Locally homogeneous finitely nondegenerate CR-manifolds

Classification of Levi degenerate homogeneous CR-manifolds in dimension 5

“…Then If (M, o) is the manifold germ associated with the CR-algebra in §9.4, then the holomorphic tangent space H/g o ⊂g/g o in the sense of (ii) can be canonically identified with the holomorphic tangent space H o M in the geometric sense. As shown in [15], the mapping q!H, w !w+σw, induces a complex linear isomorphism q/q (∞) ∼ =H/go. The former quotient is canonically isomorphic to…”

“…Then the CR-algebra (g, q) is called a CR-algebra associated with the locally homogeneous CR-germ (M, o). It is obvious that g∩q is nothing but the isotropy subalgebra In [15] it has been shown that the geometric properties of the CR-structure of the CR-germ (M, o) like minimality, k-nondegeneracy and holomorphic degeneracy can be read off every CR-algebra (g, q) associated with (M, o). The facts relevant for our classification will be discussed below.…”

“…The Levi kernel K o M and its higher-order analogues K r o M can be considered as complex linear subspaces of q/q (∞) . We will make extensive use of some of the main results of [15], such as Theorem 5.10.…”

“…The proof of Theorem II relies on a natural equivalence (see [15,Proposition 4.1]) between the category of CR-manifold germs with a locally transitive Lie algebra action and a certain purely algebraically defined category. Before we briefly outline the main steps of our proof, we recall the notion of a CR-algebra, taken from [28], and introduce some notation.…”

“…Then g o ⊂F⊂H⊂g are real subspaces stable under ad(g o ) and l o ⊂f⊂q are complex subalgebras. In [15,Lemma 5.9] it has been shown that actually F is a Lie algebra and it coincides with N g (H)∩H (here, given W ⊂g, N g (W ):={v∈g:[v, W ]⊂W }).…”