Free CR distributions

TL;DR

This paper investigates specific parabolic CR geometries with unique dimensions and codimensions, analyzing their invariants, explicit computations, and geometric properties, including analogies to Fefferman constructions.

Contribution

It provides a detailed study of CR geometries with dimensions n and n^2, including explicit invariant computations and geometric analysis for all n>1.

Findings

Fundamental invariant is of torsion type.

Explicit computation of the invariant.

Discussion of analogy to Fefferman construction.

Abstract

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be $\PSU(n+1,n)/P$ with a suitable parabolic subgroup $P$. We study the geometric properties of such real $(2n+n^2)$-dimensional submanifolds in $\mathbb C^{n+n^2}$ for all $n>1$. In particular we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.