The Flat CR Twistor Model $Q^{2,2}$ and Its Algebraic Sections

TL;DR

This paper analyzes the flat CR twistor model $Q^{2,2}$ in projective space, classifies lines and hyperplane sections, and constructs explicit non-spherical CR structures on $S^3$.

Contribution

It provides a detailed classification of projective lines, hyperplane sections, and CR geometries in the flat CR twistor model using explicit projective methods.

Findings

Classified lines in $Q^{2,2}$ into twistor fibres and transverse lines.

Related transverse lines to round 2-spheres in $S^3$ via incidence-tangency correspondence.

Constructed a family of non-spherical Levi-nondegenerate CR structures on $S^3$.

Abstract

We study the flat CR twistor model $Q^{2,2}\subset \mathbb{CP}^3$ by explicit projective methods. Using the anti-holomorphic involution $j$ associated with the twistor fibration, we classify the projective lines contained in $Q^{2,2}$ into twistor fibres and transverse lines, and relate the latter to round $2$-spheres in $S^3$ through an explicit incidence--tangency correspondence. We classify hyperplane sections under the twistor-compatible symmetry group $PSp(1,1)$ and describe the induced CR geometries on $S^3$. For smooth $j$-invariant quadric sections, we obtain a complete relative classification in terms of Coxeter's inversive distance and show that, in the disjoint case, the construction yields an explicit one-parameter family of globally defined real-analytic non-spherical Levi-nondegenerate CR structures on $S^3$.