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> [!proposition] Proposition. ([[Hopf fibration]])
> The **Hopf bundle** is the [[tautological vector bundle|tautological]] rank-$1$ [[complex projective space|complex]] [[vector bundle]] $\gamma_{1,2}^{\mathbb{C}} \to \mathbb{C}P^{1}$. It admits the [[general orthogonal group|unitary group]] $U(1)$ as a [[structure group of a real vector bundle over a smooth manifold|structure group]].
>
The [[associated vector bundle and principal bundle|associated]] [[principal bundle over a smooth manifold|principal]] $U(1)$-[[principal bundle over a smooth manifold|bundle]] is called the **Hopf fibration**. It is isomorphic to a map $\mathbb{S}^{3} \xrightarrow{\pi} \mathbb{S}^{2}$.
^proposition
> [!proof]- Proof. ([[Hopf fibration]])
> **The unitary structure group on the Hopf bundle.** [[complex projective space|As usual]], [[cover]] $\mathbb{C}P^{1}$ by [[coordinate chart|coordinate neighborhoods]] $U_{1},U_{2}$ given [[projective space as a smooth manifold|given by]] $U_{i}=\{ [z_{1}: z_{2}] \in \mathbb{C}P^{1} : z_{i} \neq 0 \}, i=1,2,$
> corresponding to charts $\begin{align}
> \varphi_{1}: U_{1} &\to \mathbb{C}^{1} \\
> [z_{1}: z_{2}] & \mapsto \frac{z_{2}}{z_{1}} =: z
> \end{align}$
> and $\begin{align}
> \varphi_{2}:U_{2} &\to \mathbb{C}^{1} \\
> [z_{1}:z_{2}] & \mapsto \frac{z_{1}}{z_{2}} =: \zeta
> \end{align}$
> with inverses $z \xmapsto{\varphi_{1} ^{-1}}[1:z]$ and $\zeta \xmapsto{\varphi_{2} ^{-1}}[\zeta:1]$.
>
>
> In other words, $z$ is a local coordinate on $U_{1}$ and $\zeta$ is a local coordinate on $U_{2}$.
>
> Every point in the fiber of $\gamma_{1,2}^{\mathbb{C}}$ over $[1:z] \in U_{1}$ is $(w,wz)$ for some unique scalar $w \in \mathbb{C}$. Similarly, every point in the fiber over $[\zeta:1] \in U_{2}$ is $(w\zeta, w)$ for some unique scalar $w \in \mathbb{C}$. It might be tempting to define local trivializations as something like $\Phi_{U_{1}}(w, wz)=([1:z], w)$. Indeed, at least the first component is enforced by commutativity in the [[vector bundle|local trivialization]] definition. But we don't quite do this, because the [[structure group of a real vector bundle over a smooth manifold|structure group]] won't be interesting. Instead, we scale the second component by $\|(1,z)\|=\sqrt{ 1^{2} + z \overline{z} }=\sqrt{ 1+|z|^{2} }$, defining $\begin{align}
> \Phi_{U_{1}}: \pi ^{-1}(U_{1}) & \to U_{1} \times \mathbb{C}^{1} \\
> (w, wz) & \mapsto ([1 : z], w \sqrt{ 1 + |z|^{2} })
> \end{align}$
> and $\begin{align}
> \Phi_{U_{2}}: \pi ^{-1}(U_{2}) & \to U_{2} \times \mathbb{C}^{1} \\
> (w\zeta, w) & \mapsto ([\zeta:1], \zeta \sqrt{ 1+ |\zeta|^{2} }).
> \end{align}$
> The [[transition functions for a vector bundle over a smooth manifold|transition functions]] can then be computed to be $\begin{align}
> \psi_{21}:\mathbb{C} &\to \mathbb{C} , [1:z] \mapsto \frac{z}{|z|} \in \mathbb{C}^{\times}=U(1)\\
> \psi_{12} : &\mathbb{C} \to \mathbb{C}, [\zeta:1] \mapsto \frac{|z|}{z}=\frac{\zeta}{|\zeta|} \in \mathbb{C}^{\times}=U(1).
> \end{align}$
> These give an invariantly defined [[conjugate symmetric|Hermitian]] [[inner product]] (in this case is really just a norm) on the fibers, corresponding to $|\cdot|$ on $\mathbb{C}$
>
>
> **The associated principal $U(1)$-bundle to the Hopf bundle (the Hopf fibration).**
>
> Put $E^{\sharp}=E - s_{0}(B)$ for $s_{0}(B)$ the image of the zero section. Then $E-E^{\sharp} \cong \mathbb{C}^{2}-\{ 0,0 \}$. Now, what is the [[associated vector bundle and principal bundle|associated]] principal $U(1)$-bundle $P$? It is the 'bundle of unit vectors in $E