Editing Connection (principal bundle) (section)

=== Connection on the complex Hopf-bundle ===
We<ref name=":0" /><sup>pg 94</sup> can construct <math>\mathbb{CP}^n</math> as a principal <math>\mathbb{C}^*</math>-bundle <math>\gamma:H_\mathbb{C} \to \mathbb{CP}^n</math> where <math>H_\mathbb{C} = \mathbb{C}^{n+1}-\{0\}</math> and <math>\gamma</math> is the projection map<blockquote><math>\gamma(z_0,\ldots,z_n) = [z_0,\ldots,z_n]</math></blockquote>Note the Lie algebra of <math>\mathbb{C}^* = GL(1,\mathbb{C})</math> is just the complex plane. The 1-form <math>\omega \in \Omega^1(H_\mathbb{C},\mathbb{C})</math> defined as<blockquote><math>\begin{align}
\omega &= \frac{\overline{z}^tdz}{|z|^2} \\
&= \sum_{i=0}^n\frac{\overline{z}_i}{|z|^2}dz_i
\end{align}</math></blockquote>forms a connection, which can be checked by verifying the definition. For any fixed <math>\lambda \in \mathbb{C}^*</math> we have<blockquote><math>\begin{align}
R_\lambda^*\omega &= \frac{\overline{(z\lambda)}^td(z\lambda)}{|z\lambda|^2} \\
&= \frac{
  \overline{\lambda}\lambda\overline{z}^tdz
}{|\lambda|^2\cdot |z|^2}

\end{align}</math></blockquote>and since <math>|\lambda|^2 = \overline{\lambda}{\lambda}</math>, we have <math>\mathbb{C}^*</math>-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any <math>z \in H_\mathbb{C}</math> we have a short exact sequence<blockquote><math>0 \to \mathbb{C} \xrightarrow{v_z} T_zH_\mathbb{C} \xrightarrow{\gamma_*} T_{[z]}\mathbb{CP}^n \to 0</math></blockquote>where <math>v_z</math> is defined as<blockquote><math>v_z(\lambda) = z\cdot \lambda</math></blockquote>so it acts as scaling in the fiber (which restricts to the corresponding <math>\mathbb{C}^*</math>-action). Taking <math>\omega_z\circ v_z(\lambda)</math> we get

<math>\begin{align}
\omega_z\circ v_z(\lambda) &= \frac{\overline{z}dz}{|z|^2}(z\lambda) \\
&= \frac{\overline{z}z\lambda}{|z|^2} \\
&= \lambda

\end{align}</math>

where the second equality follows because we are considering <math>z\lambda</math> a vertical tangent vector, and <math>dz(z\lambda) = z\lambda</math>. The notation is somewhat confusing, but if we expand out each term<blockquote><math>\begin{align}
dz &= dz_0 + \cdots + dz_n \\
z &= a_0z_0 + \cdots +a_nz_n \\
dz(z) &= a_0 + \cdots + a_n \\
dz(\lambda z) &= \lambda\cdot (a_0 + \cdots + a_n) \\
\overline{z} &= \overline{a_0} + \cdots + \overline{a_n}

\end{align}</math></blockquote>it becomes more clear (where <math>a_i \in \mathbb{C}</math>).