On geometry of $2$-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations

TL;DR

This paper develops a new geometric approach to classify and analyze 2-nondegenerate hypersurface-type CR structures of arbitrary odd dimension, extending previous results and clarifying the relationship between different prolongation methods.

Contribution

It introduces a reduction to dynamical Legendrian contact structures on leaf spaces, extending classification results to arbitrary CR symbols and providing a geometric interpretation of prolongation conditions.

Findings

Constructed canonical absolute parallelisms for broad classes of CR structures.

Extended classification to arbitrary CR symbols beyond regular cases.

Showed genericity of non-homogeneous CR symbols in high dimensions.

Abstract

We construct canonical absolute parallelisms over real-analytic manifolds equipped with $2$-nondegenerate, hypersurface-type CR structures of arbitrary odd dimension not less than $7$ whose Levi kernel has constant rank belonging to a broad subclass of CR structures that we label as recoverable. For this we develop a new approach based on a reduction to a special flag structure, called the dynamical Legendrian contact structure, on the leaf space of the CR structure's associated Levi foliation. This extends the results of Porter-Zelenko [20] from the case of regular CR symbols constituting a discrete set in the set of all CR symbols to the case of the arbitrary CR symbols for which the original CR structure can be uniquely recovered from its corresponding dynamical Legendrian contact structure. Our method clarifies the relationship between the bigraded Tanaka prolongation of regular symbols developed in Porter-Zelenko [20] and their usual Tanaka prolongation, providing a geometric interpretation of conditions under which they are equal. Motivated by the search for homogeneous models with given nonregular symbol, we also describe a process of reduction from the original natural frame bundle, which is inevitable for structures with nonregular CR symbols. We demonstrate this reduction procedure for examples whose underlying manifolds have dimension $7$ and $9$. We show that, for any fixed rank $r>1$, in the set of all CR symbols associated with 2-nondegenerate, hypersurface-type CR manifolds of odd dimension greater than $4r+1$ with rank $r$ Levi kernel, the CR symbols not associated with any homogeneous model are generic, and, for $r=1$, the same result holds if the CR structure is pseudoconvex.