Normal forms for almost non-integrable CR structures

TL;DR

This paper extends the Chern-Moser normal form to non-integrable Levi-nondegenerate almost CR structures, providing intrinsic and extrinsic constructions and analyzing their implications for equivalence and symmetry groups.

Contribution

It introduces two novel constructions for normal forms of non-integrable almost CR structures, adapting classical methods to handle non-integrability.

Findings

Partial normal forms for general almost CR structures

Extension of normal forms to almost-complex structures

Applications to CR-diffeomorphism groups

Abstract

We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense "maximally CR". One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory. Partial normal forms are obtained for general almost CR structures of any CR codimension, in particular, for almost-complex structures. Applications are given to the equivalence problem and the Lie group structure of the group of all CR-diffeomorphisms.