CR geometry and conformal foliations (Q2376976)

The authors study some properties of conformal foliations in \(3\)-space by identifying such foliations with local CR hypersurfaces in the standard Levi-indefinite hyperquadric in \(\mathbb{CP}_3\). The conformal foliations are defined using a holomorphic function of two complex variables. After recalling some specific results from CR geometry, they consider and study the hyperquadric of indefinite signature \(Q\) in \(\mathbb{CP}_3\) defined by the equation \(|Z_1|^2+ |Z_2|^2=|Z_3|^2+|Z_4|^2\). Next, they obtain the conditions under which a unit vector field on an open set \(\Omega\subset \mathbb{R}^3\) is transversally conformal. That means that the Lie derivative with respect to \(U\) preserves the conformal metric orthogonal to the integral curves of \(U\). Next, the authors study a certain twistor fibration obtained from the submersion \(\tau :\mathbb{CP}_3\to \mathbb{HP}_1=S^4\) when restricted to \(Q\to S^3\). After presenting some aspects from the theory of integrable Hermitian structures, the authors discuss the twistor theory of conformal foliations. The crucial example is that of the Hopf fibration. They present some explicit constructions and comparisons and present a certain counterexample related to the eikonal equation. Finally, they relate the notions of conformal foliation and shear-free ray congruence from relativity, and present some aspects concerning the Kerr theorem.