New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy

Abstract

We introduce new invariant tensors in CR structures which can be viewed as higher order Levi forms. Using the second and third order tensors, we give a complete formal normal form (in the sense of Chern-Moser) for a real hypersurface at a generic Levi degeneracy. (We say that M has a generic Levi degeneracy at p if the Levi determinant vanishes at p but its differential does not, and the set of Levi degenerate points of M is transverse to the Levi null space at p.) By applying a convergence theorem for formal mappings due to the author, Baouendi, and Rothschild, we conclude that the above mentioned formal normal form provides a complete set of biholomorphic invariants for real-analytic hypersurfaces at generic Levi degeneracies.