On Geometry of 2-nondegenerate, Hypersurface-Type Cauchy–Riemann Manifolds Encoded in Dynamical Legendrian Contact Structures

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 antecedent results of Curtis Porter and Igor Zelenko, for which they developed a kind of bigraded Tanaka prolongation, 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. We find an explicit criterion for this recoverability. The method developed here clarifies the relationship between the bigraded Tanaka prolongation of regular symbols and their usual Tanaka prolongation, providing a geometric interpretation of conditions under which these two constructions coincide. Motivated by the search for homogeneous models with given non-regular symbols, we describe a process of reduction of an initial natural frame bundle, which is needed to treat structures with non-regular CR symbols. We demonstrate this reduction procedure for examples whose underlying manifolds have dimensions 7 and 9. We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n^2 + 1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n^2 + 7. This supports Beloshapka’s conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate. Essential to the calculation of this upper bound is a classification of the CR symbols, which we also derive. Lastly, we classify (up to local equivalence) the 7-dimensional maximally symmetric (among structures with a given CR symbol) homogeneous, 2-nondegenerate, 7-dimensional CR manifolds, of which there are eight, and give a similar partial classification of the 9-dimensional models