Manifolds

“…The minimality of M implies that M is minimal as well. Therefore, it follows from Lemma 2.2 of [FK2] that M ∩ ϕ(E) is connected. Since M contains ϕ(M) as an open subset, part (b) of Condition ( * ) yields that M ∩ ϕ(E) = ϕ(M) and that ϕ(M) is dense in M .…”

“…Proof: Fix g ∈ BR(M), and let V ⊂ M be a non-empty domain such that V ⊂ reg(g) and g(V ) ⊂ M. By Lemma 2.2 of [FK2] the non-empty set M ∩reg(g) is connected, and therefore…”

“…the corresponding tube submanifold with base F . Then M is a generic semi-homogeneous CR-submanifold of E, and g + ig contains all constant holomorphic vector fields on E. Furthermore (see Lemma 4.1 of [FK2]), the tube manifold M is minimal at a point if and only if…”