Classification of Spherical Tube Hypersurfaces Having One Minus in the Signature of the Levi Form

Local tube realizations of CR-manifolds and maximal abelian subalgebras

“…On the other hand it is well known that every hypersurface S p,q := S 1 p,q has many local tube realizations, compare e.g. with [11,12]. whose sizes are determined by the requirement u 11 , v 11 , w 11 ∈ C m×m .…”

“…In the particular case of spherical hypersurfaces the following result has been obtained in [7] by solving a certain partial differential equation coming from the Chern-Moser theory [6]: For every r ≥ 2 there exist, up to affine equivalence, precisely r + 2 closed smooth tube submanifolds of C r that are locally CR-isomorphic to the Euclidean sphere S 2r−1 ⊂ C r . In [12,13] the same method has been used for a certain more general class of CR-flat manifolds. All the above results rely on Chern-Moser theory and therefore only apply to CR-manifolds that are Levi nondegenerate and of hypersurface type.…”

Classification of Levi degenerate homogeneous CR-manifolds in dimension 5

“…For Levi nondegenerate tube manifolds (which necessarily cannot be conical) such examples can be found in [13]. In [25] even two affinely homogeneous examples are contained which are locally affinely non-equivalent but whose associated tube manifolds are locally CR-equivalent.…”

Classification of commutative algebras and tube realizations of hyperquadrics

“…with non-degenerate Levi form of type (p−1, q−1). Therefore the classification problem for local tube realizations for both classes is the same, compare also [5], [10], [11], [12], [13], [14], [20] for partial results in this context.…”

“…The paper is organized as follows: In Section 2 we relate our results to existing results in the literature, in particular to those in [5], [11], [12]. In Section 3 we recall the necessary tools from [10] and give a short outline of the classification procedure.…”