On ?type? conditions for generic real submanifolds of ? n

On some rigidity properties of mappings between CR-submanifolds in complex space

“…Since the only known examples of points p ∈ M for which the conclusion of Theorem 1.1 fails to hold are non CR-points, one may be tempted to conjecture that the subvariety V in Theorem 1.1 is contained in the set of non CR-points: Here we recall that a real-analytic CR-submanifold M ⊂ C N is of finite type at a point p ∈ M (in the sense of Kohn [K72] and Bloom-Graham [BG77]) if the Lie algebra generated by all (1, 0) and (0, 1) vector fields tangent to M spans the complexified tangent space of M at p. It is worth mentioning that Theorems 1.1 and 1.3 are independent and not contained into each other. Indeed, given a realanalytic CR-submanifold M , the set of points of finite type can be empty, e.g.…”

Untitled

“…when X is the complexification of a real-analytic generic submanifold M ⊂ C N , the construction of X (l) yields the iterated complexification M 2l as defined in [Z97]. In this case the images µ(D l (x)) are the Segre sets in the sense of Baouendi-EbenfeltRothschild [BER96] and their finite type criterion says that M is of finite type in the sense of Kohn [K72] and Bloom-Graham [BG77] if and only if the Segre sets of sufficiently high order have nonempty interior. The last condition can also be expressed in terms of ranks (see [BER99a]).…”

Helical CR structures and sub-Riemannian geodesics

“…The connection which we identify between CR and sub-Riemannian geometry may also be of interest in the higher codimension setting. See [4] for the construction of partial normal forms for the defining equations of CR manifolds of finite type in higher codimension. The finite type conditions in [4] arise via iterated commutators of complex vector fields and their conjugates.…”