Conformal Fibrations of $S^3$ by Circles

TL;DR

This paper characterizes conformal fibrations of the 3-sphere by circles using complex geometry and shows that all such fibrations are equivalent to the Hopf fibration under conformal transformations.

Contribution

It provides a new geometric description of circle fibrations in $S^3$ and classifies conformal fibrations as equivalent to the Hopf fibration.

Findings

Conformal submersions of $S^3$ correspond to intersections with complex surfaces in $ ext{CP}^3$.

A bilinear form on the tangent sphere bundle describes the space of circles in $S^3$.

All conformal fibrations of $S^3$ by circles are conformally equivalent to the Hopf fibration.

Abstract

It is shown that analytic conformal submersions of $S^3$ are given by intersections of (not necessary closed) complex surfaces with a quadratic real hyper-surface in $\mathbb{C}P^3.$ A new description of the space of circles in the 3-sphere in terms of a natural bilinear form on the tangent sphere bundle of $S^3$ is given. As an application it is shown that every conformal fibration of $S^3$ by circles is the Hopf fibration up to conformal transformations.