Hopf fibration as a principal U(1)-bundle

Let $S^3=\{(z_1,z_2)\in\mathbb C^2:|z_1|^2+|z_2|^2=1\}$ and let $U(1)\subset\mathbb C^\times$ act on $S^3$ by scalar multiplication:

Definition (Hopf fibration)

The Hopf fibration is the quotient map

With this action, $\pi$ is a principal G-bundle with structure group $U(1)$ (a Lie group ), i.e. a principal circle bundle over $S^2$.

Concretely, one can take $\pi(z_1,z_2)=[z_1:z_2]\in\mathbb{CP}^1$, and the fiber over a point is exactly the $U(1)$-orbit of any representative.

This bundle is nontrivial; in particular it has no global smooth section (compare the section criterion for triviality ).

Examples

1. Local trivializations from the standard affine charts.
   On the open set $U_2=\{[z_1:z_2]\in\mathbb{CP}^1: z_2\neq 0\}$, write $w=z_1/z_2\in\mathbb C$. A point in $S^3$ over $w$ can be uniquely written as

   $$\frac{1}{\sqrt{1+|w|^2}}(w,1)\cdot u,\qquad u\in U(1),$$

   giving a smooth identification $\pi^{-1}(U_2)\cong U_2\times U(1)$. A similar trivialization holds on $U_1=\{z_1\neq 0\}$.

2. Clutching function.
   Using the two chart trivializations, the transition function on the overlap is the map $S^1\to U(1)$ of degree $1$ (it winds once). This winding is the basic topological reason the bundle is not isomorphic to $S^2\times U(1)$.

3. Natural connection.
   The Hopf bundle carries a canonical principal connection whose curvature is a multiple of the area form on the base; this is the geometric starting point for the Dirac monopole connection .