Conformal and elliptic models of Hopf bundle

The Hopf bundle $\pi\colon S^{2n+1}\to\mathbb{CP}^n$ is one of the most famous examples of nontrivial principal bundles. In this paper we consider two models of this bundle for $n=1$. The first (conformal) model is obtained by the stereographic mapping of $S^3$ onto the conformal space, the second one is constructed with the use of the standard two-sheeted covering $S^3\to\mathbb{B}^3$ of the elliptic space. We find the bundle connection in these models and find the curvature of this connection.
The results of the first two sections are obtained by I. A. Kuzmina, the third part is written by B. N. Shapukov.