Equivalence and embedding problems for CR-structures of any codimension

Abstract

An approach is suggested to equivalence and embedding problems for smooth CR-submanifolds of complex spaces (and, more generally, for abstract CR-manifolds) in terms of complete differential systems in jet bundles satisfied by all CR-equivalences or CR-embeddings respectively. For equivalence problems, manifolds are assumed to be of finite type and finitely nondegenerate. These are higher order generalizations of the corresponding nondegeneracy conditions for the Levi form. It is shown by a simple example that these nondegeneracy conditions cannot be even slightly relaxed to more general known conditions. In particular, for essentially finite hypersurfaces in C 2 , such a complete system does not exist in general. For embedding problems, source manifolds are assumed to be of finite type and their embeddings to be finitely nondegenerate. Sufficient conditions on CR-manifolds are given, where the last condition is automatically satisfied by all CR-embeddings.